# Stochastic Nash evolution

Dominik Inauen  Institut für mathematik, Universität Leipzig, D-04109,
Leipzig, Germany<EMAIL_ADDRESS>and  Govind Menon
Division of Applied Mathematics, Brown University, 182 George St., Providence,
RI 02912<EMAIL_ADDRESS>
###### Abstract.

This paper introduces a probabilistic formulation for the isometric embedding
of a Riemannian manifold $(M^{n},g)$ into Euclidean space $\mathbb{R}^{q}$. We
show that an embedding $u:M\to\mathbb{R}^{q}$ is isometric if and only if the
intrinsic and extrinsic constructions of Brownian motion on
$u(M)\subset\mathbb{R}^{q}$ yield processes with the same law. The equivalence
is first established for smooth embeddings and it is followed by a
renormalization procedure for $C^{1,\alpha}$ embeddings,
$\alpha>\tfrac{1}{2}$.

This formulation is based on a gedanken experiment that relates the intrinsic
and extrinsic constructions of Brownian motion on an embedded manifold to the
measurement of geodesic distance by observers in distinct frames of reference.
This viewpoint provides a thermodynamic formalism for the isometric embedding
problem that is suited to applications in geometric deep learning, stochastic
optimization and turbulence.

This work was supported by the National Science Foundation (DMS 171487 and
2107205), the Simons Foundation (Award 561041) and the Charles Simonyi
Foundation. The authors also express their gratitude to the School of
Mathematics at the Institute for Advanced Study, Princeton and the Max Planck
Institute for Mathematics in the Sciences, Leipzig for partial support during
the completion of this work.

##  1\. Introduction

This work presents a probabilistic characterization of the isometric embedding
problem for Riemannian manifolds based on the equivalence between the
intrinsic and extrinsic constructions of Brownian motion on the manifold. We
introduce these problems, state and prove our results, and then discuss their
scientific context.

###  1.1. The isometric embedding problem for Riemannian manifolds

Assume given a smooth $n$-dimensional closed differentiable manifold $M$
equipped with a metric $g$ and let $(\mathbb{R}^{q},e)$ denote $q$-dimensional
Euclidean space with the identity metric. An immersion $u:M\to\mathbb{R}^{q}$
is isometric if $u^{\sharp}e=g$, where $u^{\sharp}e$ denotes the pullback
metric. In coordinates, this is the system of partial differential equations

$\sum_{a=1}^{q}\partial_{i}u^{a}\partial_{j}u^{a}=g_{ij}.$ (1.1)

Here $a$ indexes coordinates in $\mathbb{R}^{q}$ and $1\leq i,j\leq n$ index
coordinates on $M$. An embedding is an immersion that is one to one. The
isometric embedding and immersion problems are closely related: the central
difficulty in both problems is the analysis of equation (1.1). The results in
this paper have a natural extension to isometric immersions, but we focus on
the embedding problem to be concrete. Similarly, we have assumed that $M$ is
closed only to focus on the essential ideas.

Nash’s work on the isometric embedding problem has determined the development
of the subject since the 1950s. In 1954, he proved the existence of surprising
$C^{1}$ isometric embeddings assuming that $g\in C^{0}$ and that there are no
topological obstructions that prevent smooth embeddings of $M$ into
$\mathbb{R}^{q}$ (this is always true when $q\geq 2n$) [51]. In 1956, he
established the existence of $C^{\infty}$ isometric embeddings when $g\in
C^{\infty}$ and $q$ is large enough [52]. These theorems have played a seminal
role in several areas of mathematics as discussed in the surveys [20, 27, 28,
40].

###  1.2. Brownian motion on Riemannian manifolds

Our sources for stochastic analysis are the monographs [30, 33, 60]. Let
$\triangle_{g}$ denote the Laplace-Beltrami operator on $(M,g)$. Its action on
a smooth function $f$ is given in coordinates by

$\triangle_{g}f=\frac{1}{\sqrt{|g|}}\partial_{i}\left(\sqrt{|g|}g^{ij}\partial_{j}f\right),$
(1.2)

where $|g|$ denotes the determinant of $g_{ij}$ and $g^{ij}$ denotes its
inverse. Brownian motion on $(M,g)$ is a diffusion on $M$ whose generator is
$\tfrac{1}{2}\triangle_{g}$.

####  1.2.1. Intrinsic Brownian motion

The standard intrinsic construction of Brownian motion on $(M,g)$, introduced
by Eells, Elworthy and Malliavin, proceeds as follows. Consider the bundle of
orthonormal frames $O(M)$ for $(M,g)$, let $\pi:O(M)\to M$ denote the
canonical projection and let $\\{H_{j}\\}_{j=1}^{n}$ denote the fundamental
horizontal vector fields. Let $\\{W^{j}\\}_{j=1}^{n}$ denote $n$ independent
standard Wiener processes and consider the Stratonovich SDE on $O(M)$

$dU_{t}=\sum_{j=1}^{n}\left.H_{j}\right|_{U_{t}}\circ dW^{j}_{t}.$ (1.3)

The generator for the diffusion $U_{t}$ is one half the Bochner Laplacian on
$O(M)$,

$\triangle_{O(M)}=\sum_{j=1}^{n}H_{j}^{2}.$ (1.4)

The projection $X_{t}=\pi(U_{t})$ is a Brownian motion on $(M,g)$ starting at
$x=\pi(U_{0})$.

While the Bochner Laplacian may be written as a sum of squares, there is no
such canonical decomposition for the Laplacian $\triangle_{g}$ on $(M,g)$.
Thus, $X_{t}$ itself does not admit such a natural formulation as an SDE.
However, in any coordinate patch, the Stratonovich SDE (1.3) implies the Itô
SDE

$dX^{i}_{t}=\sigma^{i}_{j}(X_{t})dW_{t}^{j}-b^{i}\,dt,$ (1.5)

where the drift and covariance of the noise are given by (see [30, Ex. 3.3.5])

$b^{i}=\frac{1}{\sqrt{|g|}}\frac{\partial\sqrt{|g|}g^{ij}}{\partial
x^{j}},\quad\sigma^{i}_{k}\sigma^{j}_{k}=g^{ij}.$ (1.6)

We may now use Itô’s formula to see that the generator of $X_{t}$ is
$\tfrac{1}{2}\triangle_{g}$.

####  1.2.2. Extrinsic Brownian motion

Assume that $u:M\to\mathbb{R}^{q}$ is a smooth isometric embedding and denote
by $\Sigma=u(M)$ the image of $M$ in $\mathbb{R}^{q}$. We may project Brownian
motion on $\mathbb{R}^{q}$ onto $\Sigma$ as follows. Let
$\\{B^{a}\\}_{a=1}^{q}$ denote $q$ independent standard Wiener processes, let
$\\{P_{a}\\}_{a=1}^{q}$ denote the orthonormal projection of the standard
basis vectors $e_{a}\in\mathbb{R}^{q}$ onto the tangent space $T\Sigma$, let
$x\in M$, and consider the Stratonovich SDE

$dZ_{t}=\sum_{a=1}^{q}P_{a}\circ dB^{a}_{t},\quad Z_{0}=u(x).$ (1.7)

The solution to this SDE also corresponds to a Brownian motion on $(M,g)$
beginning at the point $x$ (see [30, Eq. 3.2.6, p.87]).

###  1.3. Overview of results

Our main insight is that the equivalence of these constructions of Brownian
motion characterizes the isometric embedding problem for $(M,g)$. We formalize
this insight in a set of theorems that culminate in Theorem 4 below. We begin
with some remarks that explain our viewpoint.

