# Expected Signature on a Riemannian Manifold and Its Geometric Implications

X. Geng, H. Ni and C. Wang  X. Geng, H. Ni and C. Wang School of Mathematics
and Statistics, University of Melbourne, Parkville VIC 3052, Australia. Email:
<EMAIL_ADDRESS>of Mathematics, University College London,
London WC1H 0AY, United Kingdom. Email<EMAIL_ADDRESS>of
Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom.
Email<EMAIL_ADDRESS>
###### Abstract

On a compact Riemannian manifold $M,$ we show that the Riemannian distance
function $d(x,y)$ can be explicitly reconstructed from suitable asymptotics of
the expected signature of Brownian bridge from $x$ to $y$. In addition, by
looking into the asymptotic expansion of the fourth level expected signature
of the Brownian loop based at $x\in M$, one can explicitly reconstruct both
intrinsic (Ricci curvature) and extrinsic (second fundamental form) curvature
properties of $M$ at $x$. As independent interest, we also derive the
intrinsic PDE for the expected Brownian signature dynamics on $M$ from the
perspective of the Eells-Elworthy-Malliavin horizontal lifting.

###### Contents

  1. 1 Introduction and outline of main results
    1. 1.1 The signature transform
    2. 1.2 Extended signature and the basic geometric question
    3. 1.3 Intrinsic PDE for the expected signature dynamics
    4. 1.4 Reconstruction of Riemannian distance
    5. 1.5 Reconstruction of metric and curvature properties
  2. 2 Some geometric background and a notion of signature on manifolds
    1. 2.1 Notions from Riemannian geometry
      1. 2.1.1 The Laplace-Beltrami operator
      2. 2.1.2 The Malliavin-Stroock heat kernel expansion
      3. 2.1.3 Ricci curvature and the second fundamental form
    2. 2.2 Brownian motion and bridge
      1. 2.2.1 The Eells-Elworthy-Malliavin construction of Brownian motion
      2. 2.2.2 The Brownian bridge
    3. 2.3 The $\phi$-signature on manifolds
  3. 3 The expected Brownian signature dynamics
    1. 3.1 Intrinsic PDE for the expected $\phi$-signature
    2. 3.2 The Brownian bridge case
  4. 4 Signature asymptotics and reconstruction of Riemannian distance
    1. 4.1 The main theorem
    2. 4.2 Proof of Theorem 4.1
      1. 4.2.1 Computations under normal chart
      2. 4.2.2  Localisation I: the main contribution
      3. 4.2.3 Localisation II: the remainder
      4. 4.2.4 Completing the proof of Theorem 4.1
  5. 5 Small-time expansion and its connection with curvature properties
    1. 5.1 The main theorem
    2. 5.2 Some basic expansions
      1. 5.2.1 SDE representation of $X^{t,x,x}$
      2. 5.2.2 Decomposition of $\Pi_{t}$
      3. 5.2.3 Expansions of $\frac{X_{tr}}{1-r}$ and $f(X_{tr})$
      4. 5.2.4 Summary of structure and order-counting
    3. 5.3 Computation of the $t^{2}$-coefficient $\hat{\Theta}_{x}$
      1. 5.3.1 The $(II;II)$ term
      2. 5.3.2 Final result
    4. 5.4 Computation of the $t^{3}$-coefficient $\hat{\Xi}_{x}$
      1. 5.4.1 Total degree $=3$
      2. 5.4.2 Total degree $=2.5$
      3. 5.4.3 Total degree $=2$
      4. 5.4.4 Final result
    5. 5.5 Completing the proof of Theorem 5.1: geometric reduction
      1. 5.5.1 Reduction of $\hat{\Theta}_{x}$
      2. 5.5.2 Reduction of $\hat{\Xi}_{x}$
  6. A Remaining cases for $t^{2}$-coefficient
    1. A.1 The $(II;IK)$ and $(IK;II)$ terms
    2. A.2 The $(II;KI)$ and $(KI;II)$ terms
    3. A.3 The $(II;KK)$ and $(KK;II)$ terms
    4. A.4 The $(IK;KI)$ and $(KI;IK)$ terms
    5. A.5 The $(IK;KK)$ and $(KK;IK)$ terms
    6. A.6 The $(KI;KK)$ and $(KK;KI)$ terms
    7. A.7 The $(IK;IK)$ term
    8. A.8 The $(KI;KI)$ term
    9. A.9 The $(KK;KK)$ term
    10. A.10 The $(II;P)$ and $(P;II)$ terms
    11. A.11 The $(IK;P)$ and $(P;IK)$ terms
    12. A.12 The $(KI;P)$ and $(P;KI)$ terms
    13. A.13 The $(KK;P)$ and $(P;KK)$ terms
    14. A.14 The $(P;P)$ term
  7. B Remaining cases for $t^{3}$-coefficient
    1. B.1 Total degree $=3$
      1. B.1.1 The $(IJ;JI)$ and $(JI;IJ)$ terms
      2. B.1.2 The $(IJ;IJ)$ term
      3. B.1.3 The $(JI;JK)$ and $(JK;JI)$ terms
      4. B.1.4 The $(IJ;JK)$ and $(JK;IJ)$ terms
      5. B.1.5 The $(JI;KJ)$ and $(KJ;JI)$ terms
      6. B.1.6 The $(IJ;KJ)$ and $(KJ;IJ)$ terms
      7. B.1.7 The $(JK;JK)$ term
      8. B.1.8 The $(JK;KJ)$ and $(KJ;JK)$ terms
      9. B.1.9 The $(KJ;KJ)$ term
      10. B.1.10 The $(JJ;II)$ and $(II;JJ)$ terms
      11. B.1.11 The $(JJ;IK)$, $(IK;JJ)$, $(JJ;KK)$ and $(KK;JJ)$ terms
      12. B.1.12 The $(JJ;KI)$ and $(KI;JJ)$ terms
      13. B.1.13 The $(JJ;P)$ and $(P;JJ)$ terms
    2. B.2 Total degree $=2.5$
      1. B.2.1 The $(IJ;P)$ term
      2. B.2.2 The $(JK;P)$ term
      3. B.2.3 The $(KJ;P)$ term
      4. B.2.4 The $(JI;II)$ term
      5. B.2.5 The $(IJ;II)$ term
      6. B.2.6 The $(JK;II)$ term
      7. B.2.7 The $(KJ;II)$ term
      8. B.2.8 The $(IJ;IK)$ term
      9. B.2.9 The $(JI;IK)$ term
      10. B.2.10 The $(KJ;IK)$ term
      11. B.2.11 The $(JK;IK)$ term
      12. B.2.12 The $(IJ;KI)$ term
      13. B.2.13 The $(JI;KI)$ term
      14. B.2.14 The $(KJ;KI)$ term
      15. B.2.15 The $(JK;KI)$ term
      16. B.2.16 The $(IJ;KK)$ term
      17. B.2.17 The $(JI;KK)$ term
      18. B.2.18 The $(JK;KK)$ term
      19. B.2.19 The $(KJ;KK)$ term
    3. B.3 Total degree $=2$
      1. B.3.1 The $(II;II)$ term
      2. B.3.2 The $(II;IK)$ term
      3. B.3.3 The $(II;KI)$ term
      4. B.3.4 The $(II;KK)$ term
      5. B.3.5 The $(IK;IK)$ term
      6. B.3.6 The $(IK;KI)$ term
      7. B.3.7 The $(IK;KK)$ term
      8. B.3.8 The $(KI;KI)$ term
      9. B.3.9 The $(KI;KK)$ term
      10. B.3.10 The $(KK;KK)$ term
      11. B.3.11 The $(II;P)$ term
      12. B.3.12 The $(IK;P)$ term
      13. B.3.13 The $(KI;P)$ term
      14. B.3.14 The $(KK;P)$ term
      15. B.3.15 The $(P;P)$ term

##  1 Introduction and outline of main results

In this section, we discuss the main motivation, introduce some background
information and outline the main results of the present article.

###  1.1 The signature transform

The signature transform (or simply the signature) of a multidimensional path
$\gamma:[0,T]\rightarrow\mathbb{R}^{d},$ which is defined by the formal tensor
series

$\displaystyle S(\gamma)$
$\displaystyle\triangleq\big{(}1,\gamma_{T}-\gamma_{0},\int_{0<s<t<T}d\gamma_{s}\otimes
d\gamma_{t},\cdots,\int_{0<t_{1}<\cdots<t_{n}<T}d\gamma_{t_{1}}\otimes\cdots\otimes
d\gamma_{t_{m}},\cdots\big{)}$

of iterated path integrals, provides an effective summary of the essential
information encoded in the original path $\gamma$. Such a transformation, as
well as its intrinsic form defined in terms of iterated integrals against
spatial one-forms, was originally introduced by the geometer K.T. Chen [Che54]
in the 1950s for his purpose of constructing a de Rham-type cohomology theory
on loop spaces over manifolds. Similar type of iterated integrals (with
$\gamma$ being operator-valued paths) was also used by the physicist F. Dyson
[Dys49] to obtain perturbative expansions of Schrödinger equation (Dyson
series).

The signature plays a fundamental role in the analysis of rough paths and
rough differential equations. Its mathematical properties have been largely
developed and better understood over the past two decades by using modern
techniques from the rough path theory, which was founded by T. Lyons in his
seminal work [Lyo98] in 1998 and has led to far-reaching applications in
stochastic analysis. Among others, a basic result is the so-called signature
uniqueness theorem. It asserts that every rough path is uniquely determined by
its signature up to treelike equivalence. This result was already obtained in
Chen’s originial work [Che58] in 1958 for piecewise smooth paths. It was
extended to the bounded variation case by Hambly-Lyons [HL10] in 2010 and
further to the general rough path case by Boedihardjo et al. [BGLY16] in 2016.
The importance of the signature uniqueness theorem lies in the fact that it
potentially opens a gateway of studying geometric properties of a rough path
from its signature. In fact, the reconstruction of a rough path from its
signature at variaous quantitative levels (the signature inversion problem) is
among the most significant and challenging open problems in rough path theory.
Partial progress and several different methods have been developed in the
literature over the past years to understand this problem, e.g. Lyons-Xu
[LX15] by hyperbolic developments, Lyons-Xu [LX18] by symmetrisation, Chang et
al. [CDNX17] by probabilistic sampling, Chang-Lyons [CL19] by tensor
insertions, Le Jan-Qian [LQ12] and Geng [Gen17] by iterated integrals against
suitable spatial one-forms etc. If one is only interested in some particular
quantitative properties of a path instead of recovering the full trajectory,
there could be neat inversion formulae such as the relation (1.3) below which
reconstructs the length of a smooth path from its signature asymptotics in a
rather simple and elegant way.

On the other hand, if the underlying rough path is random (sample paths of a
stochastic process such as the Brownian motion) the signature becomes a
tensor-valued random variable. In this case, one naturally considers the
expectation of the signature (the expected signature). There is a
probabilistic counterpart of the signature uniqueness theorem due to Chevyrev-
Lyons [CL16] in 2016, which asserts that under suitable conditions the law of
a stochastic process is uniquely characterised by its expected signature. To
some extent, this can be viewed as an infinite dimensional analogue of the
classical moment problem for random variables. There has been several works
using a PDE approach to compute and analyse the expected signature of Brownian
motion [LV04], [LN15], diffusion processes [Ni12] and Lévy processes [FS17].
For a general stochastic process, how one can explicitly and quantitatively
reconstruct distributional properties of the process from its expected
signature is widely unknown at the moment.

On the applied side, more recently the applications of expected signature in
machine learning are gaining attention. Specifically, Sig-MMD or Sig-
Wasserstein-1 (Sig-W1) distance based on the expected signature can serve as
the maximum mean discrepancy (MMD) distance for general stochastic processes,
which have wide applications in hypothesis testing [CO22], distributional
regression [LSD21] and generative models on sequential data [NSW23, NSS21].
Compared with the vanilla MMD with vector-valued time series feature, Sig-MMD
demonstrates superior performance in the two-sample hypothesis test of
stochastic processes, as illustrated in [CO22]. This metric can be kernelized
to facilitate efficient computation [SCF21]. Moreover, the Sig-WGAN model,
which utilises Sig-W1 based discriminator, has empirically shown to improve
the accuracy and robustness of traditional GAN models for synthetic time
series generation. This approach reduces the min-max game to supervised
learning, hence significantly reducing the computational cost and yielding
better performance, in particular, for the case of limited data [NSW23,
NSS21].

###  1.2 Extended signature and the basic geometric question

In this article, we take a different perspective of signature inversion.
Instead of trying to reconstruct a generic path from the knowledge of its
signature, we study signature dymanics in a geometric setting and investigate
the following basic question.

######  Question.

Let $M$ be a Riemannian manifold. By observing the expected signature of
certain geometric dynamics on $M$ (e.g. Brownian bridges), can one explicitly
recover geometric properties (e.g. Riemannian distance, curvature properties)
of the underlying space?

Summarised in vague terms, the main finding of the present article is that the
expected signature of a Brownian bridge on $M$ with lifetime $t$ encode rich
geometric information (more precisely, metric and curvature properties) about
the underlying manifold $M$ in the asymptotics when $t\rightarrow 0^{+}$. How
such information can be extracted explicitly from the expected signature
asymptotics is the main focus of the present work. The precise statements of
our main results are stated in Theorem 4.1 (reconstruction of Riemannian
distance) and Theorem 5.1 (reconstruction of curvature properties)
respectively.

Before explaining the essential ideas, one needs to be careful about the
notion of signature for manifold-valued paths in the first place. Let
$\gamma:[0,T]\rightarrow M$ be a smooth path taking values in a differentiable
manifold $M$. In the differential-geometric setting, one cannot give intrinsic
meanings to integrals like $\int d\gamma\otimes\cdots\otimes d\gamma$ without
any additional structure. In fact, the intrinsic notion of integration along
$\gamma$ is the line integral against a one-form on $M$. More specifically,
let $\phi_{1},\cdots,\phi_{n}$ be a given family of smooth one-forms on $M$.
One can consider the iterated line integral

$\int_{0<t_{1}<\cdots<t_{n}<T}\phi_{1}(d\gamma_{t_{1}})\cdots\phi_{n}(d\gamma_{t_{n}}),$

which is globally well-defined due to the natural pairing between cotangent
and tangent vectors ($\phi(d\gamma_{t})$ is understood as the pairing between
$\phi(\gamma_{t})\in T_{\gamma_{t}}^{*}M$ and $\dot{\gamma}_{t}\in
T_{\gamma_{t}}M$). By varying $n$ and the test one-forms
$\phi_{1},\cdots,\phi_{n}$, one obtains a family of numbers associated with
the given path $\gamma.$ This collection of numbers, known as the extended
signature of $\gamma$ (cf. [LQ12]), uniquely determines $\gamma$ up to tree-
like pieces. This geometric viewpoint is classical and was already well
understood back in K.T. Chen’s original signature uniqueness theorem [Che58]
in the 1950s. The extended signature (iterated line integrals against one-
forms) has been used in various contexts, e.g. in reconstructing trajectories
from the signature [LQ12, Gen17] and in the study of rough paths on manifolds
[CDL15]. In the Euclidean case, the signature is a special case of iterated
line integrals (one simply takes $\phi_{i}=dx^{i}$) and the extended signature
contains exactly the same amount of information as the usual signature.
Indeed, it is well known that smooth one-forms can be approximated by
polynomials and iterated line integrals against polynomial forms can be
expressed as a suitable linear combination of signature coefficients (the
shuffle product formula). As a consequence, the extended signature is uniquely
determined by the usual signature.

In our study, we will assume that a collection of one-forms
$(\phi_{1},\cdots,\phi_{N})$ on $M$ (equivalently, an $\mathbb{R}^{N}$-valued
one form $\phi$) is given fixed and consider the Euclidean signature of the
$\mathbb{R}^{N}$-valued path $\int_{0}^{\cdot}\phi(d\gamma_{t})$. We call this
the $\phi$-signature of $\gamma$ (of course, this is just part of the extended
signature of $\gamma$ given by iterated line integrals along combinations of
one-forms taken from the family $\phi=(\phi_{1},\cdots,\phi_{N})$). A basic
example that will be of our primary interest is the case when $\phi$ is given
by the exterior derivative of an embedding $F:M\rightarrow\mathbb{R}^{N}$. In
this case, the $\phi$-signature of $\gamma$ is just the Euclidean signature of
$F(\gamma)$ computed in the ambient space $\mathbb{R}^{N}$.

###  1.3 Intrinsic PDE for the expected signature dynamics

Let us now consider a Riemannian manifold $M$ and let $\phi$ be an
$\mathbb{R}^{N}$-valued one-form on $M$. For most of the time, $\phi=dF$ where
$F:M\rightarrow\mathbb{R}^{N}$ is an isometric embedding (this always exists
by Nash’s isometric embedding theorems). We consider the expected
$\phi$-signature of Brownian dynamics on $M$. More specifically, let
$W_{t}^{x}$ be a Brownian motion on $M$ (i.e. a Markov process generated by
$\Delta/2$ where $\Delta$ is the Laplace-Beltrami operator) starting at $x$.
Let $\Psi(t,x)$ be the expected value of the $\phi$-signature (in the
Stratonovich sense) of $W^{x}$ up to time $t$.

Our first main result establishes the intrinsic PDE governing the dynamics of
$\Psi(t,x)$ (cf. Theorem 3.1 ).

######  Theorem.

The function $\Psi(t,x)$ satisfies the following tensor-algebra-valued
parabolic PDE on $M$:

$\frac{\partial\Psi}{\partial t}=\frac{1}{2}\Delta\Psi+{\rm
Tr}\big{(}\phi\otimes
d\Psi+\frac{1}{2}(\nabla\phi+\phi\otimes\phi)\otimes\Psi\big{)}$ (1.1)

Here $d$ is the exterior derivative operator and $\nabla$ is the Riemannian
connection. The definition of ${\rm Tr}(\cdot)$ is explained in the remarks
following the statement of Theorem 3.1 below. The PDE for the expected
signature of Euclidean Brownian motion (and also of diffusion processes) was
first established by Lyons-Ni [LN15] from the Markovian perspective. To derive
the PDE (1.1), we take an intrinsic approach from the perspective of the
Eells-Elworthy-Malliavin lifting onto the orthornormal frame bundle. It is
well known that the lifted Brownian motion $\Xi_{t}$ on the bundle satisfies a
canonical SDE governed by the fundamental horizontal vector fields. As a
result, one can write down an SDE for the joint process
$(\Xi_{t},\tilde{S}_{t})$ (cf. (3.10)), where $\tilde{S}_{t}$ is the
$\Phi$-signature of $\Xi$ up to time $t$ and $\Phi$ is the pullback of $\phi$
onto the bundle. Once the SDE for $(\Xi_{t},\tilde{S}_{t})$ is obtained, it is
standard to extract its generator and write down the associated PDE governing
the expectation of $\tilde{S}_{t}$ on the bundle (cf. Lemma 3.4). It then
remains to see how the PDE on the bundle gets projected to the intrinsic PDE
(1.1) on the base manifold $M$.

The PDE (1.1) for the Brownian signature dynamics easily yields an associated
PDE for the expected signature of Brownian bridge. Let
$\\{X_{s}^{t,x,y}:s\in[0,t]\\}$ be a Brownian bridge from $x$ to $y$ with
lifetime $t$, i.e. the Brownian motion starting at $x$ conditioned on reaching
$y$ at time $t$. Let $\psi(t,x,y)$ be its expected $\phi$-signature.

######  Theorem.

The function $\psi(t,x,y)$ satisfies the following tensor-algebra-valued
parabolic PDE on $M$:

$\displaystyle\frac{\partial\psi}{\partial t}$
$\displaystyle=\frac{1}{2}\Delta_{x}\psi+\langle\nabla_{x}\log
p(t,x,y),\nabla_{x}\psi+\phi\otimes\psi\rangle+{\rm Tr}(\phi\otimes
d_{x}\psi)$ $\displaystyle\ \ \ +\frac{1}{2}{\rm
Tr}\big{(}(\nabla\phi+\phi\otimes\phi)\otimes\psi\big{)},$

where $p(t,x,y)$ is the heat kernel on $M$ and the subscript $x$ means
differentiating with respect to the $x$-variable.

As we will see, the function $\psi(t,x,y)$ encodes rich geometric information
about the manifold $M$ in the asymptotics as $t\rightarrow 0^{+}$. In what
follows, we consider the particular case when $\phi=dF$, where $F:M\rightarrow
E\triangleq\mathbb{R}^{N}$ is an isometric embedding. We will establish two
types of reconstruction results:

(i) [Reconstruction of Riemannian distance] Through a suitable procedure of
combined asymptotics for $\psi(t,x,y)$ as $t\rightarrow 0^{+}$ and
$n\rightarrow\infty$ ($n$ is the degree of the signature), one can explicitly
recover the Riemannian distance between $x$ and $y$.

(ii) [Reconstruction of curvature properties] By considering the asymptotic
expansion of the fourth level component of $\psi(t,x,x)$ as $t\rightarrow
0^{+}$, one can explicitly recover the Riemannian metric tensor as well as
both intrinsic (Ricci curvature) and extrinsic (the second fundamental form)
curvature properties at $x$. Here we consider the fourth level projection
because this can be shown to be the first nonzero component in $\psi(t,x,x)$.

###  1.4 Reconstruction of Riemannian distance

Recall that $\psi(t,x,y)$ is the expected $dF$-signature of the Brownian
bridge $X^{t,x,y}$ from $x$ to $y$ with lifetime $t$, where
$F:M\rightarrow\mathbb{R}^{N}$ is a given isometric embedding. Our main idea
of reconstructing the Riemannian distance $d(x,y)$ is largely inspired by a
well-known open problem in rough path theory which we shall first describe.

Let $\gamma:[0,T]\rightarrow\mathbb{R}^{d}$ be a continuous path with bounded
total variation. By using the triangle inequality, it is straight forward to
see that

$\big{(}n!\big{\|}S_{n}(\gamma)\big{\|}\big{)}^{1/n}\leqslant L(\gamma)\ \ \
\forall n\geqslant 1.$ (1.2)

Here $S_{n}(\gamma)\in(\mathbb{R}^{d})^{\otimes n}$ is the $n$-th level
signature of $\gamma$, $\|\cdot\|$ is a suitable tensor norm (e.g. the
projective norm) and $L(\gamma)$ is the length of $\gamma$. The remarkable
(and rather surprising) point is that the simple estimate (1.2) becomes
asymptotically sharp as $n\rightarrow\infty$. More precisely, it was
implicitly conjectured by Hambly-Lyons [HL10] which was made explicit in
Chang-Lyons-Ni [CLN18] that under the projective tensor norm $\|\cdot\|_{{\rm
proj}}$,

$\lim_{n\rightarrow\infty}\big{(}n!\big{\|}S_{n}(\gamma)\big{\|}_{{\rm
proj}}\big{)}^{1/n}=L(\gamma)$ (1.3)

for any tree-reduced (i.e. without treelike pieces) BV path $\gamma$. In other
words, the length of a tree-reduced path is recovered from its normalised
signature asymptotics. The formula (1.3) was established for ${\cal C}^{3}$
paths by Hambly-Lyons [HL10] and ${\cal C}^{1}$ paths by Lyons-Xu [LX15]. It
was extended to the two-dimensional BV case by Boedihardjo-Geng [BG22] without
any regularity assumption but under a stronger notion of tree-reducedness. A
stronger version of (1.3) for ${\cal C}^{2}$-paths was recently obtained by
[CMT23] generalising a corresponding result in [HL10]. To our best knowledge,
proving (or disproving) the conjectural formula (1.3) for the general BV case
remains an open problem.

Returning to the geometric setting, our main idea of reconstructing the
Riemannian distance $d(x,y)$ from suitable asymptotics of the expcted
$dF$-signature $\psi(t,x,y)$ of the Brownian bridge $X^{t,x,y}$ can be
summarised as follows.

  1. 1.

When $t$ is small, one is essentially forcing the bridge $X^{t,x,y}$ to travel
from $x$ to $y$ by a short amout of time and thus $X^{t,x,y}$ behaves like a
minimising geodesic $\gamma^{x,y}$ joining $x$ to $y$. This property is
quantified through the large deviation principle (LDP) of the Brownian bridge
established by Hsu [Hsu90].

  2. 2.

As a consequence of the first point, it is reasonable to expected that
$\pi_{n}\psi(t,x,y)$ (the $n$-th level projection) is close to the $n$-th
level $dF$-signature of the geodesic $\gamma^{x,y}$. Note that the latter is
just the usual Euclidean signature of $F(\gamma^{x,y})$ in the ambient space
$\mathbb{R}^{N}.$

  3. 3.

As a consequence of the length conjecture (1.3) (which is a proven fact due to
the smoothness of geodesics), the length of $F(\gamma^{x,y})$ in
$\mathbb{R}^{N}$ can be recovered from its normalised signature asymptotics.
Note that this length is precisely the Riemannian distance $d(x,y)$ since $F$
is an isometric embedding.

  4. 4.

In view of the second and third points, one naturally expects that

$\big{(}n!\|\pi_{n}\psi(t,x,y)\|\big{)}^{1/n}\approx d(x,y)$

provided that $t$ is small and $n$ is large.

Our main reconstruction result can be roughly stated as follows.

######  Theorem.

The Riemannian distance $d(x,y)$ can be reconstructed from the following
asymptotic formula:

$\lim_{n\rightarrow\infty}\big{(}n!\|\pi_{n}\psi(t_{n},x,y)\|\big{)}^{1/n}=d(x,y)$
(1.4)

provided that $t_{n}<<n^{-6}$.

The precise and quantitative statement of such a result, which requires a few
technical assumptions, is given by Theorem 4.1. Although the underlying idea
is simple and natural, proving such a relation mathematically requires a
substantial amount of non-trivial analysis. For instance, the above second
point is not a direct consequence of the LDP since the latter was proved under
the uniform (instead of rough path) topology and it is well-known that the
signature is not a continuous functional of the driving path with respect to
the uniform topology. As a result, one has to establish the signature
approximation separately and we do so by using the semimartingale
decomposition of Brownian bridge under normal coordinate charts. In
particular, one needs to separate out the geodesic component in the
decomposition and estimate the remainder effectively.

We should mention a particularly subtle point among several other technical
challenges which will all be clear along the development of the proof in
Section 4.2. For each fixed level $n$, as long as the lifetime $t_{n}$ is
chosen to be small enough one naturally has

$\pi_{n}\psi(t_{n},x,y)\approx n\text{-th level signature of }\gamma^{x,y}.$

If one applies standard signature moment estimates for semimartingales without
much care, one is led to choosing $t_{n}=o(e^{-Cn})$ which is too small to be
practically useful (e.g. in the context of simulating a Brownian bridge with
lifetime $t_{n}$). A main contribution of our analysis is that the lifetime
needs not be exponentially small in $n$ to make the asymptotics (1.4) valid.
Indeed, a polynomial dependence $t_{n}=O(n^{-6})$ is sufficient (cf. (4.1) for
the quantitative reconstruction estimate with explicit error bound). Improving
from exponential to polynomial dependence requires much finer signature
estimates for the aforementioned semimartingale decomposition. To achieve
this, we make use of a deep result of Kallenberg-Sztencel [KS91] on dimension-
free BDG inequalities for Hilbert-space-valued martingales.

###  1.5 Reconstruction of metric and curvature properties

The asymptotic formula (1.4) provides a way of recovering the Riemannian
distance $d(x,y)$ from the expected signature asymptotics of Brownian bridge.
This can be viewed as an zeroth order result. It then becomes reasonable to
expect that curvature properties would start to appear in the higher order
asymptotic expansion of the expected signature function $\psi(t,x,y)$.

To be specific, we consider the case when $x=y$ (the Brownian loop based at
$x$) and look at the function

$\psi_{4}(t,x)\triangleq\pi_{4}\psi(t,x,x)\in(\mathbb{R}^{N})^{\otimes 4}.$

Due to symmetry considerations, it can be shown that
$\pi_{i}\psi_{4}(t,x,x)=0$ for $i=1,2,3$ (see the discussion at the start of
Section 5.1). Therefore, $\psi_{4}(t,x)$ is the first non-trivial component of
$\psi(t,x,x)$ which indeed carries rich geometric information. In Propositions
5.9 and 5.16 below, we compute the expansion of $\psi_{4}(t,x)$ up to order
$t^{3}$:

$\psi_{4}(t,x)=\hat{\Theta}_{x}t^{2}+\hat{\Xi}_{x}t^{3}+O(t^{4})$

with explicit expressions for the tensor coefficients
$\hat{\Theta}_{x},\hat{\Xi}_{x}\in(\mathbb{R}^{N})^{\otimes 4}$ in terms of
metric and curvature coefficients as well as the embedding $F$. The actual
expressions of $\hat{\Theta}_{x}$ and $\hat{\Xi}_{x}$ are too complicated to
write down here. However, after applying suitable tensor contraction
$\mathfrak{C}:(\mathbb{R}^{N})^{\otimes 4}\rightarrow(\mathbb{R}^{N})^{\otimes
2}$ (cf. (5.3) for its precise definition) their expressions are simplified
substantially and their intrinsic meanings in terms of metric and curvature
properties become clear. The main formulae are given as follows.

######  Theorem.

The contracted tensors $\Theta_{x}\triangleq\mathfrak{C}\hat{\Theta}_{x}$ and
$\Xi_{x}\triangleq\mathfrak{C}\hat{\Xi}_{x}$, being viewed as symmetric
bilinear forms on $T_{x}M$ (note that $T_{x}M$ is viewed as a subspace of
$\mathbb{R}^{N}$ under the embedding $F$), are given by the following
formulae:

$\Theta_{x}=\frac{d-1}{24}g_{x}$

and

$\Xi_{x}=\frac{{\rm
S}_{x}-18d^{2}|H_{x}|_{\mathbb{R}^{N}}^{2}}{8640}g_{x}+\frac{49d-20}{8640}{\rm
Ric}_{x}+\frac{(5-4d)d}{480}\langle B_{x},H_{x}\rangle_{\mathbb{R}^{N}}.$

Here $d$ is the dimension of $M,$ $g$ is the Riemannian metric tensor, ${\rm
S}$ is the scalar curvature, ${\rm Ric}$ is the Ricci curvature tensor,
$B_{x}$ is the second fundamental form and $H$ is the mean curvature field
with respect to $F$.

The precise formulation of the theorem is given by Theorem 5.1 below, where
the relevant geometric concepts are recalled in Section 2.1.3. As a
consequence, the tensor $\Theta_{x}$ recovers the metric tensor and $\Xi_{x}$
encodes both intrinsic and extrinsic curvature properties. It is also seen
from the formula (5.4) and Remark 5.18 that the tangent space $T_{x}M$ can be
reconstructed from the bilinear form $\Theta_{x}\in{\cal L}(E\times
E;\mathbb{R})$. In addition, by modifying the tensor contraction map
$\mathfrak{C}$ it is actually possible to recover the quantities

${\rm Ric}_{x},{\rm S}_{x},\langle
B_{x},H_{x}\rangle_{\mathbb{R}^{N}},|H_{x}|_{\mathbb{R}^{N}}$

separately. This is a simple linear algebra matter and is explained in Remark
5.23 below.

#### Comparision with heat kernel asymptotics and potential applications

The aforementioned results should be compared with the classical heat kernel
asymptotics in geometric analysis. It was a renowned theorem of Varadhan
[Var67] that the Riemannian distance function can be recovered from the small-
time asymptotics of the heat kernel:

$\lim_{t\rightarrow\infty}t\log p(t,x,y)=-\frac{1}{2}d(x,y)^{2}.$ (1.5)

In addition, it was shown by Minakshisundaram-Pleijel [MP49] that the heat
kernel admits the following asymptotic expansion:

$p(t,x,y)\sim\frac{1}{(2\pi
t)^{d/2}}e^{-\frac{d(x,y)^{2}}{2t}}\sum_{k=0}^{\infty}u_{k}(x,y)t^{k}\ \ \
\text{as }t\rightarrow 0^{+},$

where the coefficients $u_{k}(x,y)$ are certain $C^{\infty}$-functions defined
in a neighbourhood of the diagonal $\\{(x,x):x\in M\\}$. It is known that
$u_{0}(x,x)=1$, $u_{1}(x,x)={\rm S}_{x}/12$ (${\rm S}_{x}$ is the scalar
curvature at $x$) and $u_{k}(x,x)$ are certain universal polynomial functions
of curvature coefficients and their covariant derivatives at $x$. These
asymptotic results indicate that some aspects of the geometry / shape of the
underlying space can be inferred from the small-time asymptotics of the heat
kernel. On the applied side, the Varadhan asymptotics formula (1.5) provides
the theoretical foundation for the so-called heat method in manifold learning
that are widely used in problems related to learning the shape / structure of
high-dimensional data sets (cf. [CWW17, ZCX20, LS23] and the references
therein).

Our asymptotic results (Theorems 4.1 and 5.1) provide a different perspective
and mechanism for learning the shape of the underlying manifold. It is based
on the use of Brownian bridges and their associated signature features instead
of the heat kernel. From the practical viewpoint, both the simulation of
Brownian bridges on manifolds and the effective computation of path signatures
have been largely developed over the past years by various groups (cf.
[Jen22], [KL20] and the references therein). We therefore expect that our
results will potentially lead to new applications in manifold learning. As a
mathematical work on its own, we do not address applications in data sciences
in the current article and leave these exciting questions to future work.

##  2 Some geometric background and a notion of signature on manifolds

We discuss the basic geometric set-up, introduce some necessary geometric
tools and define a natural notion of geometric signature that is used in the
current work.

###  2.1 Notions from Riemannian geometry

Throughout the rest of the article, let $(M,g)$ be a $d$-dimensional, compact,
oriented Riemannian manifold without boundary. We choose to focus on the
compact case to avoid non-essential technicalities; we do not use global
geometric properties (e.g. spectral decomposition) and thus the main results
extend naturally to the non-compact case under extra technical conditions. We
begin by reviewing some basics on Riemannian geometry. A standard reference
for this part is [DoC92].

Convention. Unless otherwise stated, we always adopt Einstein’s convention
that repeated indices in an expression are summed automatically over their
range.

####  2.1.1 The Laplace-Beltrami operator

A fundamental analytic object on $M$ is the so-called Laplace-Beltrami
operator. This is a second order differential operator which admits the
following local expression:

$\Delta f=\frac{1}{\sqrt{\det g}}\partial_{i}\big{(}\sqrt{\det
g}g^{ij}\partial_{j}f\big{)}.$ (2.1)

Here $g=(g_{ij})$ is the metric tensor and $(g^{ij})$ is the inverse of $g$
under a local chart. The Laplacian $\Delta$ can be alternatively defined as
follows. Suppose that $M$ is isometrically embedding inside some Euclidean
space $\mathbb{R}^{N}$ with canonical basis $\\{e_{1},\cdots,e_{N}\\}$. Let
$V_{i}$ ($1\leqslant i\leqslant N$) be the vector field on $M$ defined by
orthogonally projecting $e_{i}$ onto the tangent space at every point of $M$.
Then one has $\Delta=\sum_{i=1}^{N}V_{i}^{2}$, where $V_{i}$ is equivalently
viewed as a differential operator on $M$ (defined by taking directional
derivative along $V_{i}$).

The Laplace-Beltrami operator $\Delta$ is a non-positive definite, unbounded,
essentially self-adjoint operator on $L^{2}(M,dx)$ ($dx$ is the volume
measure) which admits a kernel $p(t,x,y)$ known as the heat kernel. This is
the fundamental solution to the heat equation, i.e. the smallest positive
solution to the equation

$\begin{cases}\partial_{t}p(t,x,y)=\frac{1}{2}\Delta_{x}p(t,x,y),&t>0,x,y\in
M;\\\ \lim_{t\downarrow 0}p(t,x,\cdot)=\delta_{x},\end{cases}$ (2.2)

where $\delta_{x}$ denotes the Dirac delta function on $M$. The factor $1/2$
is introduced for its connection with Brownian motion; the Brownian motion
$B_{t}$ is generated by $\Delta/2$ and the heat kernel $p(t,x,y)$ is also the
transition density of Brownian motion:

$\mathbb{P}(B_{t}\in dy|B_{0}=x)=p(t,x,y)dy.$

####  2.1.2 The Malliavin-Stroock heat kernel expansion

Small-time expansions of the heat kernel are well studied in the geometric
analysis literature (cf. [BGV04] and the references therein). We are going to
use the following expansion for the logarithmic derivative of the heat kernel,
which was due to Malliavin and Stroock [MS96]. Recall that the injective
radius at $x\in M$ is the largest $\rho_{x}>0$ such that the exponential map
$\exp_{x}:T_{x}M\rightarrow M$ is a diffeomorphism on the metric ball
$B_{x}(\rho)\triangleq\\{y:d(x,y)<\rho\\}.$ Here $d(x,y)$ denotes the
Riemannian distance function. The global injective radius $\rho_{M}$ is the
minimum of $\rho_{x}$ over all $x\in M$. This is a finite positive number due
to the compactness of $M$. For example, $\rho_{M}=\pi$ if $M=S^{2}$ (the unit
sphere in $\mathbb{R}^{3}$). Define

$S_{M}\triangleq\\{(x,y):d(x,y)<\rho_{M}\\}.$

It is clear that any pair $(x,y)\in S_{M}$ can be joined by a unique
minimising geodesic.

######  Theorem 2.1.

The following expansion holds uniformly on each compact subset of $S_{M}:$

$u\nabla_{x}\log p(u,x,y)\sim\sum_{k=0}^{\infty}\nabla_{x}G_{k}(x,y)u^{k}\ \ \
\text{as }u\searrow 0,$ (2.3)

where $\nabla_{x}$ denotes the Riemannian gradient with respect to the
$x$-variable and $G_{k}(x,y)$ are intrinsic $C^{\infty}$-functions on $S_{M}$.

The following explicit expressions will be used in the sequel:

$G_{0}(x,y)=-\frac{d(x,y)^{2}}{2},\
G_{1}(x,y)=-\frac{1}{2}\log\det(d\exp_{y})_{{\bf x}},\ \ \ (x,y)\in S_{M}.$
(2.4)

Here $\exp_{y}:T_{y}M\rightarrow M$ denotes the exponential map, ${\bf
x}\triangleq\exp_{y}^{-1}(x)$ and $(d\exp_{y})_{{\bf x}}$ is the differential
of $\exp_{y}$ at ${\bf x}$, which is a linear isomorphism between Euclidean
spaces $T_{{\bf x}}T_{y}M\cong T_{y}M$ and $T_{x}M$. The functions
$G_{k}(x,y)$ ($k\geqslant 2$) can also be determined explicitly in a recursive
manner, but this is not needed for us.

####  2.1.3 Ricci curvature and the second fundamental form

Many fundamental geometric properties are related to curvature. Recall that
the Riemannian curvature tensor is defined by

$R(X,Y)Z\triangleq\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,$

where $\nabla$ is the Levi-Civita connection and $X,Y,Z$ are smooth vector
fields. Note that $R$ is a type-$(1,3)$ tensor field on $M$ and for each $x\in
M$, it gives rise to a multilinear map $R_{x}:T_{x}M\times T_{x}M\times
T_{x}M\rightarrow T_{x}M$. The Ricci curvature tensor at $x$ is the symmetric
bilinear form ${\rm Ric}_{x}:T_{x}M\times T_{x}M\rightarrow\mathbb{R}$ defined
by

${\rm Ric}_{x}(v,w)\triangleq{\rm Tr}\big{[}u\mapsto R_{x}(u,v)w\big{]},\ \ \
v,w\in T_{x}M.$

The scalar curvature at $x$ is the trace of ${\rm Ric}_{x}$ and is denoted as
${\rm S}_{x}$. Note that ${\rm Ric}$ is a type-$(0,2)$ tensor field and ${\rm
S}$ is a smooth function on $M$. All these curvature quantities are intrinsic
in the sense that their definitions only depend on the metric tensor.

On the other hand, one can also consider extrinsic curvature properties. Let
$F:M\rightarrow\mathbb{R}^{N}$ be a given isometric embedding. For each $x\in
M$, let $(T_{x}M)^{\perp}$ denote the orthogonal complement of $T_{x}M$ inside
$\mathbb{R}^{N}$ ($T_{x}M$ is viewed as a subspace of
$T_{F(x)}\mathbb{R}^{N}\cong\mathbb{R}^{N}$ through the embedding). The second
fundamental form at $x$ is the vector valued bilinear form defined by

$B_{x}(v,w)\triangleq\tilde{\nabla}_{\tilde{X}}\tilde{Y}-\nabla_{X}Y,\ \ \
v,w\in T_{x}M.$ (2.5)

Here $X,Y$ are any vector fields on $M$ such that $X_{x}=v,Y_{x}=w$ and
$\tilde{X},\tilde{Y}$ are any extensions of $X,Y$ to $\mathbb{R}^{N}$. The
operator $\tilde{\nabla}$ is the Levi-Civita connection in
$\mathbb{R}^{N}$(which is just usual Euclidean differentiation). It can be
shown that the expression (2.5) is well-defined (independent of choices of
$X,Y$ and their extensions) and $B_{x}$ is an $(T_{x}M)^{\perp}$-valued,
symmetric bilinear form on $T_{x}M\times T_{x}M$. In other words, $B$ is a
type-$(0,2)$ tensor field on $M$ taking values in the normal bundle
$(TM)^{\perp}$. It is also known that $\nabla_{X}Y$ is the tangential
component of $\tilde{\nabla}_{\tilde{X}}\tilde{Y}$ (i.e. its projection onto
$TM$). As a result, $B(X,Y)$ can also be defined as the normal component of
$\tilde{\nabla}_{\tilde{X}}\tilde{Y}$ (i.e. its projection onto
$(TM)^{\perp}$). The mean curvature vector at $x\in M$ is the normal vector
defined by

$H_{x}\triangleq d^{-1}{\rm Tr}B_{x}\in(T_{x}M)^{\perp},$ (2.6)

where $d$ is the dimension of $M$. By varying $x$, this gives rise to a smooth
section $H$ of the normal bundle $(TM)^{\perp}$. It can be shown that

$\Delta F=d\cdot H,$ (2.7)

where $\Delta$ is the Laplace-Beltrami operator on $M$. Note that the second
fundamental form is an extrinsic curvature quantity as it depends on the
embedding $F$ (it describes how $M$ is embedded / curved inside the ambient
space).