The construction of Brownian motion on $(M,g)$ using equation (1.3) is
appealing because it is an extension of the intrinsic notion of parallel
transport to the stochastic setting. The ordinary differential equations that
describe parallel transport have been replaced by Stratonovich SDE in equation
(1.3). The use of the bundle of frames allows us to think of Brownian motion
on $(M,g)$ as the limit of small, independent, isotropic random parallel
translations. It does not require embeddability of the manifold and it
requires exactly $n$ driving Brownian motions.

The extrinsic construction reduces Brownian motion on $(M,g)$ to the familiar
construction of Brownian motion on $\mathbb{R}^{q}$. This simplifies many
calculations and is amenable to numerical simulations. However, it is (at
least at first sight) less satisfactory than the intrinsic construction. The
flaws in this approach include the assumption of the existence of a smooth
embedding and the fact that the number of driving Brownian motions is $q\geq
n+1$.

The isometric embedding problem has been a subject of renewed study in the
past ten years following the creation of the embedding-turbulence analogy by
De Lellis and Székelyhidi (see [20] for a survey). An important area of
enquiry has been the critical exponent for $C^{1,\alpha}$ embeddings that
separate rigidity and flexibility, in analogy with the Onsager conjecture for
the Euler equations [13, 15, 17, 18, 19, 34] (see also [12, 14, 42] for
closely related results for the Monge–Ampère equation). The embedding-
turbulence analogy, as well as the importance of the isometric embedding
problem for several techniques in machine learning, has motivated us to
revisit Nash’s work using probabilistic techniques in order to align
scientific applications with mathematical foundations.

Theorem 1–Theorem 4 provide a rigorous foundation for the information
theoretic interpretation of isometric embedding introduced in [46]. While the
conceptual foundation remains the same, the techniques used here reveal the
probabilistic nature of the isometric embedding problem in its simplest form.
A specific observation that underlies our results is that while the extrinsic
construction of Brownian motion uses Nash’s 1956 smooth embedding theorem, the
equivalence between the two constructions of Brownian motion also applies to
$C^{1,\alpha}$ embeddings when $\alpha>\tfrac{1}{2}$ (see Theorem 4 below).
This formulation provides a new approach to critical exponents in the
isometric embedding problem. It also sheds new light on some classical topics
in stochastic differential geometry, especially the construction of Brownian
motion on $(M,g)$ and Brownian motion in $\mathrm{Diff}(M)$ [4, 41].

These theorems should be seen as the first steps towards a rigorous, and
minimal, thermodynamic formalism for the isometric embedding problem. They
have already been used for the design of algorithms for Gibbs sampling and
optimization, for the construction of models in random matrix theory, as well
as to shed new light on geometric deep learning. These applications are
discussed in Section 7. While our work does not yet provide new proofs of
Nash’s theorems, we sketch a road map that builds on these collective
insights. In particular, we stress the importance of Riemannian Langevin
equations as the natural evolution equations corresponding to the equilibrium
thermodynamics developed in this paper.

##  2\. Statement of results

###  2.1. Overview

We study carefully the manner in which we may recover geometric information
from strictly probabilistic foundations. A general theme in our work is an
unexpected harmony between critical exponent phenomena established in the PDE
literature and their probabilistic counterparts.

Our main result is Theorem 4 below. This theorem provides an equivalence
between the isometric embedding problem and the intrinsic and extrinsic
constructions of Brownian motion for $C^{1,\alpha}$ embeddings when
$\alpha>1/2$. The proof combines techniques from PDE theory (especially
elliptic regularity theory and $h$-principles) and stochastic analysis. We
approach the main theorem by first establishing results in the $C^{\infty}$
regime and then extending these results to the low-regularity regime.

Theorem 1 establishes the equivalence for $C^{\infty}$ embeddings and
$C^{\infty}$ metrics. Theorem 2 provides a probabilistic description of mean
curvature for $C^{\infty}$ embeddings. We then begin our study of the low
regularity regime. Theorem 3 replaces the Eells-Elworthy-Malliavin
construction of intrinsic Brownian with a martingale problem for $C^{2}$
metrics. Finally, we introduce a renormalization procedure to construct
extrinsic Brownian motion for $C^{1,\alpha}$ embeddings and use it to
establish our main theorem. Conjecture 5 (which holds for $C^{1,2/3+}$ convex
surfaces) suggests a renormalization of mean curvature for $C^{1,1/2+}$
embeddings using semimartingales.

These theorems are proved in the sections that follow. Section 7 has a
different character. Here we explain the conceptual foundation for our work.
In particular, we explain the manner in which our theorems provide a rigorous
Bayesian foundation for the thermodynamics of isometric embedding. This
viewpoint ties it to many applications, including geometric deep learning,
optimization and turbulence.

###  2.2. Smooth embeddings

We always assume that $M$ is a closed, connected manifold with a $C^{\infty}$
differentiable structure. In this section, we also assume that both the metric
$g$ and embedding $u$ are $C^{\infty}$.

Assume given an embedding $u:M\to\mathbb{R}^{q}$ (not necessarily isometric).
Fix $x\in M$, let $Y_{t}$ denote the process $u(X_{t})$ obtained from equation
(1.3) and assume $Z_{t}$ is as above in (1.7). Both these processes are curves
on $\Sigma$ that originate at $u(x)$.

######  Theorem 1.

Assume $u:M\to\mathbb{R}^{q}$ is a $C^{\infty}$ embedding and $g$ is a
$C^{\infty}$ metric. The processes $Y_{t}$ and $Z_{t}$ have the same law for
each $x\in M$ if and only if the embedding $u$ is isometric.

Theorem 1 provides an equivalence between the isometric embedding problem and
the construction of Brownian motion on a manifold $(M,g)$. The main ideas are
as follows. Of course, we expect to have agreement between the laws of $Y_{t}$
and $Z_{t}$ when $u$ is isometric. But this requires a proof since the
processes $U_{t}$ and $Z_{t}$ are constructed on different probability spaces,
their sample paths lie in $O(M)$ and $\Sigma\subset\mathbb{R}^{q}$
respectively, and the regularity assumptions on $u$ and $g$ play different
roles in these constructions. The converse provides a strictly probabilistic
formulation of isometric embedding: equality in law of the processes $Y_{t}$
and $Z_{t}$ determines isometry of a (sufficiently smooth) embedding.

Once the converse has been conceived, it is easy to prove. If the processes
$Y_{t}$ and $Z_{t}$ have the same law they yield the same heat kernels
$k(t,x,y)$ for pairs of points $x,y\in M$ and $t>0$. This allows us to
determine the geodesic distance $d_{g}(x,y)$ between pairs of points $x,y\in
M$, and thus the metric itself, using Varadhan’s lemma (equation (1.5) in
[64])

$d_{g}^{2}(x,y)=-2\lim_{t\to 0}t\log k(t,x,y).$ (2.1)

The following theorem makes precise an intuitive relation between extrinsic
Brownian motion and mean curvature. Let $II(z)$ denote the second fundamental
form of the embedding for $z\in\Sigma$ and define the mean curvature $H(z)$ as
the trace of $II(z)$. The correction term in the Itô-Stratonovich conversion
for equation (1.7) is as follows.