###  2.2 Brownian motion and bridge

In this section, we recall the construction of Brownian motion and Brownian
bridge on $M$. The reader is referred to [Hsu02] for more details.

####  2.2.1 The Eells-Elworthy-Malliavin construction of Brownian motion

The Brownian motion on $M$ could just be defined as the Markov process with
generator $\Delta/2$ from the Markovian perspective. However, the following
intrinsic (SDE / pathwise) construction by Eells-Elworthy-Malliavin will be
useful for our derivation of the intrinsic PDE for the expected signature
dynamics in Section 3.

A main difficulty in the intrinsic SDE construction of Brownian motion is that
the Laplacian may not always admit a decomposition $\Delta=\sum_{i}V_{i}^{2}$
with intrinsic vector fields $V_{i}$. This issue is overcome by lifting the
construction to the so-called orthonormal frame bundle (OFB) ${\cal O}(M)$
over $M$ defined by

${\cal O}(M)\triangleq\bigcup_{x\in M}{\cal F}_{x}.$

Here ${\cal F}_{x}$ denotes the set of ONBs of $T_{p}M$. A generic element in
${\cal O}(M)$ is given by a pair $(x,u)$, where $x\in M$ and
$u=(e_{1},\cdots,e_{d})$ is an ONB of $T_{x}M$. The OFB ${\cal O}(M)$ is a
principal bundle over $M$ with structure group $O(d)$ (the orthogonal group)
acting from the right:

$(x,u)\cdot Q\triangleq(x,u\cdot Q),\ \ \ x\in
M,u=(e_{1},\cdots,e_{d})\in{\cal F}_{x}.$

In particular, it is a differentiable manifold of dimension

$D=\dim M+\dim O(d)=\frac{d^{2}+d}{2}.$

Let $\pi:{\cal O}(M)\rightarrow M$ denote the canonical projection.

An essential point of considering ${\cal O}(M)$ is that one can construct
intrinsic (horizontal) vector fields on it. Let $1\leqslant i\leqslant d$ be
fixed. Given a point

$\xi=(x,u=(e_{1},\cdots,e_{d}))\in{\cal O}(M),$

note that $e_{i}$ is a tangent vector of $M$ at $x$. Let
$\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ be a curve such that
$\gamma_{0}=x$ and $\gamma^{\prime}_{0}=e_{i}.$ By parallel-translating the
ONB $u$ along $\gamma$, one obtains a frame of ONBs along $\gamma$, more
precisely a curve $\xi_{t}\triangleq(\gamma_{t},u_{t})\in{\cal O}(M)$ where
$u_{t}$ is an ONB of $T_{\gamma_{t}}M$. Define

$H_{i}(\xi)\triangleq\xi^{\prime}_{0}\in T_{\xi}{\cal O}(M).$

Since $\xi$ is arbitrary, one then obtains a global vector field $H_{i}$ on
${\cal O}(M).$ The vector fields $\\{H_{1},\cdots,H_{d}\\}$ are called
fundamental horizontal vector fields on ${\cal O}(M).$

It is well known that the second order differential operator ${\cal
L}\triangleq\sum_{i=1}^{d}H_{i}^{2}$ naturally projects to the Laplacian on
$M$. This motivates the Eells-Elworthy-Malliavin construction of Brownian
motion given by the theorem below. The essential idea is to first construct a
horizontal Brownian motion on ${\cal O}(M)$ by solving an intrinsic SDE
associated with the horizontal vector fields $\\{H_{i}\\}$ and then to obtain
an $M$-valued Brownian motion through projection.

######  Theorem 2.2.

Let $x\in M$ and $u=(e_{1},\cdots,e_{d})$ be a given fixed ONB of $T_{x}M$.
Let $\Xi_{t}$ be the solution to the following Stratonovich SDE on
$\mathcal{O}(M)$:

$\begin{cases}d\Xi_{t}=\sum_{i=1}^{d}H_{i}(\Xi_{t})\circ dB_{t}^{i},\\\
\Xi_{0}=(x,u)\in{\cal O}(M),\end{cases}$ (2.8)

where $(B^{1},\cdots,B^{d})$ denotes a $d$-dimensional Euclidean Brownian
motion. Define $W_{t}\triangleq\pi(\Xi_{t})\in M.$ Then the law of $W$ depends
only on $x$ but not on the initial frame $u$. By varying $x$ over $M$, one
obtains a Markovian family whose generator is $\Delta/2$ (thus a Brownian
family on $M$).

####  2.2.2 The Brownian bridge

Let $x,y\in M$ and $t\in(0,1]$ be given fixed. The Brownian bridge
$(X_{s}^{t,x,y})_{0\leqslant s\leqslant t}$ from $x$ to $y$ with lifetime $t$
is the Brownian motion starting at $x$ conditional on reaching $y$ at time
$t$:

$X_{s}^{t,x,y}=W_{s}^{x}|_{W_{t}^{x}=y},\ \ \ 0\leqslant s\leqslant t,$

where $W^{x}$ is a Brownian motion starting at $x$. Simple calculation shows
that this is a (time-inhomogeneous) Markov process with transition density

$\mathbb{P}\big{(}X_{s_{2}}^{t,x,y}\in
dz|X_{s_{1}}^{t,x,y}=w\big{)}=\frac{p(t-s_{2},z,y)p(s_{2}-s_{1},w,z)}{p(t-s_{1},w,y)}dz,\
\ \ 0\leqslant s_{1}<s_{2}<t$ (2.9)

and generator

$({\cal L}f)(w)=\frac{1}{2}\Delta f(w)+\nabla_{w}\log p(t-s,w,y)\cdot\nabla
f(w).$

Here $p(t,x,y)$ is the heat kernel for $\Delta/2$. One can also use the above
Markovian perspective to construct the Brownian bridge mathematically.

If the Brownian motion $W^{x}$ admits an SDE representation (e.g. under some
coordinate chart)

$dW_{s}^{x}=b(W_{s}^{x})ds+\sigma(W_{s}^{x})dB_{s},\ W_{0}^{x}=x$

in $\mathbb{R}^{d},$ then the Brownian bridge $X^{t,x,y}$ has a corresponding
SDE representation

$\begin{cases}dX_{s}^{t,x,y}=\big{(}b(X_{s}^{t,x,y})+\nabla\log
p(t-s,X_{s}^{t,x,y},y)\big{)}ds+\sigma(X_{s}^{t,x,y})dB_{s},&0\leqslant
s<t;\\\ X_{0}^{t,x,y}=x.\end{cases}$ (2.10)

By abuse of notation the above $\nabla$ is its local expression under the
given chart (i.e. the vector $(g^{ij}\partial_{j})_{1\leqslant i\leqslant d}$)
.

######  Example 2.3.

In the Euclidean case $M=\mathbb{R}^{d}$, by using the explicit formula for
the heat kernel, one finds that

$\nabla\log p(t-s,x,y)=\frac{y-x}{t-s}.$

The SDE (2.10) becomes

$\begin{cases}dX_{s}^{t,x,y}=-\frac{1}{t-s}X_{s}^{t,x,y}ds+dB_{s},\\\
X_{0}^{t,x,y}=x-y=:\xi.\end{cases}$

Its solution is explicitly given by

$X_{s}^{t,x,y}=\frac{t-s}{t}\xi+(t-s)\int_{0}^{s}\frac{dB_{u}}{t-u},\ \ \
0\leqslant s<t.$

###  2.3 The $\phi$-signature on manifolds

In this section, we introduce a natural notion of signature for paths on
differentiable manifolds (iterated integrals against one-forms). This was
first used by K.T. Chen in a geometric setting and also by Le Jan-Qian in the
study of the signature uniqueness theorem for Brownian motion (they called it
extended signature).

We first recall the definition of path signature in the Euclidean space. Let
$E=\mathbb{R}^{N}$ be equipped with the Euclidean metric. The (infinite)
tensor algebra over $E$ is the unital algebra

$T((E))\triangleq\prod_{n=0}^{\infty}E^{\otimes
n}=\big{\\{}a=(a_{0},a_{1},a_{2},\cdots):a_{i}\in E^{\otimes i}\ \forall
i\in\mathbb{N}\big{\\}},$

where addition $+$ is just the vector addition, multiplication $\otimes$ is
defined by

$(a\otimes b)_{m}\triangleq\sum_{k=0}^{m}a_{k}\otimes
b_{m-k}\in(\mathbb{R}^{N})^{\otimes m},\ \ \ a=(a_{i}),b=(b_{i})\in T((E))$

and the unit is ${\bf 1}\triangleq(1,0,0,\cdots).$ The signature of a smooth
path $\gamma:[0,T]\rightarrow E$ is the element in $T((E))$ defined by

$S(\gamma)\triangleq\big{(}1,\gamma_{T}-\gamma_{0},\int_{0<s<t<T}d\gamma_{s}\otimes
d\gamma_{t},\cdots,\int_{0<t_{1}<\cdots<t_{n}<T}d\gamma_{t_{1}}\otimes\cdots\otimes
d\gamma_{t_{n}},\cdots\big{)}.$

More generally, one could also define the signature path $t\mapsto
S_{t}(\gamma)$ by integrating up to time $t$ instead
($S(\gamma)=S_{T}(\gamma)$). The definition extends to the rough path setting
according to Lyons’ extension theorem (cf. [Lyo98, Theorem 2.2.1]). If
$\gamma$ is the Brownian motion, the above iterated integrals are equivalently
understood in the Stratonovich sense. It is standard that the signature path
$S_{t}(\gamma)$ satisfies the following differential equation on $T((E))$:

$dS_{t}(\gamma)=S_{t}(\gamma)\otimes d\gamma_{t}.$ (2.11)

If $\gamma$ is a smooth path on a differentiable manifold $M$, one cannot
integrate along $\gamma$ intrinsically without any additional structure; one
needs to integrate $\gamma$ against a one-form. Recall that a (smooth) one-
form is a smooth section $\phi$ of the cotangent bundle. In other words, it is
a map that assigns to each location $x\in M$ a linear functional $\phi(x)\in
T_{x}^{*}M$ in a smooth manner. The integral

$\int_{0}^{t}\phi(d\gamma)\triangleq\int_{0}^{t}{}_{T_{\gamma_{s}}^{*}M}\langle\phi(\gamma_{s}),\gamma^{\prime}_{s}\rangle_{T_{\gamma_{s}}M}ds$

clearly has an intrinsic meaning and is called the line integral of $\gamma$
against $\phi$.

######  Example 2.4.

Let $M$ be the torus $S^{1}\times S^{1}$ and let $\phi=(\alpha,\beta)$ be the
two canonical generators of the first homology group of $M$. Then
$\int_{0}^{t}\phi(d\gamma)\in\mathbb{R}^{2}$ describes the winding angles of
$\gamma$ up to time $t$ with respect to the vertical and horizontal circles.

Now let $E=\mathbb{R}^{N}$ and suppose that $\phi$ is a smooth $E$-valued one-
form on $M$. In other words, $\phi(x)$ is an $E$-valued linear functional on
$T_{x}M$. Equivalently, one can write $\phi=(\phi^{1},\cdots,\phi^{N})$ where
each $\phi^{i}$ is a real-valued one-form.

######  Definition 2.5.

Let $\gamma:[0,T]\rightarrow M$ be a smooth path. The $\phi$-signature of
$\gamma$ up to time $t$ is defined to be

$S_{t}^{\phi}(\gamma)\triangleq S_{t}(\int_{0}^{\cdot}\phi(d\gamma))\in
T((E)).$

This is the signature up to time $t$ of the $E$-valued path
$\int_{0}^{\cdot}\phi(d\gamma)$ . We also write $S^{\phi}(\gamma)\triangleq
S_{T}^{\phi}(\gamma)$.

######  Remark 2.6.

This notion of signature depends on the choice of $\phi$. Indeed, the usual
Euclidean signature is a special case of this: the signature of an
$\mathbb{R}^{N}$-valued path is the $\phi$-signature with respect to the
$\mathbb{R}^{N}$-valued one-form $\phi=(dx^{1},\cdots,dx^{N}).$

######  Remark 2.7.

The definition extends naturally to the rough path case. We choose not to
introduce any technicalities from general rough path theory since they are not
essential for most of our analysis. In our study, $\gamma$ will either be a
smooth path (a geodesic) or an $M$-valued semimartingale where the
Stratonovich calculus for $\gamma$ is classical (e.g. via embedding $M$ into
an ambient Euclidean space and perform the usual stochastic calculus over
there). The reader is referred to [Hsu02] for an excellent introduction to
stochastic analysis on manifolds.

######  Example 2.8.

An important example which is of our primary interest is the case when $M$ is
embedded inside some Euclidean space $\mathbb{R}^{N}$ and the one-form
$\phi=dF$ ($F$ is the embedding map). In this case, one has
$\int_{0}^{t}\phi(d\gamma_{s})=F(\gamma_{t})-F(\gamma_{0})$ and the
$\phi$-signature of $\gamma$ is just the Euclidean signature of
$F(\gamma_{t})$ in $\mathbb{R}^{N}$.

##  3 The expected Brownian signature dynamics

In this section, we consider the expected $\phi$-signature of Brownian motion
and Brownian bridge on $M$. We derive intrinsic PDEs describing their dynamics
from the perspective of the Eells-Elworthy-Malliavin lifting. Throughout the
rest, $E=\mathbb{R}^{N}$ and $\phi$ is a given fixed $E$-valued one-form on
$M$.

###  3.1 Intrinsic PDE for the expected $\phi$-signature

Let $W^{x}=\\{W_{t}^{x}:t\geqslant 0\\}$ be a Brownian motion in $M$ starting
at $x\in M$. Consider the expected $\phi$-signature of $W^{x}$ up to time $t$
defined by

$\Psi(t,x)\triangleq\mathbb{E}[S_{t}^{\phi}(W^{x})]\in T((E)),\ \ \
(t,x)\in[0,\infty)\times M.$ (3.1)

Note that $\Psi$ is well-defined due to the compactness of $M$; by embedding
$M$ into an Euclidean space $V$ one can view $W^{x}$ as the solution to an SDE
on $V$ with $C_{b}^{\infty}$-coefficients and $\phi$ also extends to a
$C_{b}^{\infty}$-form on $V$. We first derive the intrinsic PDE governing the
dynamics of $\Psi(t,x)$.

######  Theorem 3.1.

The function $\Psi(t,x)$ satisfies the following parabolic PDE on $M$:

$\frac{\partial\Psi}{\partial t}=\frac{1}{2}\Delta\Psi+{\rm
Tr}\big{(}\phi\otimes
d\Psi+\frac{1}{2}(\nabla\phi+\phi\otimes\phi)\otimes\Psi\big{)}$ (3.2)

with initial condition $\Psi(0,\cdot)={\bf 1}.$

Before getting into the proof, some explanation about the notation in (3.2) is
needed.

(i) The PDE (3.2) is $T((E))$-valued. Its projection onto the first $m$
degrees becomes a finite dimensional coupled PDE system.
(ii) Since $\Psi$ is an $T((E))$-valued function on $M$ (for fixed time $t$),
$d\Psi$ is a $T((E))$-valued one-form. The product $\phi\otimes d\Psi$ is the
$T((E))$-valued bilinear form defined by

$\phi\otimes d\Psi(U,V)\triangleq\langle\phi,U\rangle\otimes\langle
d\Psi,V\rangle$

where $U,V$ are vector fields on $M$ and the tensor product $\otimes$ is the
multiplication in $T((E))$. The trace ${\rm Tr}(\phi\otimes d\Psi)$ becomes a
$T((E))$-valued function on $M$. At each point $x\in M$, one has

${\rm Tr}(\phi\otimes
d\Psi)(x)=\sum_{i=1}^{d}\langle\phi,e_{i}\rangle(x)\otimes\langle
d\Psi(t,x),e_{i}\rangle,$

where $\\{e_{1},\cdots,e_{d}\\}$ is any ONB of $T_{x}M.$ The above expression
is clearly independent of the choice of the ONB.
(iii) $\nabla\phi$ is the covariant derivative of $\phi$ with respect to the
Levi-Civita connection. The zeroth order term in (3.2) is the $T((E))$-valued
function defined by

${\rm
Tr}\big{(}(\nabla\phi+\phi\otimes\phi)\otimes\Psi\big{)}(x)=\sum_{i=1}^{d}\big{(}\langle\nabla_{e_{i}}\phi,e_{i}\rangle(x)+\langle\phi,e_{i}\rangle(x)\otimes\langle\phi,e_{i}\rangle(x)\big{)}\otimes\Psi(t,x).$

This expression is also independent of the choice of the ONB
$\\{e_{1},\cdots,e_{d}\\}$ of $T_{x}M$.

######  Example 3.2.

In the Euclidean case $M=\mathbb{R}^{d}$ with $\phi=(dx^{1},\cdots,dx^{d}),$
the term $\nabla\phi+\phi\otimes\phi$ appearing in the PDE (3.1) is constant
in space. As a result, the PDE must also be spatially homogeneous and thus
reduces to the ODE

$\frac{d\Psi}{dt}=\frac{1}{2}\sum_{i=1}^{d}E_{i}\otimes E_{i}\otimes\Psi,\ \ \
\Psi(0,\cdot)={\bf 1},$ (3.3)

where $\\{E_{1},\cdots,E_{d}\\}$ is the canonical basis of $\mathbb{R}^{d}$.
Its explicit solution is the expected signature of Euclidean Brownian motion:

$\Psi(t,x)=\exp\big{(}\frac{t}{2}(E_{1}\otimes E_{1}+\cdots+E_{d}\otimes
E_{d})\big{)}.$ (3.4)

#### Proof of Theorem 3.1

We now proceed to prove Theorem 3.1. The main observation is that the
horizontal Brownian motion on the OFB ${\cal O}(M)$ (cf. Theorem 2.2) coupled
with its signature process satisfies an intrinsic SDE on the ${\cal
O}(M)\times T((E))$. A benefit from this is that one can easily identify the
generator from the SDE and write down the associated Kolmorogov’s forward
equation on the bundle ${\cal O}(M)$ (Feynman-Kac representation). The PDE on
${\cal O}(M)$ then naturally projects to the desired intrinsic PDE (3.2) on
the base manifold $M$. Some technical care is needed to implement this idea
precisely and we develop the steps carefully in what follows.

First of all, one has the following standard fact.

######  Lemma 3.3.

The PDE (3.2) with initial condition $\Psi(0,\cdot)={\bf 1}$ has a unique
smooth solution denoted as $\hat{\Psi}(t,x)$ (i.e. every component of
$\hat{\Psi}$ is smooth in $(t,x)$).

###### Proof.

Let $f_{0}\in{\cal C}(M)$ and $g\in{\cal C}([0,\infty)\times M)$ be given
functions. Consider the following inhomogeneous (scalar) Cauchy problem on
$M$:

$\begin{cases}\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+g,\\\
u(0,\cdot)=f_{0}.\end{cases}$ (3.5)

Since $M$ is compact, it is standard that the above PDE has a unique solution
$u(t,x)$ which is smooth for all positive times. In fact, the solution is
explicitly given by

$u(t,x)=\int_{M}p(t,x,y)f_{0}(y)dy+\int_{0}^{t}\int_{M}p(t-s,x,y)g(s,y)dyds.$

The integrals are well-defined due to the compactness of $M$ and the
continuity of $f_{0},g.$ The smoothness of $u$ follows from the smoothness of
the heat kernel $p(t,x,y)$.

To prove the lemma, it is enough to observe that the PDE (3.2) is completely
decoupled into the (scalar) form of (3.5) when one rewrites it in terms of
coordinate components of $\Psi$. In fact, one can solve the PDE (3.5)
inductively on the degree of signature where $g$ is given by lower degree
components which is presumed to be known by induction. ∎

Next, we lift the solution $\hat{\Psi}(t,x)$ given by Lemma 3.3 to the bundle
${\cal O}(M)$. Let us define

$f(t,\xi)\triangleq\hat{\Psi}(t,\pi(\xi)),\ \ \ t\geqslant 0,\xi\in{\cal
O}(M),$ (3.6)

where $\pi:{\cal O}(M)\rightarrow M$ denotes the bundle projection. We also
denote $\Phi\triangleq\pi^{*}\phi$ as the pullback of $\phi$ on the bundle.
Then one can establish the PDE for $f(t,\xi)$ on ${\cal O}(M)$.

######  Lemma 3.4.

The function $f(t,\xi)$ satisfies the following parabolic PDE:

$\frac{\partial f}{\partial
t}=\frac{1}{2}\sum_{i=1}^{d}H_{i}^{2}f+\sum_{i=1}^{d}\langle\Phi,H_{i}\rangle\otimes
H_{i}f+\frac{1}{2}\sum_{i=1}^{d}\big{(}H_{i}\langle\Phi,H_{i}\rangle+\langle\Phi,H_{i}\rangle^{\otimes
2}\big{)}\otimes f$ (3.7)

with initial condition $f(0,\xi)={\bf 1}.$

###### Proof.

Since $\hat{\Psi}(t,x)$ satisfies the PDE (3.2) on $M$ and $f$ is the pullback
of $\hat{\Psi}$ on the bundle ${\cal O}(M)$, it is sufficient to prove that
the pullback of the right hand side of (3.2) is precisely the right hand side
of (3.7). We look at each individual term separately.

(i) The horizontal Laplacian $\sum_{i=1}^{d}H_{i}^{2}$ on ${\cal O}(M)$ is the
pullback of the Laplacian $\Delta$ on $M$ (cf. [Hsu02, Proposition 3.1.2]):

$\sum_{i=1}^{d}H_{i}^{2}f=\sum_{i=1}^{d}H_{i}^{2}\pi^{*}\hat{\Psi}=\pi^{*}\Delta\hat{\Psi}.$

(ii) We claim that

$\sum_{i=1}^{d}\langle\Phi,H_{i}\rangle\otimes H_{i}f=\pi^{*}{\rm
Tr}(\phi\otimes d\hat{\Psi}).$

Indeed, let $\xi=(x,u)\in{\cal O}(M)$ with $u=\\{e_{1},\cdots,e_{d}\\}$. Then
one has

$\langle\Phi,H_{i}\rangle(\xi)=\langle\pi^{*}\phi,H_{i}\rangle(\xi)=\langle\phi,(\pi_{*})_{\xi}H_{i}\rangle(x)=\langle\phi,e_{i}\rangle(x).$

Similarly,

$(H_{i}f)(\xi)=\langle df,H_{i}\rangle(\xi)=\langle
d\hat{\Psi},e_{i}\rangle(x).$

It follows that

$\big{(}\sum_{i=1}^{d}\langle\Phi,H_{i}\rangle\otimes
H_{i}f\big{)}(\xi)=\big{(}\sum_{i=1}^{d}\langle\phi,e_{i}\rangle\otimes\langle
d\hat{\Psi},e_{i}\rangle\big{)}(x)={\rm Tr}((\phi\otimes d\hat{\Psi}))(x).$

Note that the trace, as a bilinear form on the inner product space $T_{x}M$,
is independent of the choice of the ONB $u$. In a similar way, one also has

$\sum_{i=1}^{d}\langle\Phi,H_{i}\rangle^{\otimes 2}=\pi^{*}{\rm
Tr}(\phi\otimes\phi).$

(iii) It remains to show that

$\sum_{i=1}^{d}H_{i}\langle\Phi,H_{i}\rangle=\pi^{*}{\rm Tr}(\nabla\phi).$
(3.8)

As before, let $\xi=(x,u)$ be given fixed with $u=\\{e_{1},\cdots,e_{d}\\}$.
Let $\gamma:(-\varepsilon,\varepsilon)\rightarrow M$ be a curve in $M$ such
that $\gamma_{0}=x$ and $\gamma^{\prime}_{0}=e_{i}.$ Let
$\xi_{t}=(\gamma_{t},u_{t})$ be the horizontal lifting of $\gamma$ with
$\xi_{0}=\xi.$ Then one has

$\displaystyle H_{i}\langle\Phi,H_{i}\rangle(\xi)$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}\langle\Phi,H_{i}\rangle(\xi_{t})=\frac{d}{dt}\big{|}_{t=0}\langle\phi,(\pi_{*})_{\xi_{t}}H_{i}\rangle(\gamma_{t})$
$\displaystyle=\frac{d}{dt}\big{|}_{t=0}\langle\phi(\gamma_{t}),e_{i}(t)\rangle=\langle\frac{D\phi(\gamma_{t})}{dt}\big{|}_{t=0},e_{i}\rangle+\langle\phi(x),\frac{De_{i}(t)}{dt}\big{|}_{t=0}\rangle,$

where $e_{i}(t)$ denotes the $i$-th component of $u_{t}$ and $\frac{D}{dt}$ is
the covariant derivative along $\gamma$. Since $\xi_{t}$ is horizontal, one
has

$\frac{De_{i}(t)}{dt}=0\ \ \ \forall t.$

As a result, one obtains that

$H_{i}\langle\Phi,H_{i}\rangle(\xi)=\langle\nabla_{e_{i}}\phi,e_{i}\rangle(x)$

and the claim (3.8) thus follows. ∎

Now we derive the Feynman-Kac representation for the PDE (3.7) (with no
surprise, it is given in terms of the joint process defined by the horizontal
Brownian motion and its $\Phi$-signature process). Recall that $\Xi^{\xi}$ is
the horizontal Brownian motion defined by the horizontal SDE (2.8) on the
bundle and let $\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\in T((E))$ denote the
signature path of the $E$-valued path

$t\mapsto\int_{0}^{t}\Phi(\circ d\Xi_{s}^{\xi}).$

######  Lemma 3.5.

One has the following representation:

$f(t,\xi)=\mathbb{E}\big{[}\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\big{]},\ \ \
(t,\xi)\in[0,\infty)\times{\cal O}(M).$ (3.9)

###### Proof.

Consider the Markov family

$(\Xi,\tilde{S})\triangleq\big{\\{}\big{(}\Xi_{t}^{\xi},g\otimes\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\big{)}:t\geqslant
0,(\xi,g)\in{\cal O}(M)\times T((E))\big{\\}}.$

This family is governed by the Stratonovich SDE (cf. (2.11))

$\begin{cases}d\Xi_{t}=\sum_{i=1}^{d}H_{i}(\Xi_{t})\circ dB_{t}^{i},\\\
d\tilde{S}_{t}=\sum_{i=1}^{d}\tilde{S}_{t}\otimes\langle\Phi,H_{i}\rangle(\Xi_{t})\circ
dB_{t}^{i}\end{cases}$ (3.10)

on the state space ${\cal O}(M)\times T((E))$. Let ${\cal L}$ be the generator
of $(\Xi,\tilde{S})$. It is standard that ${\cal
L}=\frac{1}{2}\sum_{i=1}^{d}{\cal V}_{i}^{2}$ where $\\{{\cal
V}_{1},\cdots,{\cal V}_{d}\\}$ are the vector fields for the SDE (3.10) and
they are viewed as differential operators on $C^{\infty}\big{(}{\cal
O}(M)\times T((E))\big{)}$. Respectively, we also denote

$\hat{{\cal
L}}F\triangleq\frac{1}{2}\sum_{i=1}^{d}H_{i}^{2}F+\sum_{i=1}^{d}\langle\Phi,H_{i}\rangle\otimes
H_{i}F+\frac{1}{2}\sum_{i=1}^{d}\big{(}H_{i}\langle\Phi,H_{i}\rangle+\langle\Phi,H_{i}\rangle^{\otimes
2}\big{)}\otimes F$

for $F\in C^{\infty}\big{(}{\cal O}(M)\times T((E))\big{)}$. Note that the
situation here is essentially finite dimensional since one can consistently
look at the truncations of the SDE (3.10) and PDE (3.7) up to any fixed level.

Consider the function

$F(t,(\xi,g))\triangleq g\otimes f(t,\xi),$ (3.11)

where we recall that $f(t,\xi)$ is defined by (3.6). We claim that ${\cal
L}F=\hat{{\cal L}}F$. Indeed, note that the vector fields for the SDE (3.10)
are given by

${\cal V}_{i}\triangleq V_{i}\oplus W_{i},\ \ \ 1\leqslant i\leqslant d,$

where

$V_{i}(\xi,g)=H_{i}(\xi),\
W_{i}(\xi,g)=g\otimes\langle\Phi,H_{i}\rangle(\xi).$ (3.12)

We first compute ${\cal V}_{i}F$. By the definition (3.11) of $F$, it is clear
that

$(V_{i}F)(t,(\xi,g))=g\otimes(H_{i}f)(t,\xi).$ (3.13)

In addition, by the definition (3.12) of $W_{i}$ one has (recall
$E=\mathbb{R}^{N}$)

$(W_{i}F)(t,(\xi,g))=\sum_{m\geqslant
1}\sum_{i_{1},\cdots,i_{m}=1}^{N}g^{i_{1},\cdots,i_{m-1}}\langle\Phi^{i_{m}},H_{i}\rangle(\xi)\partial_{i_{1},\cdots,i_{m}}(g\otimes
f(t,\xi)),$

where $g^{i_{1},\cdots,i_{m-1}}$ are the coordinate components of the tensor
$g$, $\Phi^{j}$ is the $j$-th component of $\Phi=(\Phi^{1},\cdots,\Phi^{N})$
as an $\mathbb{R}^{N}$-valued one-form and $\partial_{i_{1},\cdots,i_{m}}$ is
the partial derivative in $g$ with respect to the coordinate
$g^{i_{1},\cdots,i_{m}}.$ For any given word $J=(j_{1},\cdots,j_{l})$, the
$J$-th component of $W_{i}F$ is thus given by

$\displaystyle(W_{i}F)^{J}(t,(\xi,g))$ $\displaystyle=\sum_{m\geqslant
1}\sum_{i_{1},\cdots,i_{m}=1}^{N}g^{i_{1},\cdots,i_{m-1}}\langle\Phi_{i_{m}},H_{i}\rangle(\xi)\partial_{i_{1},\cdots,i_{m}}\big{(}\sum_{s=1}^{l}g^{j_{1},\cdots,j_{s}}f(t,\xi)^{j_{s+1},\cdots,j_{l}}\big{)}$
$\displaystyle=\sum_{s=1}^{l}g^{j_{1},\cdots,j_{s-1}}\langle\Phi_{j_{s}},H_{i}\rangle(\xi)f(t,\xi)^{j_{s+1},\cdots,j_{l}}$
$\displaystyle=\big{(}g\otimes\langle\Phi,H_{i}\rangle\otimes
f(t,\xi)\big{)}^{J}.$

In its equivalent tensor form,

$(W_{i}F)(t,(\xi,g))=g\otimes\langle\Phi,H_{i}\rangle(\xi)\otimes f(t,\xi).$
(3.14)

Combining (3.13) and (3.14), one finds that

$({\cal
V}_{i}F)(t,(\xi,g))=g\otimes\big{(}(H_{i}f)(t,\xi)+\langle\Phi,H_{i}\rangle(\xi)\otimes
f(t,\xi)\big{)}.$

By applying ${\cal V}_{i}$ again, one arrives at

$\displaystyle{\cal V}_{i}^{2}F$ $\displaystyle=g\otimes
H_{i}\big{(}H_{i}f+\langle\Phi,H_{i}\rangle\otimes
f\big{)}+g\otimes\langle\Phi,H_{i}\rangle\otimes\big{(}H_{i}f+\langle\Phi,H_{i}\rangle\otimes
f\big{)}$
$\displaystyle=g\otimes\big{(}H_{i}^{2}f+H_{i}\langle\Phi,H_{i}\rangle\otimes
f+2\langle\Phi,H_{i}\rangle\otimes H_{i}f+\langle\Phi,H_{i}\rangle^{\otimes
2}\otimes f\big{)}.$ (3.15)

This gives the relation ${\cal L}F=\hat{{\cal L}}F.$

It follows from the above relation and Lemma 3.4 that $F$ satisfies the PDE
$\frac{\partial F}{\partial t}={\cal L}F$ with initial condition
$F(0,\cdot)=\rho(\cdot),$ where $\rho:{\cal O}(M)\times T((E))\rightarrow
T((E))$ denotes the projection onto the second component. It is now a standard
application of the Feynman-Kac representation that

$g\otimes
f(t,\xi)=F(t,(\xi,g))=\mathbb{E}\big{[}\rho\big{(}\Xi_{t}^{\xi},g\otimes\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\big{)}\big{]}=g\otimes\mathbb{E}\big{[}\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\big{]}.$

By cancelling out the $g$ on both sides, one arrives at the desired relation
(3.9). ∎

Finally, the lemma below completes the proof of Theorem 3.1. Recall that
$\Psi(t,x)$ is the expected $\phi$-signature of the Brownian motion (cf.
(3.1)).

######  Lemma 3.6.

$\Psi(t,x)=\hat{\Psi}(t,x)$ for all $t\geqslant 0$ and $x\in M$.

###### Proof.

Fix $x\in M$ and let $\xi\in\pi^{-1}(x).$ According to Theorem 2.2,
$W_{t}\triangleq\pi(\Xi_{t}^{\xi})$ is a Brownian motion starting at $x$. In
addition, by the definition of $\Phi$ one has

$\langle\Phi,\circ d\Xi_{t}^{\xi}\rangle=\langle\pi^{*}\phi,\circ
d\Xi_{t}^{\xi}\rangle=\langle\phi,\circ\pi_{*}d\Xi_{t}^{\xi}\rangle=\langle\phi,\circ
dW_{t}\rangle.$

It follows from Lemma 3.5 that

$\hat{\Psi}(t,x)=f(t,\xi)=\mathbb{E}\big{[}\tilde{S}_{t}^{\Phi}(\Xi^{\xi})\big{]}=\mathbb{E}[S_{t}^{\phi}(W)]=\Psi(t,x).$

Note that the above relation is independent of the choice of
$\xi\in\pi^{-1}(x)$ since the law of $W$ is. ∎

######  Remark 3.7.

The proof of Lemma 3.5 does not use the fact that $\Phi$ is the lifting of the
one-form $\phi$. Indeed, let $\Phi$ be any given $E$-valued one-form on ${\cal
O}(M)$. For each $\xi\in{\cal O}(M)$, let $\Xi_{t}^{\xi}$ be the horizontal
Brownian motion starting at $\xi$. Then one can show that the function

$(t,\xi)\mapsto\mathbb{E}[\tilde{S}_{t}^{\Phi}(\Xi^{\xi})]$

satisfies the PDE (3.7).

###  3.2 The Brownian bridge case

Now let us consider the expected $\phi$-signature of the Brownian bridge
defined by

$\psi(t,x,y)\triangleq\mathbb{E}\big{[}S^{\phi}(X^{t,x,y})\big{]},\ \ \
t\geqslant 0,\ x,y\in M,$

where $X^{t,x,y}$ is the Brownian bridge from $x$ to $y$ with lifetime $t$. By
using the PDE (3.2), it is not hard to derive the associated PDE for
$\psi(t,x,y)$.

######  Theorem 3.8.

For each fixed $y\in M,$ the function

$(0,\infty)\times M\ni(t,x)\mapsto\psi(t,x,y)$

satisfies the following PDE:

$\displaystyle\frac{\partial\psi}{\partial t}$
$\displaystyle=\frac{1}{2}\Delta_{x}\psi+\langle\nabla_{x}\log
p(t,x,y),\nabla_{x}\psi+\phi\otimes\psi\rangle+{\rm Tr}(\phi\otimes
d_{x}\psi)$ $\displaystyle\ \ \ +\frac{1}{2}{\rm
Tr}\big{(}(\nabla\phi+\phi\otimes\phi)\otimes\psi\big{)},$ (3.16)

where $p(t,x,y)$ is the heat kernel.

######  Remark 3.9.

The term $\langle\nabla\log p,\phi\otimes\psi\rangle$ is understood as
${}_{TM}\langle\nabla\log p,\phi\rangle_{T^{*}M}\otimes\psi\in T((E))$.

###### Proof.

By the definition of Brownian bridge, one has

$\psi(t,x,y)=\frac{\mathbb{E}[S_{t}^{\phi}(W^{x})\delta_{y}(W_{t}^{x})]}{\mathbb{E}[\delta_{y}(W_{t}^{x})]}=\frac{\mathbb{E}[S_{t}^{\phi}(W^{x})\delta_{y}(W_{t}^{x})]}{p(t,x,y)},$

where $W^{x}$ is the Brownian motion starting at $x$. Given any smooth
function $f:M\rightarrow\mathbb{R},$ exactly the same argument as in the proof
of Theorem 3.1 shows that the function

$(0,\infty)\times
M\ni(t,x)\mapsto\mathbb{E}\big{[}f(W_{t}^{x})S_{t}^{\phi}(W^{x})\big{]}$

satisfies the same PDE (3.2). In particular, the function ($y$ being fixed)

$\bar{\psi}(t,x)\triangleq\mathbb{E}[S_{t}^{\phi}(W^{x})\delta_{y}(W_{t}^{x})]$

also satisfies (3.2).

Now let us write $\psi=\bar{\psi}/p$. By using the PDE for $\bar{\psi}$ and
the heat equation (2.2) for $p$, one finds that

$\displaystyle\frac{\partial\psi}{\partial t}$
$\displaystyle=\frac{1}{p}\frac{\partial\bar{\psi}}{\partial
t}+\bar{\psi}\big{(}-\frac{1}{p^{2}}\frac{\partial p}{\partial t}\big{)}$
$\displaystyle=\frac{1}{p}\big{(}\frac{1}{2}\Delta\bar{\psi}+{\rm
Tr}\big{(}\phi\otimes
d\bar{\psi}+\frac{1}{2}(\nabla\phi+\phi\otimes\phi)\otimes\bar{\psi}\big{)}-\frac{\bar{\psi}}{2p}\Delta
p\big{)}$ $\displaystyle=\frac{1}{p}\big{(}\frac{1}{2}\Delta(p\psi)+{\rm
Tr}\big{(}\phi\otimes d(p\psi)\big{)}+\frac{1}{2}{\rm
Tr}\big{(}(\nabla\phi+\phi\otimes\phi)\otimes\psi\big{)}-\frac{\psi}{2p}\Delta
p.$

Since

$\Delta(p\psi)=\Delta p\cdot\psi+p\cdot\Delta\psi+2\langle\nabla
p,\nabla\psi\rangle,$

it follows that

$\frac{\partial\psi}{\partial
t}=\frac{1}{2}\Delta\psi+\frac{1}{p}\langle\nabla
p,\nabla\psi\rangle+\frac{1}{p}{\rm Tr}(\phi\otimes\psi dp)+{\rm
Tr}\big{(}\phi\otimes
d\psi+\frac{1}{2}(\nabla\phi+\phi\otimes\phi)\otimes\psi\big{)}.$

The result follows by further noting that

$\displaystyle{\rm Tr}(\phi\otimes\psi dp)$
$\displaystyle=\sum_{i=1}^{d}\langle\nabla
p,e_{i}\rangle\phi(e_{i})\otimes\psi=\phi(\nabla p)\otimes\psi,$

where $\\{e_{1},\cdots,e_{d}\\}$ is any ONB of $T_{x}M$ and $p^{-1}\nabla
p=\nabla\log p$. ∎

A basic point to make is that rich geometric information about the underlying
space $M$ is encoded in the asymptotic behaviour of the function $\psi(t,x,y)$
as $t\rightarrow 0^{+}$. This motivates our study in the next two sections.

##  4 Signature asymptotics and reconstruction of Riemannian distance

In this section, we develop a method of reconstructing the Riemannian distance
$d(x,y)$ from the expected signature of the Brownian bridge $X^{t,x,y}$
through an explicit asymptotics procedure, at least when $d(x,y)$ is not too
far from each other.

###  4.1 The main theorem

Suppose that $F:M\rightarrow E=\mathbb{R}^{N}$ is a given fixed isometric
embedding (which always exists by Nash’s embedding theorem). For each
$n\geqslant 1$, we equip $E^{\otimes n}$ with the Hilbert-Schmidt tensor norm
$\|\cdot\|_{{\rm HS}}$. In other words, viewing $E$ as an Euclidean space, the
space $E^{\otimes n}$ is equipped with an inner product structure induced by

$\langle v_{1}\otimes\cdots\otimes v_{n},w_{1}\otimes\cdots\otimes
w_{n}\rangle_{{\rm HS}}\triangleq\langle v_{1},w_{1}\rangle_{E}\cdots\langle
v_{n},w_{n}\rangle_{E},\ \ \ v_{i},w_{j}\in E.$

Recall that $X^{t,x,y}$ is the Brownian bridge from $x$ to $y$ with lifetime
$t$ and $S^{dF}(X^{t,x,y})$ is its $\phi$-signature with $\phi=dF$. Our main
result in this section is stated as follows.