######  Theorem 2.

Assume $u:M\to\mathbb{R}^{q}$ is $C^{\infty}$. The Itô form of the
Stratonovich SDE (1.7) is

$dZ_{t}=\sum_{a=1}^{q}P_{a}dB^{a}_{t}+\frac{1}{2}H(Z_{t})\,dt.$ (2.2)

While we have not found Theorem 2 in the literature in this specific form,
closely related discussions may be found in [30, §2.6] and [60, Thm 4.42].

Our main interest lies in the extension of Theorem 2 to $C^{1,\alpha}$
embeddings (see Conjecture 5 below). However, this version of Theorem 2 has
also proved to be of value in the development of algorithms, matrix models and
geometric stochastic flows. Typically, in each of these applications we
compute the mean curvature of a group orbit explicitly and use this as the
basis for a random matrix model or numerical scheme [31, 48, 49, 69].

###  2.3. Background on critical exponents in the $C^{1,\alpha}$ regime

We briefly review the main ideas in Nash’s 1954 proof for the existence of
$C^{1}$ embeddings and the manner in which these ideas persist in the
embedding-turbulence analogy and related critical exponent problems.

Let us simplify matters by assuming that the topology of $M$ is such that it
admits a $C^{\infty}$ embedding into $q=n+2$. By rescaling, we may then assume
that $M$ admits a $C^{\infty}$ short embedding $u_{0}$. This is a subsolution
to equation (1.1), i.e $u_{0}^{\sharp}e<g$ in the order on symmetric quadratic
forms. Nash established the existence of $C^{1}$ isometric embeddings by
constructing a sequence $\\{u_{k}\\}_{k=1}^{\infty}$ of $C^{\infty}$
subsolutions such that the pullback metrics increase monotonically to $g$,

$u_{0}^{\sharp}e<\ldots u_{k}^{\sharp}e<u^{\sharp}_{k+1}e<\ldots<g.$

This family is constructed through a careful addition of small, high-frequency
oscillations. A fundamental step is the Nash lemma, decomposing a $C^{\infty}$
metric $h$ into a finite ‘sum of squares’ of the form

$h=\sum_{l=1}^{N}a_{l}^{2}\,d\psi_{l}\otimes d\psi_{l}.$

Here $a_{l}$ and $\psi_{l}$ are $C^{\infty}$ functions on $M$. The integer $N$
and the set $\\{\psi_{l}\\}_{l=1}^{N}$ are determined by a simplicial
decomposition of the manifold. Given $h$, the Nash lemma provides the
coefficients $\\{a_{l}\\}_{l=1}^{N}$ [51, Lemma 1].

This lemma is used in the following way. Nash’s iteration consists of an outer
loop (‘stages’) indexed by $k=0,1,2,\ldots$ and an inner loop (‘steps’)
indexed by $l=1,\ldots,N$. At the $k$-th stage, the residual
$h=g-u_{k}^{\sharp}e$ is decomposed using the Nash lemma. Then $u_{k}$ is
modified in $N$ steps, with the $l$-th step correcting the pullback metric by
(approximately) $a_{l}^{2}d\psi_{l}\otimes d\psi_{l}$ through the addition of
small, high-frequency, normal and binormal fluctuations.

The embedding-turbulence analogy relies on an analogous, and rather subtle,
notion of subsolutions for the Euler equations which involves replacing the
metric $g$ with the Reynolds stress [21]. This is followed by the construction
of a sequence of subsolutions that converge ‘upwards’ to a weak solution to
the Euler equations. Hölder regularity is obtained by interweaving the
addition of oscillations with a smoothing step. The estimates are challenging
and the resolution of the Onsager conjecture by Isett [34] builds on several
patient improvements on the first such scheme in [21]. The improvements that
provide Hölder regularity for both the Euler equations and isometric embedding
problem rely on commutator estimates and smoothing [13, 15, 17, 18, 19]. Other
$h$-principles that follow [21] rely on similar decompositions; a key step in
the construction is an analog of the Nash lemma.

Theorem 1 allows a different approach to these questions. By viewing the
equivalence of two constructions of Brownian motion as the main characteristic
of isometric embedding, we seek a critical exponent $\alpha_{*}$ such that (a
modification of) Theorem 1 holds for $C^{1,\alpha}$ embeddings with
$\alpha>\alpha_{*}$. In the spirit of Stroock and Varadhan’s introduction of
the martingale problem for diffusions in $\mathbb{R}^{d}$ [62], we also ask if
Theorem 1 may be used to define a martingale characterization of the isometric
embedding problem. This requires an extension of both theorems into the low
regularity regime for $u$ and $g$ and a switch from the use of stochastic
differential equations to martingale problems. Let us now consider the
regularity assumptions that underlie Theorem 1 and Theorem 2 more carefully.

###  2.4. Construction of intrinsic Brownian motion for $g\in C^{2}$

The construction of the process $U_{t}$ using the SDE (1.3) relies only on the
smoothness of the differential structure of $M$ (which we have assumed
$C^{\infty}$), as well as the smoothness of the metric $g$. The standard
monographs in the area assume $g\in C^{\infty}$ for simplicity, although a
more careful argument shows that $g\in C^{3}$ suffices. In Theorem 3 below, we
construct intrinsic Brownian motion for $g\in C^{2}$ as the solution to a
martingale problem. Let us explain the issue.

When $g\in C^{2}$, equation (1.3) is well defined in every coordinate chart,
but the coefficients in the equation are not regular enough to guarantee
strong uniqueness to the corresponding Itô SDE in a given chart. This prevents
us from constructing a solution to (1.3) by patching together the strong
solutions on different coordinate charts (see e.g. the proof of Theorem V.1.1.
in [33]).

Instead, we construct intrinsic Brownian motion on $(M,g)$ by approximation.
In what follows, we consider a sequence of $C^{\infty}$ Riemannian metrics
denoted $g_{k}$. For any $x\in M$, let $X^{k,x}$ be the intrinsic Brownian
motion on $(M,g_{k})$ starting at $x$ (see subsection 1.2). For any $k\geq 1$
we consider the law of the process $X^{k,x}$ as a probability measure
$P^{k,x}$ on the path space $C([0,+\infty[,M)$. We then have

######  Theorem 3.

Assume $g$ is a $C^{2}$ metric and assume that $\\{g_{k}\\}_{k=1}^{\infty}$ is
a sequence of $C^{\infty}$ metrics such that
$\lim_{k\to\infty}\|g_{k}-g\|_{C^{1}}=0$. The sequence $\\{P^{k,x}\\}_{k\geq
1}$ converges weakly to a solution $P^{x}$ of the martingale problem for
$\frac{1}{2}\Delta_{g}$ starting at $x$. Further, this solution to the
martingale problem is unique.

As a consequence there exists a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, a filtration $\mathcal{F}_{t}$ and an
adapted process $\\{X_{t}\\}_{t\geq 0}:\Omega\to M$ such that $X_{0}=x$ almost
surely and for every $f\in C^{\infty}(M)$ it holds that

$f(X_{t})-f(x)-\frac{1}{2}\int_{0}^{t}\Delta_{g}f(X_{s})\,ds$ (2.3)

is a martingale with respect to $\mathbb{P}$ and $\mathcal{F}_{t}$, i.e.
$\\{X_{t}\\}_{t\geq 0}$ is intrinsic Brownian motion starting at $x$.