######  Theorem 4.1.

Let $x,y\in M$ be given fixed such that $d(x,y)<\rho_{M}/2$ where $\rho_{M}$
is the global injective radius of $M$. Then there exist geometric constants
$C,\kappa$ depending on $x,y,M$, such that

$\big{|}\big{(}n!\big{\|}\mathbb{E}\big{[}\pi_{n}S^{dF}(X^{\kappa
n^{-6},x,y})\big{]}\big{\|}_{{\rm
HS}}\big{)}^{1/n}-d(x,y)\big{|}\leqslant\frac{C}{n},\ \ \ n\geqslant 1.$ (4.1)

In particular, with $t_{n}=o(n^{-6})$ one has

$\lim_{n\rightarrow\infty}\big{(}n!\big{\|}\mathbb{E}\big{[}\pi_{n}S^{dF}(X^{t_{n},x,y})\big{]}\big{\|}_{{\rm
HS}}\big{)}^{1/n}=d(x,y).$ (4.2)

######  Remark 4.2.

The assumption that $\phi=dF$ with some isometric embedding $F$ plays no
essential role; the only relevant fact is that $d(x,y)$ is precisely the
Euclidean length of the path $\int_{0}^{\cdot}\phi(d\gamma^{x,y})$
($\gamma^{x,y}$ is the unique minimising geodesic from $x$ to $y$). For a
general $\phi$, the theorem remains valid as long as one replaces $d(x,y)$
with the length of the path $\int_{0}^{\cdot}\phi(d\gamma^{x,y})$ in $E$.

######  Remark 4.3.

We expect that the same type of result holds (possibly with a different
condition on $t_{n}$) even if $y$ is on the cut-locus of $x.$ This may cause
extra technical difficulties in the current argument due to the non-uniqueness
of minimising geodesics connecting $x,y$.

###  4.2 Proof of Theorem 4.1

As we explained in the introduction, the asymptotics formula (4.2) is largely
inspired by the length conjecture (which is an actual theorem for smooth paths
such as the geodesic $\gamma^{x,y}$) as well as the fact that
$X^{t,x,y}\approx\gamma^{x,y}$ when $t$ is small. However, there are several
technical challenges to implement such an idea precisely. Essentially, given a
degree $n$ one wants to identify a lifetime scale $t_{n}$ such that the
(normalised) $n$-th level expected signature of the Brownian bridge with
lifetime $t_{n}$ yields the desired limit $d(x,y)$ as $n\rightarrow\infty$. If
one applies standard signature and semimartingale estimates without much care,
one is led to requiring that $t_{n}=o(e^{-Cn}),$ which is way too small to be
useful in practice. Improving it from exponential to polynomial scale $n^{-6}$
is the most delicate point in the argument (cf. Section 4.2.2 below).

In what follows, we develop the main ingredients for proving Theorem 4.1 in a
precise mathematical way. Basically, we localise the problem on a nice
coordinate chart (a normal chart) in which Euclidean stochastic calculus can
be applied. The part that the Brownian bridge exits the chart before its
lifetime is negligible (due to Hsu’s large deviation principle), while the
part within the chart yields the main contribution.

Throughout the rest, $\phi\triangleq dF$ and $x,y\in M$ with
$d(x,y)<\rho_{M}/2$ are all given fixed.

####  4.2.1 Computations under normal chart

We will perform local calculations under normal coordinates. For this purpose,
we first collect some useful geometric properties of normal charts.

We fix an orthonormal basis (ONB) $\\{e_{i}(0):i=1,\cdots,d\\}$ of $T_{y}M$.
Let $U\subseteq T_{y}M$ be a fixed open ball centered at the origin with
radius $r\leqslant\rho_{y}$ (the injective radius at $y$) so that the
exponential map $\exp_{y}:U\rightarrow M$ is a diffeomorphism. This gives rise
to a local parametrisation

${\bf x}=(x^{1},\cdots,x^{d})\mapsto\exp_{y}\big{(}x^{i}e_{i}(0)\big{)}$

of the ball $B(y,r)=\exp_{y}U=:V,$ which is known as the normal coordinate
chart based at $y$. As a convention, a bold letter always refers to an
Euclidean vector (e.g. ${\bf x}=(x^{1},\cdots,x^{d})$) that is also identified
with an element in $U$, while a regular letter (e.g. $x$) refers to a point on
$M$. For instance, one can legally write $x=\exp_{y}({\bf x}).$

The natural coordinate vector fields on $V$ are denoted as
$\\{\partial_{i}:i=1,\cdots,d\\}$ as usual. It is useful to introduce another
local frame of vector fields $\\{e_{i}:1,\cdots,d\\}$ as follows. Given ${\bf
x}\in U$, we define $e_{i}({\bf x})$ to be the parallel translation of
$e_{i}(0)$ along the geodesic $t\mapsto\exp_{y}(t{\bf x})$ to the endpoint
${\bf x}$ (at $t=1$). By varying ${\bf x}$, one obtains a vector field $e_{i}$
on $V$. It is clear that $\\{e_{1}({\bf x}),\cdots,e_{d}({\bf x})\\}$ is an
ONB of $T_{\exp_{y}{\bf x}}M$ for every ${\bf x}\in U.$ Note that
$e_{i}(0)=\partial_{i}|_{{\bf x}=0}$. In general, one can write
$e_{i}=\sigma_{i}^{j}\partial_{j}$ where $(\sigma_{i}^{j})_{1\leqslant
i,j\leqslant d}$ is a smooth matrix-valued function on $U$ that is everywhere
invertible.

Recall that the metric tensor under the normal chart $V$ is defined by
$g_{ij}\triangleq\langle\partial_{i},\partial_{j}\rangle$. The following fact
will be useful to us later on (cf. [BGV04, Proposition 1.27 (iii)]).

######  Lemma 4.4.

On the normal chart $V$, one has

$x^{i}g_{ij}=x^{j},\ x^{i}g^{ij}=x^{j}.$

It is well known that under the normal chart near the origin, the Riemannian
metric $g_{ij}$ agrees with Euclidean metric $\delta_{ij}$ up to the second
order with error given in terms of curvature coefficients (cf. [BGV04,
Proposition 1.28]).

######  Proposition 4.5.

On the normal chart $V$, one has

$g_{ij}({\bf x})=\delta_{ij}-\frac{1}{3}R_{ikjl}({\bf 0})x^{k}x^{l}+O(|{\bf
x}|^{3})\ \ \ \text{as\ }{\bf x}=(x^{1},\cdots,x^{d})\rightarrow 0,$

where $R_{ijkl}\triangleq\langle
R(\partial_{k},\partial_{l})\partial_{j},\partial_{i}\rangle$ are the
curvature coefficients in $V$.

We now express $\Delta/2$ on $V$ as

$\frac{1}{2}\Delta=\frac{1}{2}a^{ij}\partial_{ij}^{2}+b^{i}\partial_{i}.$
(4.3)

From the local expression (2.1), it is plain to check that

$a^{ij}=g^{ij},\ b^{i}=\frac{g^{ij}}{4\det g}\partial_{j}\det
g+\frac{1}{2}\partial_{j}g^{ij}.$ (4.4)

We also recall that $G_{0}(x,y),G_{1}(x,y)$ are the first two terms appearing
in the Malliavin-Stroock expansion (2.3). The following explicit formulae for
the local expressions of these quantities will be needed for us.

######  Lemma 4.6.

On the normal chart $V$, one has

$a^{ij}({\bf 0})=\delta^{ij},\ b^{i}({\bf 0})=0;$ (4.5)
$\partial_{p}a^{ij}({\bf 0})=0,\ \partial_{i}b^{j}({\bf
0})=\frac{1}{3}R_{jppi},\ \partial_{pp}^{2}a^{ij}({\bf
0})=\frac{2}{3}R_{ipjp};$ (4.6) $\nabla_{x}G_{0}({\bf x},{\bf
0})=-x^{j}\partial_{j};$ (4.7) $\partial_{j}G_{1}({\bf 0},{\bf 0})=0,\
\partial_{ij}^{2}G_{1}({\bf 0},{\bf 0})=\frac{1}{6}R_{pipj}.$ (4.8)

Here $R_{ijkl}$ is the curvature coefficient as in Proposition 4.5 and the
derivatives in the last line are taken with respect to the first variable in
$G_{1}.$

###### Proof.

The curvature coefficients satisfy the following symmetries:

$R_{ijkl}=-R_{jikl}=-R_{ijlk}=R_{klij}.$ (4.9)

The relations (4.5) and (4.6) follow directly from Proposition 4.5 and (4.4).
In addition, by the local expression of $\nabla$ and Lemma 4.4 one has

$\nabla_{x}G_{0}({\bf x},{\bf
0})=-x^{i}g^{ij}\partial_{j}=-x^{j}\partial_{j}.$

This gives the relation (4.7). Finally, recall that $G_{1}$ is defined by
(2.4). Standard geometric consideration shows that

$\det(d\exp_{y})_{{\bf x}}=\sqrt{\det g({\bf x})}$ (4.10)

for any ${\bf x}=\exp_{y}^{-1}(x)\in U$ ($x\in V$). The relation (4.8) then
easily follows from Proposition 4.5 and (4.10). ∎

######  Remark 4.7.

For the proof of Theorem 4.1, only (4.5) and (4.7) are relevant. The relations
(4.6) and (4.8) will only be needed for the proof of Theorem 5.1 in the next
section.

####  4.2.2  Localisation I: the main contribution

Our analysis is essentially localised on a normal chart around $y$. Throughout
the rest, we take $V\triangleq B(y,\rho_{M}/2-\varepsilon)$ where
$\varepsilon\in(0,\rho_{M}/2-d(x,y))$ is fixed (note that $x\in V$). Define

$\tau\triangleq\inf\\{s\in[0,t]:X_{s}^{t,x,y}\notin V\\}$

and we set $\tau=t$ if such $s$ does not exist. We shall consider the
decomposition

$\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y})\big{]}=\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau=t\big{]}+\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau<t\big{]}.$

One naturally expects that the first term provides the main contribution and
the second term is negligible. In this section, we establish the main estimate
for the first term. The second term will be handled in the next section.

Note that the term $\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau=t\big{]}$
corresponds to the situation where $X^{t,x,y}$ does not leave the normal chart
$B$ around $y$ during its lifetime. For this part, one can apply Euclidean
stochastic calculus to the SDE (2.10) of the Brownian bridge. We break down
the analysis into several steps.

#### A representation of $X^{t,x,y}$

In the Malliavin-Stroock expansion (2.3), recall that
$G_{0}(x,y)=-d(x,y)^{2}/2$. By applying Lemma 4.6 on the normal chart $V$
defined previously, one can write (using Euclidean variables on
$U\triangleq\exp_{y}^{-1}V$)

$\nabla_{z}\log p(u,{\bf z},{\bf 0})=-\frac{{\bf z}}{u}+Q(u,{\bf z}),$ (4.11)

where $\nabla_{z}$ denotes the Euclidean vector
$(g^{ij}\partial_{j})_{1\leqslant i\leqslant d}$ and

$Q(u,{\bf z})\triangleq\sum_{k=0}^{\infty}\nabla_{z}G_{k+1}({\bf z},{\bf
0})u^{k}$

satisfies

$\sup_{u\in[0,1],{\bf z}\in\bar{U}}|Q(u,z)|<\infty.$

The SDE (2.10) of $X^{t,x,y}$ on $V$ is thus expressed as

$\begin{cases}dX_{s}=\big{(}-\frac{X_{s}}{t-s}+b(X_{s})+Q(t-s,X_{s})\big{)}ds+\sigma(X_{s})dB_{s},&s<\tau;\\\
X_{0}={\bf x}\triangleq\exp_{y}^{-1}(x).\end{cases}$ (4.12)

where $(\sigma,b)$ are local coefficients of $\Delta/2$ satisfying the
relation (4.3) with $a=\sigma^{T}\sigma.$ We give a representation of $X_{s}$
that will be used frequently in the sequel.

######  Lemma 4.8.

On the event $\\{\tau=t\\}$, one has

$\frac{X_{tr}}{1-r}={\bf
x}+t\int_{0}^{tr}\frac{\sigma(X_{u})dB_{u}}{t-u}+t\int_{0}^{r}\frac{1}{1-\eta}\big{(}b(X_{t\eta})+Q(t(1-\eta),X_{t\eta})\big{)}d\eta,\
\ \ r\in[0,1).$ (4.13)

###### Proof.

We introduce the integrating factor $x_{s}\triangleq t-s$ ($s\in[0,t]$), which
satisfies

$dx_{s}=-\frac{1}{t-s}x_{s}ds\in\mathbb{R}.$

Setting $Z_{s}\triangleq x_{s}^{-1}X_{s},$ one finds that

$dZ_{s}=\frac{1}{t-s}\big{(}b(X_{s})+Q(t-s,X_{s})\big{)}ds+\frac{1}{t-s}\sigma(X_{s})dB_{s}.$

By integrating both sides from $0$ to $s$ and noting that $Z_{0}={\bf x}/t,$
one has

$\frac{X_{s}}{t-s}=\frac{{\bf
x}}{t}+\int_{0}^{s}\frac{\sigma(X_{u})dB_{u}}{t-u}+\int_{0}^{s}\frac{1}{t-u}\big{(}b(X_{u})+Q(t-u,X_{u})\big{)}du.$

The result follows by taking $s=tr$. ∎

#### A semimartingale decomposition for $\phi(\circ dX_{tr})$

On the event $\\{\tau=t\\}$, the signature component
$\pi_{n}S^{\phi}(X^{t,x,y})$ is given by

$\displaystyle\pi_{n}S^{\phi}(X^{t,x,y})$
$\displaystyle=\int_{0<s_{1}<\cdots<s_{n}<t}\phi(\circ
dX_{s_{1}})\otimes\cdots\otimes\phi(\circ dX_{s_{n}})$
$\displaystyle=\int_{0<r_{1}<\cdots<r_{n}<1}\phi(\circ
dX_{tr_{1}})\otimes\cdots\otimes\phi(\circ dX_{tr_{n}}),$ (4.14)

where $X_{s}$ satisfies the SDE (4.12). To relate it with the $\phi$-signature
of the minimising geodesic

$\gamma_{r}^{x,y}\triangleq\exp((1-r){\bf x}),\ \ \ 0\leqslant r\leqslant 1,$

an important step is to factor out the geodesic component in the
semimartingale decomposition of $\phi(\circ dX_{tr}).$ We summarise the main
structure in the lemma below. Let $(\Gamma_{r})_{0\leqslant r\leqslant 1}$ be
the $E$-valued path defined by
$\Gamma_{r}\triangleq\int_{0}^{r}\phi(d\gamma^{x,y})$.

######  Lemma 4.9.

On the event $\\{\tau=t\\}$, one has the following decomposition of
$\phi(\circ dX_{tr})$:

$\phi(\circ
X_{tr})=\dot{\Gamma}_{r}dr+\sqrt{t}\big{(}A_{r}^{t}dr+dM_{r}^{t}\big{)},\ \ \
r\in[0,1),$ (4.15)

where $\\{A_{r}^{t},M_{r}^{t}:0\leqslant r<1\\}$ are $E$-valued stochastic
processes satisfying the following properties:

(i) $\\{A_{r}^{t}\\}$ is a continuous, $\\{{\cal F}_{tr}^{B}\\}$-adapted
semimartingale such that

$\|A_{r}^{t}\|_{p}\leqslant\frac{C\sqrt{p}}{\sqrt{1-r}},\ \ \ \forall
r\in[0,1),\ p\geqslant 2,$ (4.16)

where $\|\cdot\|_{p}$ denotes the $L^{p}$-norm.
(ii) $\\{M_{r}^{t}\\}$ is an $\\{{\cal F}_{tr}^{B}\\}$-martinagle whose
components satisfy

$\langle M^{t,i}\rangle_{r}\leqslant Cr,\ \ \ \forall r\in[0,1),\
i=1,\cdots,N.$ (4.17)

In the above inequalities, $C$ denotes a geometric constant depending only on
$M,\phi$ and the localisation $V$.

###### Proof.

We write $\phi=\phi_{i}dx^{i}$ on $V$ and also set $\hat{Q}(u,{\bf
z})\triangleq b({\bf z})+Q(u,{\bf z})$ to ease notation. According to the SDE
(4.12) and the Itô-Stratonovich conversion, one has

$\displaystyle\phi(\circ dX_{s})$
$\displaystyle=-\phi_{i}(X_{s})\frac{X_{s}^{i}}{t-s}ds+\big{(}\phi_{i}(X_{s})\hat{Q}^{i}(t-s,X_{s})$
$\displaystyle\ \ \
+\frac{1}{2}\partial_{j}\phi_{i}(X_{s})a^{ij}(X_{s})\big{)}ds+\phi_{i}(X_{s})\sigma_{\alpha}^{i}(X_{s})dB_{s}^{\alpha}$

for $s\in[0,\tau).$ By taking $s=tr$ and substituting (4.13) into the first
term, it is easily seen that the decomposition (4.15) holds with

$\displaystyle A_{r}^{t}\triangleq$ $\displaystyle-\phi_{i}(X_{\tau\wedge
tr})\big{(}\sqrt{t}\int_{0}^{\tau\wedge
tr}\frac{\sigma_{\alpha}^{i}(X_{u})dB_{u}^{\alpha}}{t-u}+\sqrt{t}\int_{0}^{(\tau/t)\wedge
r}\frac{1}{1-\eta}\hat{Q}^{i}(t(1-\eta),X_{t\eta})d\eta\big{)}$ (4.18)
$\displaystyle\ \ \ -\big{(}\phi_{i}(X_{\tau\wedge tr})-\phi_{i}((1-r){\bf
x})\big{)}x^{i}+\sqrt{t}\big{(}\phi_{i}(X_{\tau\wedge
tr})\hat{Q}^{i}(t(1-r),X_{\tau\wedge tr})$ $\displaystyle\ \ \
+\frac{1}{2}\partial_{j}\phi_{i}(X_{\tau\wedge tr})a^{ij}(X_{\tau\wedge
tr})\big{)}$ (4.19)

and

$M_{r}^{t}\triangleq\int_{0}^{(\tau/t)\wedge
r}\phi_{i}(X_{t\eta})\sigma_{\alpha}^{i}(X_{t\eta})dB_{\eta}^{t,\alpha},$

where $B_{\eta}^{t,\alpha}\triangleq B_{t\eta}^{\alpha}/\sqrt{t}$ is a
rescaled Brownian motion. Since $\phi_{i},\sigma_{\alpha}^{i}\in
C_{b}^{\infty}(\bar{U})$, it is clear that the bound (4.17) holds.

We now estimate each term of $A_{r}^{t}.$ Since $\hat{Q}^{i}$ is also
uniformly bounded on $\bar{U}$, the entire expression of (4.19) is uniformly
bounded by some deterministic constant which is in turn trivially enlarged to
the form of (4.16). To estimate the first term of (4.18), one notes from the
Burkholder-Davis-Gundy (BDG) inequality (cf. [CK91, Theorem A] with BDG
constant $2\sqrt{p}$) that

$\big{\|}\sqrt{t}\int_{0}^{\tau\wedge tr}\frac{(\sigma
dB)_{u}^{i}}{t-u}\big{\|}_{p}\leqslant
2\sqrt{p}\cdot\big{\|}\sqrt{t\int_{0}^{tr}\frac{C}{(t-u)^{2}}du}\big{\|}_{p}=C^{\prime}\sqrt{p}\cdot\sqrt{\frac{r}{1-r}},$

which is dominated by (4.16) since $r\leqslant 1.$ The second term of (4.18)
is uniformly bounded by $C|\log(1-r)|$ which is also trivially enlarged to
(4.16). The desired bound (4.16) thus follows. ∎

#### The signature remainder estimate

In the decomposition (4.15), let us denote

$dY_{r}^{0}\triangleq\dot{\Gamma}_{r}dr,\ dY_{r}^{1}\triangleq
A_{r}^{t}dr+dM_{r}^{t},$

where we omitted the script $t$ to ease notation. Then on the event
$\\{\tau=t\\}$ one can write

$\displaystyle\int_{0<r_{1}<\cdots<r_{n}<1}\phi(\circ
dX_{tr_{1}})\otimes\cdots\otimes\phi(\circ dX_{tr_{n}})$
$\displaystyle=\sum_{I=(\omega_{1},\cdots,\omega_{n})}t^{\frac{n-|I|_{0}}{2}}\int_{0<r_{1}<\cdots<r_{n}<1}\circ
dY_{r_{1}}^{\omega_{1}}\otimes\cdots\otimes\circ dY_{r_{n}}^{\omega_{n}},$
(4.20)

where the summation is taken over all words $I=(\omega_{1},\cdots,\omega_{n})$
with $\omega_{i}=0,1$ and $|I|_{0}$ denotes the number of zero’s in $I$ (the
number of geodesic components).

Given such a word $I,$ we set

$J_{n}(\rho;I)\triangleq\int_{0<r_{1}<\cdots<r_{n}<\rho}\circ
dY_{r_{1}}^{\omega_{1}}\otimes\cdots\otimes\circ dY_{r_{n}}^{\omega_{n}},\ \ \
\rho\in[0,1].$ (4.21)

A key step for proving Theorem 4.1 is a proper estimate of $J_{n}(\rho;I)$
that respects the number of geodesic components in $I$. The main result for
this purpose is stated below. In what follows, $|\cdot|$ denotes the Hilbert-
Schmidt tensor norm and
$\|\cdot\|_{p}\triangleq\mathbb{E}[|\cdot|^{p};\tau=t]^{1/p}$ denotes the
$L^{p}$-norm on $\\{\tau=t\\}$ for tensor-valued random variables.

######  Proposition 4.10.

There exists a positive constant $\Lambda$ depending only on $M,\phi,N$ and
the localisation $V$, such that the following estimate holds true

$\big{\|}J_{n}(\rho;I)\big{\|}_{p}\leqslant\frac{(\Lambda
p)^{n-k}}{\sqrt{(n-k)!}}\frac{\|\Gamma\|_{1\text{-var};[0,\rho]}^{k}}{k!}(1-\sqrt{1-\rho})^{\frac{n-k}{2}}$
(4.22)

for any $p\geqslant 2,\rho\in[0,1],$ any $n\in\mathbb{N}$, $1\leqslant
k\leqslant n$ and any word $I=(\omega_{1},\cdots,\omega_{n})$ with
$|I|_{0}=k$.

######  Remark 4.11.

A delicate point in the proof is to make sure that one only introduces an
exponential factor like $C^{n-k}\|\Gamma\|_{1\text{-var}}^{k}$ (as seen in
(4.22)) instead of something like $C^{n}.$ This is the key point for reducing
the lifetime scale $t_{n}$ to polynomial dependence in $n$ in Theorem 4.1; a
more “standard” signature estimate would introduce a factor $C^{n}$ forcing
$t_{n}$ to decay exponentially in order to make (4.2) work.

To prove Proposition 4.10, we first recall a basic result of Kallenberg-
Sztencel (cf. [KS91, Theorem 3.1]) which leads to a dimension-free BDG
inequality. This also avoids the introduction of $C^{n}$ when estimating
moments of $E^{\otimes n}$-valued martingales.

######  Lemma 4.12.

Let $S=(S^{1},\cdots,S^{l})$ be a collection of continuous local martingales.
Then there exists a two-dimensional martingale $T=(T^{1},T^{2})$ defined
possibly on an extended filtered probability space, such that

$|S|=|T|,\ \langle S\rangle=\langle T\rangle,$

where

$|S|\triangleq\sqrt{(S^{1})^{2}+\cdots+(S^{l})^{2}},\ \langle
S\rangle\triangleq\langle S^{1}\rangle+\cdots+\langle S^{l}\rangle$

and similarly for $T.$

Next, we present a lemma that provide the basic estimates for proving
Proposition 4.10 by induction.

######  Lemma 4.13.

There exist positive constants $K_{1},K_{2},K_{3}$ depending only on
$M,\phi,N$ and the localisation $V$, such that the following estimates hold
true for any $p\geqslant 2,n\in\mathbb{N}$ and any word
$I=(\omega_{1},\cdots,\omega_{n+1})$:

(i) If $\omega_{n+1}=0$, then one has

$\|J_{n+1}(\rho;I)\|_{p}\leqslant\int_{0}^{\rho}\|J_{n}(\rho;I^{\prime})\|_{p}|\dot{\Gamma}_{\eta}|d\eta,\
\ \ \rho\in[0,1].$ (4.23)

(ii) If $\omega_{n+1}=1$, then one has

$\displaystyle\|J_{n+1}(\rho;I)\|_{p}$ $\displaystyle\leqslant
K_{1}\sqrt{r}\int_{0}^{\rho}\|J_{n}(\eta;I^{\prime})\|_{q}\frac{1}{\sqrt{1-\eta}}d\eta+K_{2}\sqrt{p}\big{(}\int_{0}^{\rho}\|J_{n}(\eta;I^{\prime})\|_{p}^{2}d\eta\big{)}^{1/2}$
$\displaystyle\ \ \
+K_{3}\int_{0}^{\rho}\|J_{n-1}(\eta;I^{\prime\prime})\|_{p}d\eta,$ (4.24)

where $q,r$ are any positive numbers such that $1/p=1/q+1/r$ and $I^{\prime}$
(resp. $I^{\prime\prime}$) is the word obtained by removing the last entry
(resp. last two entries) of $I$.

###### Proof.

If $\omega_{n+1}=0,$ one has

$J_{n+1}(\rho;I)=\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes\dot{\Gamma}_{\eta}d\eta$

and the estimate (4.23) follows immediately. If $\omega_{n+1}=1$, one has

$\displaystyle J_{n+1}(\rho;I)$
$\displaystyle=\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes
A_{\eta}^{t}d\eta+\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes\circ
dM_{\eta}^{t}$ $\displaystyle=\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes
A_{\eta}^{t}d\eta+\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes\cdot
dM_{\eta}^{t}$ $\displaystyle\ \ \ +\frac{1}{2}\int_{0}^{\rho}\cdot
dJ_{n}(\eta;I^{\prime})\otimes\cdot dM_{\eta}^{t}$
$\displaystyle=:I_{n}^{1}(\rho)+I_{n}^{2}(\rho)+I_{n}^{3}(\rho).$

where $\cdot dM$ denotes Itô’s integral. We now estimate each term on the
right hand side separately. Note that the last term comes from the Itô-
Stratonovich correction and is present only when $\omega_{n}=1$.

Estimation of $I_{n}^{1}.$ According to (4.16) and Young’s inequality, one has

$\|I_{n}^{1}(\rho)\|_{p}\leqslant\int_{0}^{\rho}\|J_{n}(\eta;I^{\prime})\|_{q}\|A_{\eta}^{t}\|_{r}d\eta\leqslant
K_{1}\sqrt{r}\int_{0}^{\rho}\|J_{n}(\eta;I^{\prime})\|_{q}\frac{1}{\sqrt{1-\eta}}d\eta.$

Estimation of $I_{n}^{2}.$ We are going to applying Lemma 4.12 to the
multidimensional martingale

$I_{n}^{2}(\rho)\triangleq\int_{0}^{\rho}J_{n}(\eta;I^{\prime})\otimes\cdot
dM_{\eta}^{t}\in(\mathbb{R}^{N})^{\otimes(n+1)}.$

By the definition of the Hilbert-Schmidt norm, one has

$|I_{n}^{2}(\rho)|=\big{(}\sum_{i_{1},\cdots,i_{n+1}=1}^{N}\big{(}\int_{0}^{\rho}J_{n}(\eta;I)^{i_{1},\cdots,i_{n}}dM_{\eta}^{t,i_{n+1}}\big{)}^{2}\big{)}^{1/2}.$

According to (4.17), one also has

$\displaystyle\langle I_{n}^{2}\rangle_{\rho}$
$\displaystyle=\sum_{i_{1},\cdots,i_{n+1}=1}^{N}\int_{0}^{\rho}\big{(}J_{n}(\eta;I)^{i_{1},\cdots,i_{n}}\big{)}^{2}d\langle
M^{t,i_{n+1}}\rangle_{\eta}$ $\displaystyle\leqslant
NC\int_{0}^{\rho}\sum_{i_{1},\cdots,i_{n}=1}^{N}\big{(}J_{n}(\eta;I)^{i_{1},\cdots,i_{n}}\big{)}^{2}d\eta=NC\int_{0}^{\rho}\big{|}J_{n}(\eta;I)\big{|}^{2}d\eta.$

It follows from Lemma 4.12 as well as the 2D BDG-inequality that

$\|I_{n}^{2}(\rho)\|_{p}\leqslant 4\sqrt{2}\sqrt{p}\|\langle
I_{n}^{2}\rangle_{\rho}^{1/2}\|_{p}\ \ \ \forall p\geqslant 2.$

Here the constant $4\sqrt{2}\sqrt{p}$ is easily derived from the 1D case whose
BDG constant is $2\sqrt{p}$ (cf. [CK91, Theorem A]). Therefore, one finds that

$\|I_{n}^{2}(\rho)\|_{p}\leqslant
4\sqrt{2NC}\sqrt{p}\big{\|}\big{(}\int_{0}^{\rho}\big{|}J_{n}(\eta;I)\big{|}^{2}d\eta\big{)}^{1/2}\big{\|}_{p}.$

In addition, note that

$\big{\|}\big{(}\int_{0}^{\rho}\big{|}J_{n}(\eta;I)\big{|}^{2}d\eta\big{)}^{1/2}\big{\|}_{p}=\big{\|}\int_{0}^{\rho}\big{|}J_{n}(\eta;I^{\prime})\big{|}^{2}d\eta\big{\|}_{p/2}^{1/2}\leqslant\big{(}\int_{0}^{\rho}\big{\|}J_{n}(\eta;I^{\prime})\big{\|}_{p}^{2}d\eta\big{)}^{1/2}.$

Consequently, one arrives at

$\|I_{n}^{2}(\rho)\|_{p}\leqslant
K_{2}\sqrt{p}\big{(}\int_{0}^{\rho}\big{\|}J_{n}(\eta;I^{\prime})\big{\|}_{p}^{2}d\eta\big{)}^{1/2}$
(4.25)

with $K_{2}\triangleq 4\sqrt{2NC}.$

Estimation of $I_{n}^{3}$. This is only relevant when
$\omega_{n}=\omega_{n+1}=1$. In this case, by (4.17) one has

$\big{|}I_{n}^{3}(\rho)\big{|}\leqslant\frac{1}{2}\int_{0}^{\rho}\big{|}J_{n-1}(\eta;I^{\prime\prime})\big{|}\cdot|d\langle
M^{t},\otimes M^{t}\rangle_{\eta}\big{|}\leqslant
K_{3}\int_{0}^{\rho}\big{|}J_{n-1}(\eta;I^{\prime\prime})\big{|}d\eta.$

It follows that

$\|I_{n}^{3}(\rho)\|_{p}\leqslant
K_{3}\int_{0}^{\rho}\big{\|}J_{n-1}(\eta;I^{\prime\prime})\big{\|}_{p}d\eta.$

By adding up the previous three inequalities, one obtains the desired estimate
(4.24) and concludes the proof of the lemma. ∎

We are now in a position to prove Proposition 4.10.

###### Proof of Proposition 4.10.

We are going to prove the inequality (4.22) by induction on $n$. Consider
first the base case $n=1$. If $I=(0)$, then one has

$\big{|}J_{1}(\rho;I)\big{|}\leqslant\int_{0}^{\rho}\big{|}\dot{\Gamma}_{\eta}\big{|}d\eta=\|\Gamma\|_{1\text{-var};[0,\rho]}$

and the estimate (4.10) holds trivially with $n=k=1$. If $I=(1)$, then

$J_{1}(\rho;I)=\int_{0}^{\rho}A_{\eta}^{t}d\eta+M_{\rho}^{t}.$

The first term is estimated by (4.16) as

$\big{\|}\int_{0}^{\rho}A_{r}^{t}dr\big{\|}_{p}\leqslant
C\sqrt{p}\int_{0}^{\rho}\frac{d\eta}{\sqrt{1-\eta}}=2C\sqrt{p}(1-\sqrt{1-\rho}).$

The second term is estimated by the BDG inequality and (4.17) as

$\|M_{\rho}^{t}\|_{p}\leqslant\sqrt{2NCp}\sqrt{\rho}.$

It follows that

$\|J_{1}(\rho;I)\|_{p}\leqslant\big{(}2C(1-\sqrt{1-\rho})+\sqrt{2NC}\sqrt{\rho}\big{)}p.$

To make the estimate (4.22) work with $n=1,k=0$, it suffices to require
$\Lambda$ to satisfy

$2C(1-\sqrt{1-\rho})+\sqrt{2NC}\sqrt{\rho}\leqslant\Lambda\big{(}1-\sqrt{1-\rho}\big{)}^{1/2}\
\ \ \forall\rho\in[0,1].$

Such a $\Lambda$ clearly exists.

In what follows, we fix the choice of

$\Lambda\triangleq\max\big{\\{}\sup_{\rho\in[0,1]}\frac{2C(1-\sqrt{1-\rho})+\sqrt{2NC}\sqrt{\rho}}{\big{(}1-\sqrt{1-\rho}\big{)}^{1/2}},4eK_{1}+\sqrt{2}K_{2}+4K_{3}\big{\\}},$
(4.26)

where $K_{1},K_{2},K_{3}$ are the constants appearing in Lemma 4.13. We just
showed the base case $n=1$ and are going to establish the induction step with
the same $\Lambda$. Suppose that the estimate (4.22) is valid for all words
with length $\leqslant n$. Now let $I=(\omega_{1},\cdots,\omega_{n+1})$ be a
given word of length $n+1$ and let $k$ be the number of zero’s in $I$. We
write

$L_{k}(\rho)\triangleq\int_{0<r_{1}<\cdots<r_{k}<\rho}\big{|}\dot{\Gamma}_{r_{1}}\big{|}\cdots\big{|}\dot{\Gamma}_{r_{k}}\big{|}dr_{1}\cdots
dr_{k}=\frac{\|\Gamma\|_{1\text{-var};[0,\rho]}^{k}}{k!}.$

Case I: $\omega_{n+1}=0.$

According to (4.23) and the induction hypothesis, one has

$\displaystyle\|J_{n+1}(\rho;I)\|_{p}$
$\displaystyle\leqslant\int_{0}^{\rho}\frac{(\Lambda
p)^{n-k+1}}{\sqrt{(n-k+1)!}}(1-\sqrt{1-\eta})^{\frac{n-k+1}{2}}L_{k}(\eta)\cdot\big{|}\dot{\Gamma}_{\eta}\big{|}d\eta$
$\displaystyle\leqslant\frac{(\Lambda
p)^{n-k+1}}{\sqrt{(n-k+1)!}}(1-\sqrt{1-\rho})^{\frac{n-k+1}{2}}\int_{0}^{\rho}L_{k}(\eta)\cdot\big{|}\dot{\Gamma}_{\eta}\big{|}d\eta$
$\displaystyle=\frac{(\Lambda
p)^{n-k+1}}{\sqrt{(n-k+1)!}}(1-\sqrt{1-\rho})^{\frac{n-k+1}{2}}L_{k+1}(\rho).$

Case II: $\omega_{n+1}=1.$

In this case, we apply the induction hypothesis to each term on the right hand
side of (4.24). Recall that $I^{\prime}$ (resp. $I^{\prime\prime}$) is the
word obtained by removing the last entry (resp. last two entries) of $I$.
Firstly, one has

$\displaystyle{\rm I}$ $\displaystyle\triangleq
K_{1}\sqrt{r}\int_{0}^{\rho}\big{\|}J_{n}(\eta;I^{\prime})\big{\|}_{q}\frac{1}{\sqrt{1-\eta}}d\eta$
$\displaystyle\leqslant K_{1}\sqrt{r}\int_{0}^{\rho}\frac{(\Lambda
q)^{n-k}}{\sqrt{(n-k)!}}(1-\sqrt{1-\eta})^{\frac{n-k}{2}}L_{k}(\eta)\frac{1}{\sqrt{1-\eta}}d\eta$
$\displaystyle\leqslant K_{1}\sqrt{r}\frac{(\Lambda
q)^{n-k}}{\sqrt{(n-k)!}}L_{k}(\rho)\int_{0}^{\rho}(1-\sqrt{1-\eta})^{\frac{n-k}{2}}\frac{d\eta}{\sqrt{1-\eta}}.$

By evaluating the last integral explicitly, it is easily seen that

${\rm I}\leqslant\frac{4K_{1}\sqrt{r}(\Lambda
q)^{n-k}}{\sqrt{(n-k)!}(n+2-k)}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$
(4.27)

We now choose $r=p(n+1-k)$ and $q=\frac{p(n+1-k)}{n-k}$ (so that
$1/p=1/q+1/r$). It follows that

$\displaystyle\sqrt{r}q^{n-k}$
$\displaystyle=\sqrt{p(n+1-k)}\big{(}\frac{p(n+1-k)}{n-k}\big{)}^{n-k}$
$\displaystyle=p^{n-k+1/2}\sqrt{n+1-k}\big{(}1+\frac{1}{n-k}\big{)}^{n-k}\leqslant
e\sqrt{n+1-k}p^{n+1-k}.$

By substituting this into (4.27), one obtains that

${\rm I}\leqslant
4eK_{1}\frac{\Lambda^{n-k}p^{n+1-k}}{\sqrt{(n+1-k)!}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$
(4.28)

Next, we estimate the second term in (4.24); one has

$\displaystyle{\rm II}$ $\displaystyle\triangleq
K_{2}\sqrt{p}\big{(}\int_{0}^{\rho}\big{\|}J_{n}(\eta;I^{\prime})\big{\|}_{p}^{2}d\eta\big{)}^{1/2}$
$\displaystyle\leqslant
K_{2}\sqrt{p}\big{(}\int_{0}^{\rho}\big{(}\frac{(\Lambda
p)^{n-k}}{\sqrt{(n-k)!}}(1-\sqrt{1-\eta})^{\frac{n-k}{2}}L_{k}(\eta)\big{)}^{2}d\eta\big{)}^{1/2}$
$\displaystyle\leqslant K_{2}\sqrt{p}\frac{(\Lambda
p)^{n-k}}{\sqrt{(n-k)!}}L_{k}(\rho)\big{(}\int_{0}^{\rho}(1-\sqrt{1-\eta})^{n-k}d\eta\big{)}^{1/2}.$

The last integral is estimated as

$\displaystyle\int_{0}^{\rho}(1-\sqrt{1-\eta})^{n-k}d\eta$
$\displaystyle\leqslant\int_{0}^{\rho}(1-\sqrt{1-\eta})^{n-k}\frac{1}{\sqrt{1-\eta}}d\eta$
$\displaystyle=\frac{2}{n+1-k}(1-\sqrt{1-\rho})^{n+1-k}.$

Therefore, one obtains that

$\displaystyle{\rm II}$ $\displaystyle\leqslant K_{2}\sqrt{p}\frac{(\Lambda
p)^{n-k}}{\sqrt{(n-k)!}}L_{k}(\rho)\times\frac{\sqrt{2}}{\sqrt{n+1-k}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}$
$\displaystyle\leqslant\sqrt{2}K_{2}\frac{\Lambda^{n-k}p^{n+1-k}}{\sqrt{(n+1-k)!}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$
(4.29)

Finally, we estimate the third term in (4.24), which is denoted as

${\rm III}\triangleq
K_{3}\int_{0}^{\rho}\big{\|}J_{n-1}(\eta;I^{\prime\prime})\big{\|}_{p}d\eta.$

Note that this term comes from Itô-Stratonovich correction and is present only
when $\omega_{n}=\omega_{n+1}=1$. In particular, there are $k$ geodesic
components in $I^{\prime\prime}$ for this scenario. As a result, by the
induction hypothesis one has

$\displaystyle{\rm III}$ $\displaystyle\leqslant
K_{3}\int_{0}^{\rho}\frac{(\Lambda
p)^{n-1-k}}{\sqrt{(n-1-k)!}}(1-\sqrt{1-\eta})^{\frac{n-1-k}{2}}L_{k}(\eta)d\eta$
$\displaystyle\leqslant
4K_{3}\frac{\Lambda^{n-k}p^{n+1-k}}{\sqrt{(n+1-k)!}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$
(4.30)

By putting (4.28), (4.29), (4.30) together, one sees that

$\big{\|}J_{n+1}(\rho;I)\big{\|}_{p}\leqslant(4eK_{1}+\sqrt{2}K_{2}+4K_{3})\frac{\Lambda^{n-k}p^{n+1-k}}{\sqrt{(n+1-k)!}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$

In view of the choice of $\Lambda$ given by (4.26), the right hand side of the
above inequality is further bounded by

$\frac{(\Lambda
p)^{n+1-k}}{\sqrt{(n+1-k)!}}(1-\sqrt{1-\rho})^{\frac{n+1-k}{2}}L_{k}(\rho).$

This completes the induction step.

∎

####  4.2.3 Localisation II: the remainder

Our aim here is to estimate the term
$\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau<t\big{]}$ and the main
result is stated as follows.

######  Proposition 4.14.

There exist constants $C_{1},C_{2},\delta>0$ depending only on $x,y,M$, such
that

$\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau<t\big{]}\big{\|}_{{\rm
HS}}\leqslant\frac{C_{1}^{n}}{\sqrt{n!}}e^{-C_{2}/t}$ (4.31)

for all $n\geqslant 1$ and $t\in(0,\delta).$

The main idea of the proof is to estimate the probability of the event
$\\{\tau<t\\}$ and moments of $\pi_{n}S^{\phi}(X^{t,x,y})$ separately. For the
first part, we rely on Hsu’s large deviation principle for the Brownian
bridge. For the second part, we make use of several heat kernel estimates in
geometric analysis. In what follows, we divide our proof of Proposition 4.14
into several steps.

#### The exit probability

We first estimate the probability that the Brownian bridge exits the chart $V$
before its lifetime $t$.