######  Remark 1.

The gap between $C^{3}$ and $C^{2}$ metrics (for the SDE construction compared
with the martingale problem), and $C^{1}$ convergence to a $C^{2}$ limit, is
reminiscent of a poorly understood transition at $C^{2}$ for the PDE (1.1).
Källen extended Nash’s $C^{1}$ embedding theorem to $C^{1,\alpha}$ for
$0<\alpha<1$ by choosing sufficiently large $q(\alpha)$ [38]. For smooth
embeddings, Nash assumes $g\in C^{3}$ [52]; this was reduced to $C^{2,\beta}$
for any $\beta>0$ (but not $\beta=0$) by Jacobowitz [35]. These coincidences
are suggestive, but we have no deeper explanation for them.

###  2.5. Renormalization and the main theorem

The process $Z_{t}$ is defined using the Stratonovich SDE (1.7). Standard SDE
theory shows that $u\in C^{3}$ is the natural hypothesis for the existence of
strong solutions to (1.7). However, the natural minimum regularity threshold
for isometric maps is when $u$ is Lipschitz. Thus, we have a large regularity
gap between the intrinsic and extrinsic constructions. The intrinsic process
$Y_{t}=u(X_{t})$ has a natural meaning when $u$ is Lipschitz and $g\in C^{2}$.
However, the extrinsic process $Z_{t}$ seems to require $u\in C^{3}$. In
Theorem 4 below, we resolve this mismatch for $C^{1,\alpha}$ embeddings,
$\alpha>\tfrac{1}{2}$. Our argument also leads us to a (conjectural)
renormalization of Theorem 2.

We are guided by a geometric interpretation of Tanaka’s formula. The map
$x\mapsto|x|$ is an isometric folding of the line onto the half-line. Let
$X_{t}$ be Brownian motion on $\mathbb{R}$, set $Y_{t}=|X_{t}|$, and let
$L_{t}$ be the local time of $X_{t}$ at zero. Then Tanaka’s formula tells us
that $Y_{t}$ is a semimartingale and

$dY_{t}=\mathrm{sgn}(X_{t})\,dX_{t}+dL_{t}.$ (2.4)

Further, by following the process $(X_{t},Y_{t})$ on its graph in
$\mathbb{R}^{2}$, we see that the constraint of mean curvature has been
replaced by random normal kicks at $(0,0)$ determined by the local time
$L_{t}$.

This interpretation of Tanaka’s fromula suggests the following renormalization
procedure. For small $\varepsilon>0$ let $u^{\varepsilon}:M\to\mathbb{R}^{q}$
denote the mollified embedding $u^{\varepsilon}=u*\varphi_{\varepsilon}$ (see
Lemma 12). Let $g^{\varepsilon}$ denote the pullback metric
$(u^{\varepsilon})^{\sharp}e$. Fix $x\in M$ and let $X^{\varepsilon}_{t}$
denote intrinsic Brownian motion on $(M,g^{\varepsilon})$ with
$X^{\varepsilon}_{0}=x$. Let
$Y^{\varepsilon}_{t}=u^{\varepsilon}(X^{\varepsilon}_{t})$ denote its image on
$u^{\varepsilon}(M)$. Let us denote the extrinsic Brownian motion beginning at
$u^{\varepsilon}(x)$ by $Z^{\varepsilon}_{t}$. Since $u^{\varepsilon}$ is a
$C^{\infty}$ embedding and $Y_{t}^{\varepsilon}$ is constructed with the
pullback metric $(u^{\varepsilon})^{\sharp}e$, Theorem 1 implies that the
processes $Y_{t}^{\varepsilon}$ and $Z_{t}^{\varepsilon}$ have the same law,
absolutely continuous with respect to the Wiener measures for
$W_{t}\in\mathbb{R}^{n}$ and $B_{t}\in\mathbb{R}^{q}$ respectively. We call
these the intrinsic and extrinsic probability spaces respectively. We then
have

######  Theorem 4.

Assume $g\in C^{2}$ and $u:M\to\mathbb{R}^{q}$ is a $C^{1,\alpha}$ embedding
with $\alpha>\tfrac{1}{2}$.

  1. (1)

If $u$ is isometric then the processes $Z_{t}^{\varepsilon}$ converge in law
to a process $Z_{t}$ defined on the extrinsic probability space and $Z_{t}$
has the same law as $Y_{t}$ for every $x\in M$.

  2. (2)

Conversely, if the processes $Z_{t}^{\varepsilon}$ converge in law to a
process $Z_{t}$ defined on the extrinsic probability space such that $Z_{t}$
has the same law as $Y_{t}$ for every $x\in M$, then $u$ is isometric.

###  2.6. Remarks on Theorem 4

The proof of Theorem 4 relies on a subtle interplay between analysis and
probability. We summarize these issues in the remarks below.

######  Remark 2 (Commutator estimates).

The proof of Theorem 4 relies on the following structure. Since
$Z_{t}^{\varepsilon}$ and $Y^{\varepsilon}_{t}$ have the same law for
$\varepsilon>0$, it is only necessary to show that the law of
$Y_{t}^{\varepsilon}$ converges to that of $Y_{t}$. The important observation
is that, even though $u$ is only $C^{1,\alpha}$, the pullback metrics
$g_{\varepsilon}$ converge in $C^{1}$ to $g$ when $u$ is isometric (see Lemma
5). This step reduces again to a use of commutator estimates for
mollifications, as in [16, 17]. Hence by Theorem 3 the processes
$X_{t}^{\varepsilon}$ converge in law to $X_{t}$, and since $u_{\varepsilon}$
converge to $u$ in $C^{1}$, the theorem follows.

######  Remark 3 (Regularity of $u^{\sharp}e$ and martingale problems).

The pullback metric $u^{\sharp}e$ is only $C^{0,\alpha}$. This means that the
construction of intrinsic Brownian motion on $(M,u^{\sharp}e)$ as the unique
solution to a martingale problem lies outside the realm of Theorem 3.
Similarly, we cannot define the martingale problem for $\triangle_{\Sigma}$
and paths on $\Sigma$ either.

######  Remark 4 (Varadhan’s lemma in the $C^{1,\alpha}$ regime).

Norris has established Varadhan’s lemma for Lipschitz Riemannian manifolds
[54], providing an important step for the converse in Theorem 4. However, we
establish equality of metrics, not just equality of geodesic distances. The
assumption that $g\in C^{2}$ is necessary to bootstrap from Norris’ theorem to
the assertion $g=u^{\sharp}e$; see Lemma 11. Again the issue is that a priori
$u^{\sharp}e$ is only $C^{0,\alpha}$.

######  Remark 5 (Semimartingales and the renormalization of mean curvature).

Theorem 2 and Tanaka’s formula suggest a renormalization of mean curvature for
$C^{1,\alpha}$ embeddings, provided one can establish

######  Conjecture 5.

The processes $Y_{t}$ and $Z_{t}$ are $\mathbb{R}^{q}$-valued semimartingales
for $\alpha>\tfrac{1}{2}$.

This conjecture provides a rigorous framework within which we may replace the
idea of ‘constraint force determined by mean curvature’ with ‘random kicks in
the right direction to maintain a constraint’ for $C^{1,\alpha}$ embeddings.
Indeed, if $Y_{t}$ and $Z_{t}$ are semimartingales, we may write the
(conjectural) Doob-Meyer decomposition of $Y_{t}$ and $Z_{t}$ as

$dY_{t}=Du(X_{t})dX_{t}+dK_{t},\quad
dZ_{t}=\sum_{a=1}^{q}P_{a}dW_{t}^{a}+dM_{t}.$ (2.5)

When $u$ is smooth we have $dK_{t}=\tfrac{1}{2}H(Y_{t})\,dt$ and
$dM_{t}=\tfrac{1}{2}H(Z_{t})\,dt$. Thus, should such a decomposition hold, the
processes $K_{t}$ and $M_{t}$ renormalize the mean curvature as a geometric
analog for $C^{1,\alpha}$ isometric embeddings of the local time.