######  Lemma 4.15.

There are constants $\delta,K>0$ depending only on $x,y,$ such that

$\mathbb{P}(\tau<t)\leqslant e^{-K/t}\ \ \ \forall t\in(0,\delta).$

###### Proof.

Let $\mathbb{Q}_{x,y}^{t}$ denote the law of the process
$\\{X_{st}^{t,x,y}:s\in[0,1]\\}$ on the path space

$\Omega_{x,y}\triangleq\big{\\{}w:[0,1]\rightarrow M\big{|}w\
\text{continuous, }w_{0}=x,w_{1}=y\big{\\}}.$

It is obvious that

$\mathbb{P}(\tau<t)\leqslant\mathbb{Q}_{x,y}^{t}\big{(}\\{w\in\Omega_{x,y}:D(w,y)\geqslant\rho_{M}/2-\varepsilon\\}\big{)}$

where $D(w,y)\triangleq\sup\\{d(w_{s},y):s\in[0,1]\\}.$ According to the large
deviation principle for $\\{\mathbb{Q}_{x,y}^{t}\\}$ (cf. [Hsu90, Theorem
2.2]), one has

$\displaystyle\underset{t\rightarrow 0}{\overline{\lim}}\ t\log\mathbb{P}(\tau
t)$ $\displaystyle\leqslant\underset{t\rightarrow 0}{\overline{\lim}}\
t\log\mathbb{Q}_{x,y}^{t}\big{(}\\{w\in\Omega_{x,y}:D(w,y)\geqslant\rho_{M}/2-\varepsilon\\}\big{)}$
$\displaystyle\leqslant-\inf_{\begin{subarray}{c}w\in\Omega_{x,y}\\\
D(w,y)\geqslant\rho_{M}/2-\varepsilon\end{subarray}}J_{x,y}(w),$

where

$J_{x,y}(w)\triangleq\begin{cases}\dfrac{1}{2}\big{(}\int_{0}^{1}|\dot{w}_{s}|^{2}ds-d(x,y)^{2}\big{)},&|\dot{w}|\in
L^{2}([0,1]);\\\ \infty,&\text{otherwise}.\end{cases}$

For those $w$ with $|\dot{w}|\in L^{2}([0,1])$ and
$D(w,y)\geqslant\rho_{M}/2-\varepsilon$, one has

$\int_{0}^{1}|\dot{w}_{s}|^{2}ds\geqslant\big{(}\int_{0}^{1}|\dot{w}_{s}|ds\big{)}^{2}\geqslant(\rho_{M}/2-\varepsilon)^{2}.$

It follows that

$\underset{t\rightarrow 0}{\overline{\lim}}\
t\log\mathbb{P}(\tau<t)\leqslant\frac{1}{2}d(x,y)^{2}-\frac{1}{2}(\rho_{M}/2-\varepsilon)^{2}.$

The result follows by noting that the right hand side is a negative number. ∎

#### Signature moment estimates of Brownian bridge

Next, we estimate moments of $\pi_{n}S^{\phi}(X^{t,x,y})$. The main result is
stated below. Here we also use the Hilbert-Schmidt norm on tensors.

######  Lemma 4.16.

There exist positive constants $\Lambda,L$ depending only on $M$ and $\phi$,
such that

$\big{\|}\pi_{n}S^{\phi}(X^{t,x,y})\big{\|}_{p}\leqslant\frac{(\Lambda
p)^{n/2}}{\sqrt{n!}}e^{L/pt}$

for all $p\geqslant 1,n\in\mathbb{N},t\in(0,1]$ and $x,y\in M$.

The argument is quite similar to the proof of Proposition 4.10. Since the
estimate here is global, we take an extrinsic perspective and make use of a
well-known estimate for the logarithmic derivative of the heat kernel.

#### I. Extrinsic construction of the Brownian bridge

Let $M$ be isometrically embedding in some Euclidean space $\mathbb{R}^{m}$
(which is always possible due to Nash’s embedding theorem). A tubular
neighbourhood of $M$ in $\mathbb{R}^{m}$ is defined by the normal bundle

${\cal B}_{\varepsilon}\triangleq\bigsqcup_{x\in M}B^{\perp}(x,\varepsilon).$

Here
$B^{\perp}(x,\varepsilon)\triangleq\\{x^{\prime}\in\mathbb{R}^{m}:\overrightarrow{xx^{\prime}}\in(T_{x}M)^{\perp},|x^{\prime}-x|_{\mathbb{R}^{m}}<\varepsilon\\}$.
Since $M$ is compact, it is a standard fact in differential topology (cf.
[Lee06, Chap. 10]) that when $\varepsilon$ is small, ${\cal B}_{\varepsilon}$
is diffeomorphic to the $\varepsilon$-neighbourhood of $M$ in
$\mathbb{R}^{m}.$ We fix such an $\varepsilon.$ There is a canonical
projection $\pi:{\cal B}_{\varepsilon}\rightarrow M$ taking $x\in{\cal
B}_{\varepsilon}$ onto its unique closest point $\pi(x)$ in $M$.

Functions on $M$ can be extended to $\mathbb{R}^{m}$ by using a bump function
on ${\cal B}_{\varepsilon}$. More precisely, let
$\eta:[0,\infty)\rightarrow\mathbb{R}$ be a smooth bump function such that
$0\leqslant\eta\leqslant 1$, $\eta\equiv 1$ on $[0,\varepsilon/2]$ and
$\eta\equiv 0$ on $(\varepsilon,\infty)$. Given any function $f\in
C^{\infty}(M),$ we define

$\bar{f}(x)\triangleq\eta(d_{\mathbb{R}^{m}}(x,\pi(x)))f(\pi(x)),\ \ \
x\in{\cal B}_{\varepsilon}.$

It is clear that $\bar{f}\in C_{c}^{\infty}(\mathbb{R}^{m})$. Vector fields
and one-forms on $M$ are extended to $\mathbb{R}^{m}$ in a similar way. We
always use the notation $\bar{\cdot}$ to denote the extended object.

For each $x\in M$ and $1\leqslant\alpha\leqslant m,$ let $V_{\alpha}(x)$ be
the tangent vector to $M$ defined by orthogonally projecting the $\alpha$-th
canonical basis vector of $\mathbb{R}^{m}$ onto $T_{x}M$. Then $V_{\alpha}$
defines a smooth vector field on $M$ and we take its extension
$\bar{V}_{\alpha}$ to $\mathbb{R}^{m}$ as before. Note that the Laplacian on
$M$ can be expressed as $\Delta_{M}=\sum_{\alpha=1}^{m}V_{\alpha}^{2}$. Given
fixed $t\in(0,1]$ and $x,y\in\mathbb{R}^{m}$, let
$\\{\bar{X}_{s}^{t,x,y}:0\leqslant s<t\\}$ be the solution to the SDE

$d\bar{X}_{s}=\sum_{\alpha=1}^{m}\bar{V}_{\alpha}(X_{s})dB_{s}+\bar{\nabla}_{x}\bar{p}(t-s,\bar{X}_{s},y)ds,\
\ \ \bar{X}_{0}=x.$

Here $\bar{\nabla}$ denotes the Euclidean gradient in $\mathbb{R}^{m}$.
Equivalently, $\bar{X}_{\cdot}^{t,\cdot,y}$ is a Markov family with generator

$\bar{{\cal
L}}_{s}^{t,y}=\frac{1}{2}\sum_{\alpha=1}^{m}\bar{V}_{\alpha}^{2}+\bar{\nabla}_{x}\bar{p}(t-s,\cdot,y)\cdot\bar{\nabla}.$

The following fact is of no surprise.

######  Lemma 4.17.

Suppose that $x,y\in M$. Then with probability one, the process
$\bar{X}^{t,x,y}$ lives on $M$ for all time. In this case,
$\bar{X}_{\cdot}^{t,x,y}$ has the same law as the Brownian bridge from $x$ to
$y$ on $M$ with lifetime $t$.

###### Proof.

Consider the function $F(z)\triangleq d_{\mathbb{R}^{m}}(z,M)^{2}$ on
$\mathbb{R}^{m}$. For $x\in{\cal B}_{2\varepsilon},$ one has $F(x)=0$ iff
$x\in M$. Define

$\sigma\triangleq\inf\\{s\in[0,t):\bar{X}_{s}^{t,x,y}\notin{\cal
B}_{\varepsilon/2}\\}.$

By our choices of extensions, it is easily seen that $(\bar{{\cal
L}}_{s}^{t,y}F)(\bar{X}_{s}^{t,x,y})=0$ on $[0,\sigma]$. It follows from the
martingale characterisation that

$\mathbb{E}\big{[}F(\bar{X}_{s\wedge\sigma}^{t,x,y})\big{]}=F(x)=0\implies\bar{X}_{s\wedge\sigma}^{t,x,y}\in
M\ \ \ \text{a.s. }$

By continuity, the process lives on $M$ for all time before $\sigma\wedge t$
which also indicates that $\sigma=t$ (i.e. no exiting ${\cal
B}_{\varepsilon/2}$ occurs). For the second part of the lemma, one first notes
that the process

$s\mapsto F(\bar{X}_{s}^{t,x,y})-F(x)-\int_{0}^{s}\bar{{\cal
L}}_{r}^{t,y}F(\bar{X}_{r}^{t,x,y})dr$

is a martingale for all $F\in C_{c}^{2}(\mathbb{R}^{m})$. In addition, the law
of the Brownian bridge $X^{t,x,y}$ on $M$ is characterised by the fact that

$s\mapsto
f(X_{s}^{t,x,y})-f(x)-\int_{0}^{s}\big{(}\frac{1}{2}\Delta_{M}f(X_{r}^{t,x,y})+\nabla_{x}^{M}p(t-r,X_{r}^{t,x,y},y)\cdot\nabla^{M}f(X_{r}^{t,x,y})\big{)}dr$

is a martingale for all $f\in C^{2}(M)$. The result follows from the fact that
$\bar{X}_{r}^{t,x,y}\in M$ as well as the relation that

$\bar{{\cal
L}}_{r}^{t,y}\bar{f}\big{|}_{M}=\frac{1}{2}\Delta_{M}f+\nabla_{x}^{M}p(t-r,\cdot,y)\cdot\nabla^{M}f\
\ \ \forall f\in C^{2}(M).$

The proof of the lemma is thus complete. ∎

#### II. Heat kernel bounds and a moment estimate for the Brownian bridge

Our proof of Lemma 4.16 relies on a few heat kernel estimates which we now
recall.

######  Proposition 4.18.

Let $M$ be a compact Riemannian manifold.

(i) (cf. [Hsu02, Theorem 5.3.4]) There are positive constants
$K_{1},K_{2},K_{3}$ such that

$K_{1}e^{-K_{2}/t}\leqslant
p(t,x,y)\leqslant\frac{K_{3}}{t^{d-1/2}}e^{-d(x,y)^{2}/2t}\ \ \
\forall(t,x,y)\in(0,1]\times M\times M.$ (4.32)

(ii) (cf. [ST98, Inequality (1.1)]) There exists a positive constant $K_{4},$
such that

$\big{|}\nabla_{x}\log p(t,x,y)\big{|}\leqslant
K_{4}\big{(}\frac{1}{\sqrt{t}}+\frac{d(x,y)}{t}\big{)}\ \ \
\forall(t,x,y)\in(0,1]\times M\times M.$ (4.33)

The following moment estimate of the Brownian bridge will also be needed for
us. It is not in the sharpest form but is enough for our purpose.

######  Lemma 4.19.

Let $\beta\in(0,1/2)$ be a given fixed parameter. There exist positive
constants $K_{5},K_{6}$ such that

$\|d(X_{s}^{t,x,y},y)\|_{p}\leqslant K_{5}\sqrt{p}e^{K_{2}/pt}(t-s)^{\beta}$

for all $p\geqslant 1$, $s<t\in(0,1]$ and $x,y\in M.$ Here $K_{2}$ is the same
constant appearing in (4.32).

###### Proof.

According to the transition density formula (2.9), one has

$\displaystyle\mathbb{E}\big{[}d(X_{s}^{t,x,y},y)^{p}\big{]}$
$\displaystyle=\frac{1}{p(t,x,y)}\int_{M}d(z,y)^{p}p(s,x,z)p(t-s,z,y)dz$
$\displaystyle=\frac{1}{p(t,x,y)}\big{(}\int_{\\{z:d(z,y)\leqslant\sqrt{p}(t-s)^{\beta}\\}}+\int_{\\{z:d(z,y)\geqslant\sqrt{p}(t-s)^{\beta}\\}}\big{)}$
$\displaystyle\ \ \ d(z,y)^{p}p(s,x,z)p(t-s,z,y)dz$
$\displaystyle\leqslant\sqrt{p}^{p}(t-s)^{\beta p}+D_{M}^{p}\cdot
K_{1}^{-1}e^{K_{2}/t}\cdot\frac{K_{3}}{(t-s)^{d-1/2}}e^{-p(t-s)^{2\beta-1}/2},$

where $D_{M}$ is the diameter of $M$ and we also used the semigroup property
as well as the estimate (4.32) to reach the last inequality. One can assume
without loss of generality that $K_{1}<1$ and $K_{3}>1$. Since $p\geqslant 1,$
one obtains that

$\big{\|}d(X_{s}^{t,x,y},y)\big{\|}_{p}\leqslant\sqrt{p}(t-s)^{\beta}+D_{M}K_{1}^{-1}K_{3}e^{K_{2}/pt}\cdot\frac{e^{-(t-s)^{2\beta-1}/2}}{(t-s)^{d-1/2}}.$

The desired estimate follows by noting that

$\frac{e^{-(t-s)^{2\beta-1}/2}}{(t-s)^{d-1/2}}\leqslant
C_{d,\beta}(t-s)^{\beta}$

for all $s<t\in(0,1].$ ∎

#### III. Proof of Lemma 4.16

We now proceed to prove Lemma 4.16. The argument is similar to (but not as
fine as) the proof of Proposition 4.10 and some repeated calculations will
thus be omitted. Recall that $\beta\in(0,1/2)$ is a fixed number appearing in
Lemma 4.19. For simplicity, we just write $X=X^{t,x,y}$ (which is also the
process $\bar{X}\triangleq\bar{X}^{t,x,y}$ by Lemma 4.17). We postulate the
following estimate on signature moments of $\bar{X}$ and will prove it by
induction on $n$:

$\big{\|}\int_{0<r_{1}<\cdots<r_{n}<\rho}\bar{\phi}(\circ
d\bar{X}_{tr_{1}})\otimes\cdots\otimes\bar{\phi}(\circ
d\bar{X}_{tr_{n}})\big{\|}_{p}\leqslant\frac{(\Lambda
p)^{n/2}}{\sqrt{n!}}e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{n/2}$ (4.34)

for all $p\geqslant 1,n\in\mathbb{N},t\in(0,1],$ $\rho\in[0,1].$ Here $K_{2}$
is the same constant appearing in Part II and $\Lambda$ is a universal
constant that will be determined in the induction argument. Lemma 4.16 follows
immediately by taking $\rho=1.$ We set

$J_{n}(\rho)\triangleq\int_{0<r_{1}<\cdots<r_{n}<\rho}\bar{\phi}(\circ
d\bar{X}_{tr_{1}})\otimes\cdots\otimes\bar{\phi}(\circ d\bar{X}_{tr_{n}}).$

Semi-martingale decomposition:

From the extrinsic perspective in Part I, one can write

$\displaystyle\bar{\phi}(\circ d\bar{X}_{tr})$
$\displaystyle=\sqrt{t}\bar{\phi}_{i}(\bar{X}_{tr})\bar{V}^{i}(\bar{X}_{tr})dB_{r}^{t}+\frac{t}{2}\partial_{j}\bar{\phi}_{i}(\bar{X}_{tr})\bar{a}^{ij}(\bar{X}_{tr})dr$
$\displaystyle\ \ \
+t\bar{\phi}_{i}(\bar{X}_{tr})\partial_{i}\log\bar{p}(t(1-r),\bar{X}_{tr},y)dr,\
\ \ r\in[0,1],$ (4.35)

where $B_{r}^{t}\triangleq B_{tr}/\sqrt{t}$ and
$\bar{a}\triangleq\bar{V}\cdot\bar{V}^{T}.$ Note that
$\bar{\phi},\bar{V},\bar{a}\in C_{c}^{\infty}$ by our choice of extension. In
what follows, we will use $C_{i}$ to denote constants depending only on
$M,\phi,\beta$ and the embedding.

Base step:

By using the BDG inequality, one has

$\big{\|}\sqrt{t}\int_{0}^{\rho}\bar{\phi}_{i}(\bar{X}_{tr})\bar{V}^{i}(\bar{X}_{tr})dB_{r}^{t}\big{\|}_{p}\leqslant
C_{1}\sqrt{p}\cdot\sqrt{\rho}.$ (4.36)

It is also obvious that

$\big{\|}\int_{0}^{\rho}\frac{t}{2}\partial_{j}\bar{\phi}_{i}(\bar{X}_{tr})\bar{a}^{ij}(\bar{X}_{tr})dr\big{\|}_{p}\leqslant
C_{2}\rho.$ (4.37)

For the last term of (4.35), by using the Stroock-Turestsky estimate (4.33)
and Lemma 4.19, one has

$\displaystyle\big{\|}\int_{0}^{\rho}t\bar{\phi}_{i}(\bar{X}_{tr})\partial_{i}\log\bar{p}(t(1-r),\bar{X}_{tr},y)dr\big{\|}_{p}$
$\displaystyle\leqslant
C_{3}t\int_{0}^{\rho}K_{4}\big{(}\frac{1}{\sqrt{t(1-r)}}+\frac{\|d(\bar{X}_{tr},y)\|_{p}}{t(1-r)}\big{)}dr$
$\displaystyle\leqslant
C_{3}K_{4}\int_{0}^{\rho}\frac{dr}{\sqrt{1-r}}+C_{3}K_{4}\int_{0}^{\rho}\frac{K_{5}\sqrt{p}e^{K_{2}/pt}(t(1-r))^{\beta}}{1-r}dr.$

After evaluating the above integrals explicitly and combining with (4.36,
4.37), one finds that

$\|J_{1}(\rho)\|_{p}\leqslant
C_{4}\sqrt{p}e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{1/2}$ (4.38)

with suitably chosen constant $C_{4}$.

Induction step:

Suppose that the induction hypothesis (4.34) holds. According to the
decomposition (4.35), one can write

$\displaystyle J_{n+1}(\rho)$
$\displaystyle=\int_{0}^{\rho}J_{n}(r)\otimes\bar{\phi}(\circ d\bar{X}_{tr})$
$\displaystyle=\sqrt{t}\int_{0}^{\rho}J_{n}(r)\otimes\bar{\phi}_{i}(\bar{X}_{tr})\bar{V}^{i}(\bar{X}_{tr})dB_{r}^{t}+\frac{t}{2}\int_{0}^{\rho}J_{n}(r)\otimes\partial_{j}\bar{\phi}_{i}(\bar{X}_{tr})\bar{a}^{ij}(\bar{X}_{tr})dr$
$\displaystyle\ \ \
+\frac{t}{2}\int_{0}^{\rho}J_{n-1}(r)\otimes\bar{\phi}_{i}(\bar{X}_{tr})\otimes\bar{\phi}_{j}(\bar{X}_{tr})\bar{a}^{ij}(\bar{X}_{tr})dr$
$\displaystyle\ \ \
+t\int_{0}^{\rho}J_{n}(r)\otimes\bar{\phi}_{i}(\bar{X}_{tr})\partial_{i}\log
p(t(1-r),\bar{X}_{tr},y)dr$
$\displaystyle=:I_{n}^{1}(\rho)+I_{n}^{2}(\rho)+I_{n}^{3}(\rho)+I_{n}^{4}(\rho).$

By using the dimension-free BDG inequality (the same way leading to (4.25)),
one has

$\|I_{n}^{1}(\rho)\|_{p}\leqslant
C_{5}\sqrt{p}\big{(}\int_{0}^{\rho}\|J_{n}(r)\|_{p}^{2}dr\big{)}^{1/2}.$

By substituting the estimate (4.34) into the right hand side and evaluating
the resulting $dr$-integral explicitly, one obtains that

$\|I_{n}^{1}(\rho)\|_{p}\leqslant\frac{C_{6}\Lambda^{n/2}p^{\frac{n+1}{2}}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$ (4.39)

The estimation of $I_{n}^{2}(\rho)$ and $I_{n}^{3}(\rho)$ follows the same
route and one finds that

$\|I_{n}^{2}(\rho)\|_{p}\vee\|I_{n}^{3}(\rho)\|_{p}\leqslant\frac{C_{7}(\Lambda
p)^{n/2}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$ (4.40)

We now consider $I_{n}^{4}(\rho).$ By using the gradient estimate (4.33), one
has

$\|I_{n}^{4}(\rho)\|_{p}\leqslant
C_{8}\big{(}\int_{0}^{\rho}\frac{1}{\sqrt{1-r}}\|J_{n}(r)\|_{p}dr+\int_{0}^{\rho}\frac{1}{1-r}\|d(\bar{X}_{tr},y)\cdot
J_{n}(r)\|_{p}dr\big{)}.$ (4.41)

After applying the induction hypothesis (4.34), the first integral is
majorised as

$\frac{2}{\beta}\frac{(\Lambda
p)^{n/2}}{\sqrt{(n+1)!}}e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$

For the second integral in (4.41), one applies Hölder’s inequality

$\|d(\bar{X}_{tr},y)\cdot
J_{n}(r)\|_{p}\leqslant\|d(\bar{X}_{tr},y)\|_{p_{1}}\cdot\|J_{n}(r)\|_{p_{2}}$

with $p_{1}\triangleq p(n+1)$ and $p_{2}\triangleq p(n+1)/n$. After further
applying Lemma 4.19 and the induction hypothesis, it follows that

$\int_{0}^{\rho}\frac{1}{1-r}\|d(\bar{X}_{tr},y)\cdot
J_{n}(r)\|_{p}dr\leqslant\frac{2K_{5}\sqrt{e}}{\beta}\cdot\frac{\Lambda^{n/2}p^{\frac{n+1}{2}}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$

Therefore, one arrives at

$\|I_{n}^{4}(\rho)\|_{p}\leqslant\frac{C_{9}\Lambda^{n/2}p^{\frac{n+1}{2}}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$ (4.42)

Putting the estimates (4.39), (4.40), (4.42) together, one concludes that

$\|J_{n+1}(\rho)\|_{p}\leqslant(C_{6}+2C_{7}+C_{9})\cdot\frac{\Lambda^{n/2}p^{\frac{n+1}{2}}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$

By taking the base step estimate (4.38) into account, one only needs to choose

$\Lambda=\max\\{C_{4}^{2},C_{6}+2C_{7}+C_{9}\\}.$

It then follows that

$\|J_{1}(\rho)\|_{p}\leqslant(\Lambda
p)^{1/2}e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{1/2}$

and

$\|J_{n+1}(\rho)\|_{p}\leqslant\frac{(\Lambda
p)^{\frac{n+1}{2}}}{\sqrt{(n+1)!}}\cdot
e^{K_{2}/pt}\big{(}1-(1-\rho)^{\beta}\big{)}^{\frac{n+1}{2}}.$

This finishes the induction argument.

#### Proof of Proposition 4.14

We now complete the proof of Proposition 4.14. Let $p,q>1$ be such that
$1/p+1/q=1.$ According to Lemma 4.15, Lemma 4.16 and Hölder’s inequality, one
has

$\displaystyle\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau<t\big{]}\big{\|}_{{\rm
HS}}$
$\displaystyle\leqslant\big{\|}\pi_{n}S^{\phi}(X^{t,x,y})\big{\|}_{p}\cdot\mathbb{P}(\tau<t)^{1/q}$
$\displaystyle\leqslant\frac{(\Lambda p)^{n/2}}{\sqrt{n!}}\cdot e^{L/pt}\cdot
e^{-K/qt}=\frac{(\Lambda p)^{n/2}}{\sqrt{n!}}e^{-(K/q-L/p)/t}.$

One can choose (and fix) $p,q$ so that $K/q-L/p>0.$ The desired estimate
(4.31) thus follows with $C_{1}\triangleq\sqrt{\Lambda p}$ and
$C_{2}\triangleq K/q-L/p.$

####  4.2.4 Completing the proof of Theorem 4.1

Putting together all previously developed ingredients, we are now in a
position to prove Theorem 4.1. We will continue to use the notation introduced
earlier.

Let us define

$b_{n}(t)\triangleq
n!\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y})\big{]}\big{\|}_{{\rm
HS}},\ L\triangleq\|\Gamma\|_{1\text{-var};[0,1]}.$

Recall that in the isometric embedding context (i.e. $\phi=dF$) one has
$L=d(x,y)$. Our goal is to estimate $b_{n}(t)^{1/n}-L$. To this end, one
writes

$b_{n}(t)^{1/n}-L=L\big{(}\exp\big{(}\frac{1}{n}\log\frac{b_{n}(t)}{L^{n}}\big{)}-1\big{)}.$
(4.43)

The proof of Theorem 4.1 will be completed as long as the following lemma is
established.

######  Lemma 4.20.

There exist positive constants $\kappa,K$, $R_{1},R_{2}$, such that

$\frac{b_{n}(t)}{L^{n}}\in[R_{1},R_{2}]$

for all $n\geqslant K$ and $t\in(0,\kappa n^{-6}].$

###### Proof.

One begins by expressing

$\frac{b_{n}(t)}{L^{n}}=\frac{n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}+\frac{b_{n}(t)-n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}.$ (4.44)

Since $\gamma^{x,y}$ is smooth, the following asymptotics for the first term
is a direct consequence of [HL10, Theorem 8] (see also [CMT23, Theorem 22]):

$\lim_{n\rightarrow\infty}\frac{n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}=c(\Gamma),$ (4.45)

where $c(\Gamma)>0$ is some explicit constant depending on the geometry of
$\Gamma$.

Our task is now reduced to estimating the second term on the right hand side
of (4.44). Trivially, one has

$\displaystyle\big{|}\frac{b_{n}(t)-n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}\big{|}$
$\displaystyle\leqslant\frac{n!}{L^{n}}\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y})\big{]}-\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}$ $\displaystyle=:A_{n}(t)+B_{n}(t),$ (4.46)

where

$\displaystyle A_{n}(t)$
$\displaystyle\triangleq\frac{n!}{L^{n}}\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y})-\pi_{n}S^{\phi}(\gamma^{x,y});\tau<t\big{]}\big{\|}_{{\rm
HS}},$ $\displaystyle B_{n}(t)$
$\displaystyle\triangleq\frac{n!}{L^{n}}\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y})-\pi_{n}S^{\phi}(\gamma^{x,y});\tau=t\big{]}\big{\|}_{{\rm
HS}}.$

According to Proposition 4.14 and Lemma 4.15, one has

$\displaystyle A_{n}(t)$
$\displaystyle\leqslant\frac{n!}{L^{n}}\big{\|}\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,y});\tau<t\big{]}\big{\|}_{{\rm
HS}}+\frac{n!}{L^{n}}\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}\cdot\mathbb{P}(\tau<t)$
$\displaystyle\leqslant\frac{n!}{L^{n}}\cdot\frac{C_{1}^{n}}{\sqrt{n!}}e^{-C_{2}/t}+\frac{n!}{L^{n}}\cdot\frac{L^{n}}{n!}\cdot
e^{-C_{3}/t}$
$\displaystyle\leqslant\big{(}\sqrt{n!}(C_{1}/L)^{n}+1\big{)}e^{-C_{4}/t}$
(4.47)

for all $n\geqslant 1$ and $t\in(0,\delta)$ with some positive constant
$\delta.$

To estimate $B_{n}(t),$ we first recall from (4.20) that on the event
$\\{\tau=t\\}$ one has the decomposition

$\pi_{n}S^{\phi}(X^{t,x,y})=\pi_{n}S^{\phi}(\gamma^{x,y})+\sum_{k=0}^{n-1}\sum_{I\in{\cal
I}_{n}(k)}t^{\frac{n-k}{2}}J_{n}(1;I),$

where ${\cal I}_{n}(k)$ denotes the set of length-$n$ words $I$ with
$|I|_{0}=k$ and $J_{n}(1;I)$ is defined by (4.21). According to Proposition
4.10 and (4.20), one finds that

$\displaystyle B_{n}(t)$
$\displaystyle\leqslant\frac{n!}{L^{n}}\sum_{k=0}^{n-1}\sum_{I\in{\cal
I}_{n}(k)}t^{\frac{n-k}{2}}\big{\|}J_{n}(1;I)\big{\|}_{2}$
$\displaystyle\leqslant\frac{n!}{L^{n}}\sum_{k=0}^{n-1}\sum_{I\in{\cal
I}_{n}(k)}\frac{(2\Lambda)^{n-k}}{\sqrt{(n-k)!}}\cdot\frac{L^{k}}{k!}\cdot
t^{\frac{n-k}{2}}=n!\sum_{k=0}^{n-1}{n\choose
k}\frac{1}{\sqrt{(n-k)!}k!}\big{(}\frac{2\Lambda\sqrt{t}}{L}\big{)}^{n-k}.$

To analyse the last expression, let us define

$F_{n}(x)\triangleq n!\sum_{k=0}^{n-1}{n\choose
k}\frac{x^{n-k}}{\sqrt{(n-k)!}k!}=\sum_{r=1}^{n}{n\choose
r}\frac{n!x^{r}}{(n-r)!\sqrt{r!}}$

where one takes $x=2\Lambda\sqrt{t}/L$ in our context. Setting

$a_{r}\triangleq{n\choose r}\frac{n!x^{r}}{(n-r)!\sqrt{r!}},$

one easily sees that

$\frac{a_{r+1}}{a_{r}}=x\cdot\frac{(n-r)^{2}}{(r+1)^{3/2}}\leqslant n^{2}x$

for all $r=1,\cdots,n-1$. By requiring

$n^{2}x=\frac{2\Lambda n^{2}\sqrt{t}}{L}\leqslant 1\iff
t\leqslant\big{(}\frac{L}{2\Lambda}\big{)}^{2}n^{-4},$ (4.48)

one concludes that $\\{a_{r}\\}_{r=1}^{n}$ is a decreasing sequence, thus
yielding that

$F_{n}(x)\leqslant na_{1}=n^{3}x=\frac{2\Lambda n^{3}\sqrt{t}}{L}.$

Therefore, one has

$B_{n}(t)\leqslant\frac{2\Lambda n^{3}\sqrt{t}}{L}$ (4.49)

for all $n$ and $t$ satisfying the relation (4.48).

By substituting the estimates (4.47) and (4.49) into (4.46), one obtains that

$\big{|}\frac{b_{n}(t)-n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}\big{|}\leqslant\big{(}\sqrt{n!}(C_{1}/L)^{n}+1\big{)}e^{-C_{4}/t}+\frac{2\Lambda
n^{3}\sqrt{t}}{L}$

for all $n,t$ provided that (4.48) holds. Note that the right hand side is of
constant magnitude if $t=O(n^{-6})$. More specifically, there exist positive
constants $\kappa,K$ such that

$\big{|}\frac{b_{n}(t)-n!\big{\|}\pi_{n}S^{\phi}(\gamma^{x,y})\big{\|}_{{\rm
HS}}}{L^{n}}\big{|}\leqslant\frac{1}{2}c(\Gamma)$ (4.50)

for all $n\geqslant K$ and $t\in(0,\kappa n^{-6}]$ (the relation (4.48) is
also satisfied with such choice of $t$). Here $c(\Gamma)$ is the Hambly-Lyons
limit appearing in (4.45). Now the conclusion of the lemma follows from the
decomposition (4.44), the asymptotics (4.45) and the inequality (4.50) (with
enlarged $K$ if necessary). ∎

###### Proof of Theorem 4.1.

Under the situation of Lemma 4.20, one has

$\big{|}\log\frac{b_{n}(t)}{L^{n}}\big{|}\leqslant\max\big{\\{}|\log
R_{1}|,|\log R_{2}|\big{\\}}.$

for all $n,t$ such that $n\geqslant K,t\leqslant\kappa n^{-6}.$ It follows
from the relation (4.43) that

$\big{|}b_{n}(t)^{1/n}-L\big{|}=L\big{|}\exp\big{(}\frac{1}{n}\log\frac{b_{n}(t)}{L^{n}}\big{)}-1\big{|}\leqslant\frac{C}{n}$

with a suitable constant $C$. The proof of Theorem 4.1 is now complete.

∎

##  5 Small-time expansion and its connection with curvature properties

Theorem 4.1 shows that the Riemannian distance can be recovered from suitable
signature asymptotics of the Brownian bridge. In this section, we are going to
show that curvature properties of the manifold $M$ (both intrinsic and
extrinsic) can also be recovered explicitly from the small-time expansion of
the signature.

###  5.1 The main theorem

From now on, we consider the Brownian loop based at $x\in M$ (i.e. the
Brownian bridge $X^{t,x,x}$). Recall that an isometric embedding
$F:M\rightarrow E=\mathbb{R}^{N}$ is given fixed. For each $n\geqslant 1$, we
define (with $\phi=dF$ here)

$\psi_{n}(t,x)\triangleq\mathbb{E}\big{[}\pi_{n}S^{\phi}(X^{t,x,x})\big{]}\in
E^{\otimes n},\ \ \ t>0,x\in M.$

This is the $n$-th level expected $\phi$-signature of the Brownian loop
$X^{t,x,x}.$

Let us denote $S_{n}\triangleq\pi_{n}S^{\phi}(X^{t,x,x})$. Since
$\int_{0}^{s}\phi(dX^{t,x,x})=F(X_{s}^{t,x,x})-F(x)$, it is clear that

$S_{1}=F(X_{t}^{t,x,x})-F(x)=F(x)-F(x)=0\implies\psi_{1}(t,x)=0.$

From the symmetry of the heat kernel, it is easily seen that $X^{t,x,x}$ has
the same law as its time reversal
$s\mapsto\overleftarrow{X}_{s}^{t,x,x}\triangleq X_{t-s}^{t,x,x}$. As a
result, the paths $\Gamma_{\cdot}\triangleq F(X_{\cdot}^{t,x,x})-F(x)$ and its
reversal $\overleftarrow{\Gamma}$ have the same law. Since the signature of
$\overleftarrow{\Gamma}$ is the inverse of the signature of $\Gamma$ (in the
tensor algebra $T((E))$), it follows that

$S^{\phi}\big{(}X^{t,x,x}\big{)}^{-1}\stackrel{{\scriptstyle{\rm
law}}}{{=}}S^{\phi}(X^{t,x,x}).$ (5.1)

By taking its second level projection, one has

$-S_{2}+S_{1}\otimes S_{1}\stackrel{{\scriptstyle{\rm law}}}{{=}}S_{2}.$

But one already knows $S_{1}=0.$ Therefore,
$\psi_{2}(t,x)=\mathbb{E}[S_{2}]=0.$ Similarly, the third level projection of
(5.1) gives

$-S_{3}=-S_{3}+S_{1}\otimes S_{2}+S_{2}\otimes S_{1}-S_{1}^{\otimes
3}\stackrel{{\scriptstyle{\rm
law}}}{{=}}S_{3}\implies\psi_{3}(t,x)=\mathbb{E}[S_{3}]=0.$

The above calculation suggests that, in order to extract nontrivial geometric
information, one has to at least consider $\psi_{4}(t,x)$. A similar
calculation shows that

$\psi_{4}(t,x)=\mathbb{E}[S_{4}]=\frac{1}{2}\mathbb{E}[S_{2}\otimes S_{2}].$
(5.2)

This needs not be a zero tensor; indeed it contains the second moment of Lévy
areas of the path $\Gamma$.

In this section, we study the small-time expansion of the function
$\psi_{4}(t,x)$. We first define a contraction on $4$-tensors to reduce
dimension. Let $\xi\in E^{\otimes 4}$ be a given tensor. Being viewed as a
$4$-linear map $\xi:E\times E\times E\times E\rightarrow\mathbb{R}$, we define
$\mathfrak{C}\xi:E\times E\rightarrow\mathbb{R}$ to be the $2$-tensor obtained
from $\xi$ by taking trace on the $(2,4)$-positions. More precisely,

$(\mathscr{\mathfrak{C}}\xi)(\cdot,\cdot)\triangleq\sum_{\alpha=1}\xi(\cdot,\varepsilon_{\alpha},\cdot,\varepsilon_{\alpha}),$
(5.3)

where $\\{\varepsilon_{\alpha}:\alpha=1,\cdots,N\\}$ is any ONB of $E$. One
also needs to recall the basic curvature quantities introduced in Section
2.1.3. Our main theorem for this part is stated as follows.

######  Theorem 5.1.

One has the following small-time expansion:

$\mathfrak{C}\psi_{4}(t,x)=\Theta_{x}\cdot t^{2}+\Xi_{x}\cdot t^{3}+O(t^{4})\
\ \ \text{as }t\rightarrow 0^{+}.$

In addition, the coefficients $\Theta_{x},\Xi_{x}\in E^{\otimes 2}$ are
symmetric $2$-tensors that admit the following explicit representation.

(i) The tensor $\Theta_{x}$ recovers the Riemannian metric tensor $g_{x}$.
More precisely, one has

$\Theta_{x}(v,w)=\frac{d-1}{24}\langle\pi_{x}v,\pi_{x}w\rangle_{T_{x}M},\ \ \
v,w\in E,$ (5.4)

where $\pi_{x}:E\rightarrow T_{x}M$ denotes the orthogonal projection onto the
tangent space $T_{x}M.$

(ii) The tensor $\Xi_{x}$ encodes both intrinsic (Ricci curvature) and
extrinsic (second fundamental form) curvature properties of $M$ at $x$. More
precisely, its restriction to $T_{x}M\times T_{x}M$ as a symmetric bilinear
form on $T_{x}M$ is explicitly given by

$\Xi_{x}|_{T_{x}M\times T_{x}M}=\frac{{\rm
S}_{x}-18d^{2}|H_{x}|_{E}^{2}}{8640}g_{x}+\frac{49d-20}{8640}{\rm
Ric}_{x}+\frac{(5-4d)d}{480}\langle B_{x},H_{x}\rangle_{E}.$ (5.5)

Here $g_{x}$ is the metric tensor, ${\rm Ric}_{x},{\rm S}_{x}$ are the Ricci
tensor and scalar curvature, $B_{x},H_{x}$ are the second fundamental form and
the mean curvature vector, and $\langle\cdot,\cdot\rangle_{E}$ is the
Euclidean inner product in $E$.

######  Remark 5.2.

It can be shown that the $4$-tensor $\psi_{4}(t,x)$ is anti-symmetric on the
first-two components and also on the last-two components. Therefore, the
operation (5.3) is basically the only nontrivial way of contraction into a
$2$-tensor.

To prove Theorem 5.1, we will actually compute the third-order expansion of
the $4$-tensor

$\psi_{4}(t,x)=\hat{\Theta}_{x}t^{2}+\hat{\Xi}_{x}t^{3}+O(t^{4})$ (5.6)

with explicit tensor coefficients $\hat{\Theta}_{x},\hat{\Xi}_{x}$ (cf.
Propositions 5.9, 5.16 below). By some elementary algebraic manipulation, it
is possible to recover all the quantities ${\rm Ric}_{x},{\rm S}_{x},\langle
B_{x},H_{x}\rangle_{E}$ and $|H_{x}|_{E}$ in (5.5) separately from the
coefficients $\hat{\Theta}_{x},\hat{\Xi}_{x}$ (cf. Remark 5.23 for a
discussion). But the corresponding formulae are more involved. For simplicity,
we choose to present the formula in the form of (5.5) through the contraction
$\mathfrak{C}\psi_{4}(t,x)$.

In the following sections, we develop the proof of Theorem 5.1. Our strategy
follows the same spirit as the proof of Theorem 4.1; we perform local
calculations in the normal chart around $x$ based on the local SDE
representation of $X^{t,x,x}$. The actual calculation turns out to be so
involved that it is impossible to achieve entirely by hand. Therefore, our
full calculation is computer-assisted (by Wolfram Mathematica). We will
develop the explicit analysis for a few representative cases in the
computation of the coefficients $\hat{\Theta}_{x},\hat{\Xi}_{x}$ and leave all
other parallel cases to the appendix with documented Mathematica codes
provided in GitHub.

###  5.2 Some basic expansions

As in the proof of Theorem 4.1, we will localise the analysis in a normal
chart $V$ around $x$ (cf. Section 4.2.1 for the construction with $x=y$ in the
current situation). As we have seen in Section 4.2.3, the case that
$X^{t,x,x}$ leaves the chart $V$ before its lifetime gives a negligible
contribution (it is of order $e^{-C/t}$ which is smaller than any power of
$t$; cf. Proposition 4.14). Therefore, one only needs consider the situation
that $X^{t,x,x}$ remains in the chart $V$ for all time (and we will always
assume this is the case). In this case, the analysis reduces to Euclidean
stochastic calculus.

We will use the relation (5.2) to compute $\psi_{4}(t,x).$ Namely, we write

$\psi_{4}(t,x)=\frac{1}{2}\mathbb{E}[\Pi_{t}^{x}\otimes\Pi_{t}^{x}],$ (5.7)

where

$\displaystyle\Pi_{t}^{x}$
$\displaystyle\triangleq\int_{0}^{t}\big{(}\int_{0}^{v}\phi(\circ
dX_{u}^{t,x,x})\big{)}\otimes\circ\phi\big{(}\circ dX_{v}^{t,x,x}\big{)}$
$\displaystyle=\int_{0}^{1}\big{(}\int_{0}^{\rho}\phi(\circ
dX_{tr}^{t,x,x})\big{)}\otimes\circ\phi(\circ dX_{t\rho}^{t,x,x}),$

and we made the change of variables $v=t\rho,u=tr$ to normalise the processes
on the unit interval. To evaluate the expectation (5.7) (in particular, its