It is an interesting fact that the decomposition (2.5) holds when $u$ is a
$C^{1,2/3+}$ embedding of a convex surface into $\mathbb{R}^{3}$. Here we
combine Aleksandrov’s notion of delta-convex surfaces (surfaces that may be
represented as a difference of convex functions) with the Meyer-Itô
decomposition of semimartingales (see [1] and [57, Thm. 70]). These surfaces
are an interesting class of examples in the theory of $C^{1,\alpha}$ isometric
embeddings [7, 8, 9, 10, 17, 55].

##  3\. Proof of Theorem 1

For the proof of Theorem 1 we first establish the Itô SDE’s of the processes
$f(Y_{t}),f(Z_{t})$ for general $f\in C^{\infty}(\Sigma)$ in Lemma 1. If the
embedding is isometric, Lemma 2 implies that the drift terms of these
processes agree. We then use the uniqueness of solutions to the martingale
problem (see (3.9)) shown in Lemma 3 to deduce that the laws of $Y_{t}$ and
$Z_{t}$ must agree.

We begin by introducing some notation. For a Riemannian manifold $(N,h)$ and a
function $f\in C^{\infty}(N)$ the gradient $\nabla_{h}f$ is the vectorfield on
$N$ defined through $h(\nabla_{h}f,X)=X(f)$ for any vectorfield
$X\in\mathfrak{X}(N)$, $|\nabla_{h}f|_{h}$ denotes its length with respect to
the metric $h$, and $\Delta_{h}$ denotes the Laplace–Beltrami operator on
$(N,h)$. The volume element $\sqrt{\det h}$ is denoted by $\sqrt{|h|}$. In the
following, we equip the submanifold $\Sigma=u(M)$ with the induced (pullback)
metric $h=\iota^{\sharp}e$, where $\iota:\Sigma\hookrightarrow\mathbb{R}^{q}$
denotes the inclusion and $e$ the Euclidean metric on $\mathbb{R}^{q}$. We
denote indices for coordinates on $M$ by $i,j,k$, so that $1\leq i\leq n$; we
denote indices for coordinates on $\mathbb{R}^{q}$ by $a,b,c$, so that $1\leq
a\leq q$ etc. We denote the coordinates on $\mathbb{R}^{q}$ by $y^{a}$ and we
typically use the summation convention.

######  Lemma 1.

For $f\in C^{\infty}(\Sigma)$ it holds

$\displaystyle df(Y_{t})$
$\displaystyle=dM^{f}_{t}+\frac{1}{2}\Delta_{g}(f\circ u)(X_{t})\,dt$ (3.1)
$\displaystyle df(Z_{t})$
$\displaystyle=dN^{f}_{t}+\frac{1}{2}\Delta_{h}f(Z_{t})\,dt\,,$ (3.2)

where $M^{f}$ and $N^{f}$ are martingales with quadratic variation

$\displaystyle\langle M^{f}\rangle_{t}$
$\displaystyle=\int_{0}^{t}|\nabla_{g}(f\circ u)(X_{s})|_{g}^{2}\,ds,$
$\displaystyle\langle N^{f}\rangle_{t}$
$\displaystyle=\int_{0}^{t}|\nabla_{h}f(Z_{s})|^{2}\,ds\,.$

###### Proof.

We first show (3.1). Recall that the intrinsic Brownian motion on $M$ is
constructed using the horizontal Brownian motion $U_{t}$ on the bundle of
orthonormal frames $O(M)$. The horizontal Brownian motion satisfies the SDE

$dU_{t}=H_{i}|_{U_{t}}\circ dW^{i}_{t}$ (3.3)

where the sum runs over $i=1,\ldots,n$, and $H_{i}$ are the fundamental
horizontal vectorfields. In particular, they have the property that for $f\in
C^{\infty}(M)$ and $r=(x,[E_{1},\ldots,E_{n}])\in O(M)$ it holds

$H_{i}|_{r}(f\circ\pi)=E_{i}(f)\,,$ (3.4)

and moreover, crucially,

$\sum_{i=1}^{n}H_{i}|_{r}(H_{i}(f\circ\pi))=\Delta_{g}f(x)\,,$ (3.5)

where $\pi:O(M)\to M$ denotes the projection. Since by definition
$X_{t}=\pi(U_{t})$ and thus, for $f\in C^{\infty}(\Sigma)$, $f(Y_{t})=f\circ
u\circ\pi(U_{t})$ we find by the conversion formula (cf. Theorem 2.3.5 in
[41])

$\displaystyle df(Y_{t})$ $\displaystyle=H_{i}|_{U_{t}}(f\circ u\circ\pi)\circ
dW^{i}_{t}=H_{i}|_{U_{t}}(f\circ u\circ\pi)dW^{i}_{t}+\frac{1}{2}\langle
H_{i}|_{U_{\cdot}}(f\circ u\circ\pi),W^{i}\rangle_{t}dt$
$\displaystyle=H_{i}|_{U_{t}}(f\circ
u\circ\pi)dW^{i}_{t}+\frac{1}{2}H_{j}|_{U_{t}}\left(H_{i}(f\circ
u\circ\pi)\right)d\langle W^{j},W^{i}\rangle_{t}$
$\displaystyle=E_{i,t}(f\circ u)\,dW^{i}_{t}+\frac{1}{2}\Delta_{g}(f\circ
u)(X_{t})\,dt$

thanks to (3.4) and (3.5), where we denoted
$U_{t}=(X_{t},[E_{1,t},\ldots,E_{n,t}])$. Thus we have (3.1) with

$M^{f}_{t}=\int_{0}^{t}E_{i,s}(f\circ u)\,dW^{i}_{s}\,,$

which is a martingale with quadratic variation

$\langle M^{f}\rangle_{t}=\sum_{i=1}^{n}\int_{0}^{t}\left(E_{i,s}(f\circ
u)\right)^{2}\,ds=\int_{0}^{t}|\nabla_{g}(f\circ u)(X_{s})|_{g}^{2}\,ds\,,$

since $\\{E_{i,s}\\}_{i}$ is an orthonormal basis of $T_{X_{s}}M$.

To show (3.2) we apply the conversion formula to (1.7) and find in the same
way

$df(Z_{t})=P_{a}|_{Z_{t}}(f)dB^{a}_{t}+\frac{1}{2}\sum_{a=1}^{q}P_{a}|_{Z_{t}}\left(P_{a}(f)\right)\,dt\,.$
(3.6)

Now fix a point $p\in\Sigma$ and let $\\{F_{i}\\}_{i=1}^{n}$ be a geodesic
frame for $T\Sigma$ in a neighbourhood $U\subset\Sigma$ of $p$, i.e.,
$\\{F_{i}|_{q}\\}_{i=1}^{n}$ is an orthonormal (with respect to
$h=\iota^{\sharp}e$) basis of $T_{q}\Sigma$ for any $q\in U$, and
$\nabla^{\Sigma}_{F_{i}}F_{j}|_{p}=0$ for every $i,j$, where $\nabla^{\Sigma}$
is the Levi-Civita connection for the metric $h$ (which is simply
$\nabla^{\Sigma}=\overline{\nabla}^{T}$ for the Euclidean connection
$\overline{\nabla}$). It then holds that

$P_{a}=\sum_{i=1}^{n}\langle F_{i},\frac{\partial}{\partial y^{a}}\rangle
F_{i}$

on $U$. Thus we find

$\displaystyle\sum_{a=1}^{q}\left(P_{a}(f)\right)^{2}$
$\displaystyle=\sum_{a=1}^{d}\sum_{i,j=1}^{n}\langle
F_{i},\frac{\partial}{\partial y^{a}}\rangle\langle
F_{j},\frac{\partial}{\partial y^{a}}\rangle
F_{i}(f)F_{j}(f)=\sum_{i=1}^{n}F_{i}(f)^{2}$
$\displaystyle=|\nabla_{\iota^{*}e}f|^{2}\,.$

On the other hand, at $p$, it holds

$\displaystyle\sum_{a=1}^{q}P_{a}\left(P_{a}(f)\right)$
$\displaystyle=\sum_{a=1}^{q}\sum_{i=1}^{n}\langle
F_{i},\frac{\partial}{\partial y^{a}}\rangle F_{i}\left(P_{a}(f)\right)$
$\displaystyle=\sum_{i,j}\sum_{a=1}^{q}\langle F_{i},\frac{\partial}{\partial
y^{a}}\rangle\left(\langle F_{j},\frac{\partial}{\partial y^{a}}\rangle
F_{i}\left(F_{j}(f)\right)+F_{i}(\langle F_{j},\frac{\partial}{\partial
y^{a}}\rangle)F_{j}(f)\right)$
$\displaystyle=\sum_{i=1}^{n}F_{i}(F_{i}(f))+\sum_{i,j}\langle
F_{i},\overline{\nabla}_{F_{i}}F_{j}\rangle F_{j}(f)$
$\displaystyle=\Delta_{h}f+\sum_{i,j}\langle
F_{i},\nabla^{\Sigma}_{F_{i}}F_{j}\rangle F_{j}(f)=\Delta_{h}f\,.$

Since $p\in\Sigma$ is arbitrary this shows (3.2) with

$N_{t}^{f}:=\int_{0}^{t}P_{a}|_{Z_{s}}(f)dB^{a}_{s}\,.\qed$

######  Lemma 2.

If $u$ is isometric it holds

$|\nabla_{g}(f\circ u)(p)|_{g}=|\nabla_{h}f(u(p))|\,,$ (3.7)
$\Delta_{g}(f\circ u)(p)=\Delta_{h}f(u(p))$ (3.8)

for any $p\in M$ and $f\in C^{\infty}(\Sigma)$.

###### Proof.

Fix $p\in M$ and a local orthonormal frame $\\{E_{i}\\}$ for $TM$ around $p$.
We denote by $u_{*}E_{i}\in\mathfrak{X}(\Sigma)$ the pushforward vectorfield
(acting on $f\in C^{\infty}(\Sigma)$ by
$u_{*}E_{i}|_{u(p)}(f)=E_{i}|_{p}(f\circ u)$). Then

$|\nabla_{g}(f\circ u)(p)|^{2}=\sum_{i=1}^{n}\left(E_{i}|_{p}(f\circ
u)\right)^{2}=\sum_{i=1}^{n}\left(u_{*}E_{i}|_{u(p)}(f)\right)^{2}=|\nabla_{h}f(u(p))|^{2}\,,$

since $u$ is isometric and thus $\\{u_{*}E_{i}\\}$ is a local orthonormal
frame for $T\Sigma$ around $u(p)$. Similarly we can show (3.8). Recall that
for a function $f\in C^{\infty}(M)$

$\Delta_{g}f=\mathrm{trace}_{g}\nabla^{2}f=\sum_{i=1}^{n}E_{i}(E_{i}(f))-\nabla_{E_{i}}E_{i}(f)\,,$

for any orthonormal frame $\\{E_{i}\\}$. If, as in the proof of Lemma 1, we
choose $\\{E_{i}\\}$ to be a local geodesic frame around $p$ we find that

$\Delta_{g}(f\circ u)(p)=\sum_{i=1}^{n}E_{i}|_{p}(E_{i}(f\circ
u))=\sum_{i=1}^{n}u_{*}E_{i}|_{u(p)}\left(u_{*}E_{i}(f)\right)=\Delta_{h}f(u(p))\,,$

since $\\{u_{*}E_{i}\\}$ is a local geodesic frame for $T\Sigma$ around
$u(p)$. Indeed, by the Gauss formula it holds

$\nabla^{\Sigma}_{u_{*}E_{i}}u_{*}E_{j}|_{u(p)}=du_{p}\left(\nabla^{M}_{E_{i}}E_{j}\right)=0\,$

for every $i,j$, finishing the proof of the lemma. ∎

Combining Lemma 1 and Lemma 2 it is apparent that if the embedding $u$ is
isometric, then the laws of the processes $Y_{t}$ and $Z_{t}$ are both
solutions to the martingale problem for $\frac{1}{2}\Delta_{h}$ starting at
$u(x)$. More precisely (analogous to [62] (where $N=\mathbb{R}^{d}$)), for a
smooth Riemannian manifold $(N,h)$ with metric $h$ of class $C^{1}$ and $y\in
N$, a _solution of the martingale problem for $\frac{1}{2}\Delta_{h}$ starting
at $y$_ is a probability measure $P^{y}$ on $(C([0,\infty[,N),\mathcal{B})$
such that

  1. (1)

$P^{y}(\gamma(0)=y)=1$

  2. (2)

for any $f\in C^{\infty}(N)$, the process

$M^{f}_{t}=f(\eta_{t})-f(y)-\frac{1}{2}\int_{0}^{t}\Delta_{h}f(\eta_{s})\,ds$
(3.9)

is a martingale with respect to $P^{y}$ and the filtration $B_{t}$.

Here, $\mathcal{B}$ is the Borel $\sigma$-algebra on the path space
$C([0,\infty[,N)$, $\mathcal{B}_{t}$ is the canonical filtration and
$\eta_{t}:C([0,\infty[,N)\to N$ be the continuous map
$\eta_{t}(\gamma)=\gamma(t)$ (see Subsection 5.1.1). The following uniqueness
statement for solutions of the martingale problem then yields the equality of
the laws of $Y_{t}$ and $Z_{t}$ if $u$ is isometric.

######  Lemma 3.

Let $(N,h)$ be a smooth, closed Riemannian manifold with $C^{1,\alpha}$ metric
$h$ for some $\alpha\in]0,1]$. Then for any $y\in N$ the martingale problem
for $\frac{1}{2}\Delta_{h}$ starting at $y$ has at most one solution.

###### Proof.

Observe that for any $\varphi\in C^{\infty}(M)$ and any $\lambda>0$ it is
straightforward to show the existence of a weak solution $u_{\lambda}\in
W^{1,2}(M)$ to

$\lambda u-\frac{1}{2}\Delta_{h}u=\varphi\,$

by energy methods (see e.g. [3]). Since the coefficients of
$\frac{1}{2}\Delta_{h}$ in any coordinate chart are $C^{0,\alpha}$-functions,
Schauder theory implies that in fact $u_{\lambda}\in C^{2,\alpha}(M)$. We can
then argue as in the proof of Theorem 5.1.3 of [61] (which treats the case
$N=\mathbb{R}^{n}$) to conclude. ∎

Conversely, assume the processes $Y_{t}$ and $Z_{t}$ have the same law for
each $x\in M$. Since $u$ is an embedding, this implies that the processes
$X_{t}$ and $\bar{Z}_{t}:=u^{-1}(Z_{t})$ have the same law. On the other hand,
from Lemmata 1 and 2 we find that $\bar{Z}_{t}$ is Brownian motion on
$(M,u^{\sharp}e)$.

Recall that if $h$ is a smooth Riemannian metric and $X_{t}^{x}$ is Brownian
motion on $(M,h)$ starting at $x$, then the function
$v(t,x)=\mathbb{E}[f(X^{x}_{t})]$ solves the heat equation on $(M,h)$ with
initial datum $f\in C^{\infty}(M)$ (see Theorem 3.1. in Chapter V of [33]).
Thanks to the uniqueness of solutions it therefore holds

$\mathbb{E}[f(X^{x}_{t})]=\int_{M}k_{h}(t,x,y)f(y)d\mathrm{vol}_{h}(y)\,,$
(3.10)

where $k_{h}$ is the heat kernel of $h$ (see also Subsection 6.1.1). For a
given $f\in C^{\infty}(M)$ it therefore follows from (3.10) and the equality
in law of the processes $\bar{Z}_{t}$ and $X_{t}$ that

$\int_{M}k_{g}(t,x,y)f(y)d\text{vol}_{g}(y)=\int_{M}k_{u^{\sharp}e}(t,x,y)f(y)d\text{vol}_{u^{\sharp}e}(y)\,,$

and hence by localization

$k_{g}(t,x,y)\sqrt{|g|}(y)=k_{u^{\sharp}e}(t,x,y)\sqrt{|u^{\sharp}e|}(y)\,$

for any $t\geq 0$ and any $x,y\in M$ close enough. Varadhan’s lemma then
implies

$d^{2}_{g}(x,y)=-2\lim_{t\to 0}t\log k_{g}(t,x,y)=-2\lim_{t\to
0}t\log\left(k_{g}(t,x,y)\sqrt{|g|}(y)\right)=d^{2}_{u^{\sharp}e}(x,y)\,.$

Using that the Riemannian metrics $g$ and $u^{\sharp}e$ are smooth, one can
recover them from the geodesic distances through differentiation separately in
$x$ and $y$, followed by passage to the limit $x=y$. The reader unfamiliar
with this calculation may find the details in [6, Lemma,p.380], observing that
$g,u^{\sharp}e\in C^{2}$ suffices to justify the interchange of limits. We
show how to conclude without differentiating the distance functions in Lemma
11 below.

######  Remark 6.

Hsu has established a large deviation principle (LDP) in the context studied
here [29, Thm. 2.2]. However, we do not use this LDP because of the need to
explicitly monitor the regularity assumptions on $g$ and $u$ in our work, as
well as the fact that short-time asymptotics of heat kernels suffice for our
needs.

##  4\. Proof of Theorem 2

It follows from (3.2) (choosing $f=y^{b}$) that

$dZ^{b}_{t}=P_{a}(y^{b})dB^{a}_{t}+\frac{1}{2}\Delta_{\Sigma}y^{b}dt\,.$

The proof of Theorem 2 then follows from the identity

$\Delta_{\Sigma}y=H$ (4.1)

found in the mean curvature flow literature (see e.g. [23]). For the reader’s
convenience we give a proof of the latter identity. Fix $p\in\Sigma$ and a
local orthonormal frame $\\{F_{i}\\}_{i=1}^{n}$ for $T\Sigma$ around $p$. By
definition, it holds at $p$

$\displaystyle\Delta_{\Sigma}y^{b}$
$\displaystyle=\sum_{i=1}^{n}\left(F_{i}(F_{i}(y^{b}))-\nabla^{\Sigma}_{F_{i}}F_{i}(y^{b})\right)$
$\displaystyle=\sum_{i=1}^{n}\left(F_{i}(F_{i}(y^{b}))-\overline{\nabla}_{F_{i}}F_{i}(y^{b})\right)+H(y^{b})\,,$

where $\overline{\nabla}$ denotes the covariant derivative in
$\mathbb{R}^{q}$. On the other hand, writing
$F_{i}=F_{i}^{c}\frac{\partial}{\partial y^{c}}$, we see

$\overline{\nabla}_{F_{i}}F_{i}(y^{b})=F_{i}(F_{i}^{c})\frac{\partial}{\partial
y^{c}}(y^{b})+F_{i}^{c}\overline{\nabla}_{F_{i}}\frac{\partial}{\partial
y^{c}}(y^{b})=F_{i}(F_{i}^{b})\,,$

since $\overline{\nabla}$ is flat. But $F_{i}^{b}=F_{i}(y^{b})$, so that
$\Delta_{\Sigma}y^{b}=H(y^{b})=H^{b}$ at $p$ for any $b=1,\ldots,q$, which
concludes the proof.

##  5\. Proof of Theorem 3

###  5.1. Preliminaries

####  5.1.1. Path space

For a given $(M,g)$ Riemannian manifold we let $C([0,\infty[,M)$ denote the
space of continuous paths $\gamma:[0,\infty[\to M$, as usual equipped with the
metric

$\rho_{g}(\gamma,\tilde{\gamma})=\sum_{n=1}^{\infty}\frac{1}{2^{n}}\max_{0\leq
t\leq n}\min\\{\mathrm{d}_{g}\left(\gamma(t),\tilde{\gamma}(t)\right),1\\}\,,$
(5.1)

(where $d_{g}$ denotes the Riemannian distance function), under which
$C([0,\infty[,M)$ is a complete, seperable metric space with the topology of
compact convergence. The Borel-$\sigma$ algebra on $C([0,\infty[,M)$ is
denoted by $\mathcal{B}=\mathcal{B}(C([0,\infty[,M))$. The canonical
filtration is given by $\mathcal{B}_{t}=\varphi_{t}^{-1}\mathcal{B}$ for
$\varphi_{t}:C([0,\infty[,M)\to C([0,\infty[,M)$ defined through
$\varphi_{t}\gamma(s)=\gamma(\min\\{s,t\\})$. For any $t\geq 0$ we let
$\eta_{t}:C([0,\infty[,M)\to M$ be the continuous map
$\eta_{t}(\gamma)=\gamma(t)$.

####  5.1.2. Hölder norms

Let $\Omega\subset\mathbb{R}^{n}$ be an open set and $f$ a real valued, vector
valued, or tensor valued map defined on $\Omega$. In every case, the target is
equipped with the Euclidean norm, denoted by $|f(x)|$. The Hölder norms are
then defined as follows:

$\|f\|_{0}=\sup_{\Omega}|f|,~{}~{}\|f\|_{m}=\sum_{j=0}^{m}\max_{|\beta|=j}\|\partial^{\beta}f\|_{0},,$

where $\beta$ denotes a multi-index, and

$[f]_{\theta}=\sup_{x\neq
y}\frac{|f(x)-f(y)|}{|x-y|^{\theta}},~{}~{}[f]_{m+\theta}=\max_{|\beta|=m}\sup_{x\neq
y}\frac{|\partial^{\beta}f(x)-\partial^{\beta}f(y)|}{|x-y|^{\theta}},0<\theta\leq
1.$

Then the Hölder norms are given as

$\|f\|_{m+\theta}=\|f\|_{m}+[f]_{m+\theta}.$

####  5.1.3. Mollification estimates

In the proof of Theorem 4 we will regularize the embedding by convolution with
a standard mollifier, i.e., a radially symmetric $\varphi_{\varepsilon}\in
C^{\infty}_{c}(B_{\varepsilon}(0))$ with $\int\varphi_{\varepsilon}=1$, where
$\varepsilon>0$ denotes the length-scale. Such a regularization of Hölder
functions enjoys the following estimates (for a proof, see for example
[17],[19]).

######  Lemma 4.

For any $p\geq 0,$ and $0<\alpha\leq 1,$ we have

$\displaystyle\|f-f*\varphi_{\varepsilon}\|_{0}\leq
C\varepsilon^{\alpha}\|f\|_{\alpha},,$
$\displaystyle\|(fg)*\varphi_{\varepsilon}-(f*\varphi_{\varepsilon})(g*\varphi_{\varepsilon})\|_{p}\leq
C\varepsilon^{2\alpha-p}\|f\|_{\alpha}\|g\|_{\alpha}$ (5.2)

with constant $C$ depending only on $d,p,\beta,\varphi.$

####  5.1.4. Hölder norms and mollification on manifolds

Using a partition of unity, Hölder spaces and mollification can be defined on
the compact manifold $\mathcal{M}$ as follows. We fix a finite atlas of
$\mathcal{M}$ with charts $(\Omega_{i},\phi_{i})$, we let $\\{\chi_{i}\\}$ be
a partition of unity subordinate to $\\{\Omega_{i}\\}$ and set

$\|f\|_{k}=\sum_{i}\|(\chi_{i}f)\circ\phi_{i}^{-1}\|_{k}\,,$

and

$f\ast\varphi_{\varepsilon}=\sum_{i}\left((\chi_{i}f)\circ\phi_{i}^{-1}\ast\varphi_{\varepsilon}\right)\circ\phi_{i}\,.$

One can check that the estimates of Lemma 4 still hold (with constants which
may depend on the fixed charts).

A straightforward consequence of the commutator estimate (5.2) is the
following

######  Lemma 5.

Let $(M,g)$ be a smooth compact manifold with Riemannian metric $g\in C^{2}$
and let $u:M\to\mathbb{R}^{q}$ be an isometric embedding of regularity
$C^{1,\alpha}$. Denote by $u^{\varepsilon}$ the mollification of $u$ as
described above. Then

$\|g-(u^{\varepsilon})^{\sharp}e\|_{1}\leq C\varepsilon^{2\alpha-1}\,.$ (5.3)

The short proof of the previous lemma is contained in [17] Proposition 1.

###  5.2. Proof of Theorem 3

######  Lemma 6.

For any $x\in M$, the sequence $\\{P^{k,x}\\}_{k\in\mathbb{N}}$ is tight and
therefore relatively compact in the set of probability measures on
$C([0,\infty[,M)$ with respect to weak convergence.

We give the proof of the preceding lemma in the next subsection and proceed
with the proof of Theorem 3.

######  Lemma 7.

Let $\\{P^{k_{j},x}\\}_{j\in\mathbb{N}}$ be a subsequence which converges
weakly to a probability measure $P^{x}$. Then $P^{x}$ is a solution to the
martingale problem for $\frac{1}{2}\Delta_{g}$ starting at $x$.

Combined with the uniqueness assertion of Lemma 3 this shows that the full
sequence $\\{P^{k,x}\\}_{k\in\mathbb{N}}$ converges weakly to the solution of
the martingale problem for $\frac{1}{2}\Delta_{g}$ starting at $x$, finishing
the proof of Theorem 3.

###### Proof of Lemma 7.

The argument is similar the one used to prove the existence of weak solutions
of stochastic differential equations with bounded, continuous coefficients
(see e.g. Theorem 5.4.22 in [39]). For the reader’s convenience we present the
argument. Firstly, let us abbreviate $P^{j}:=P^{k_{j},x}$ and observe that
since $P^{j}$ is the law of Brownian motion $X^{k_{j},x}$ starting at $x$ it
holds $P^{j}(\gamma(0)=x)=1$. By the weak convergence we therefore find for
any $f\in C^{\infty}(M)$ that

$\int_{C([0,\infty[,M)}f(\gamma(0))\,dP^{x}=\lim_{j\to\infty}\int_{C([0,\infty[,M)}f(\gamma(0))\,dP^{j}=f(x)\,,$

from which $P^{x}(\gamma(0)=x)=1$ follows. It remains to show (3.9), or,
equivalently, that for any $0\leq s<t<\infty$ and any $\mathcal{B}_{s}$
measurable $F\in C_{b}(C([0,\infty[,M))$ it holds

$E^{P}\left[\left(M^{f}_{t}-M^{f}_{s}\right)F\right]=0\,,$

where $M^{f}_{t}$ is as in (3.9) with $h=g$. Fix then $0\leq s<t<\infty$ and
$F\in C_{b}(C([0,\infty[,M))$ which is $\mathcal{B}_{s}$ measurable. Observe
that since $X^{k}$ is Brownian motion on $(M,g_{k})$, it holds

$E^{P^{k_{j}}}\left[\left(M^{f,j}_{t}-M^{f,j}_{s}\right)F\right]=0\,,$ (5.4)

where $M^{f,j}_{t}:C([0,\infty[,M)\to\mathbb{R}$ is the process defined by

$M^{f,j}_{t}(\gamma)=f(\gamma(t))-f(x)-\frac{1}{2}\int_{0}^{t}\Delta_{g_{k_{j}}}f(\gamma(s))\,ds\,.$

We will now show that $G_{j}:C([0,\infty[,M)\to\mathbb{R}$ given by

$G_{j}(\gamma)=M^{f,j}_{t}(\gamma)-M^{f,j}_{s}(\gamma)\,$

is a uniformly bounded sequence of continuous functions converging uniformly
on $C([0,\infty[,M)$ to $G(\gamma)=M^{f}_{t}(\gamma)-M^{f}_{s}(\gamma)$. We
can then pass to the limit in (5.4) to conclude (see e.g. Problem 2.4.12 in
[39]). Indeed, observe first that if $d_{g}(\rho(t),\tilde{\rho}(t))$ is
smaller than the injectivity radius of $M$ (which is strictly positive since
$M$ is compact without boundary), we get

$|M^{f}_{t}(\gamma)-M^{f}_{t}(\tilde{\gamma})|\leq\left(\|f\|_{1}+Ct\|f\|_{3}\right)d_{g}(\gamma(t),\tilde{\gamma}(t))\leq
Ct\|f\|_{3}\rho_{g}(\gamma,\tilde{\gamma})$

for some constant $C$ depending on $M$ and $g$. A similar estimate holds for
$M^{f,j}_{t}$, thus $G_{j}$ is uniformly continuous. For the convergence we
observe that for any $k\geq 1$ large enough and any function $h\in C^{2}(M)$
we can estimate

$\|(\Delta_{g}-\Delta_{g_{k}})h\|_{0}\leq C\|g-g_{k}\|_{1}\|h\|_{2}\,,$ (5.5)

for some constant depending on $M$ and $g$, as can be seen from the coordinate
expression of the Laplace-Beltrami operator. This implies

$\sup_{\gamma\in C([0,\infty[,M)}|G_{j}(\gamma)-G(\gamma)|\leq
C|t-s|\|g-g_{k}\|_{1}\|f\|_{2}\to 0\,$

for $k\to\infty$, finishing the proof.

∎

###  5.3. Proof of Lemma 6

We use the following characterization of tightness following [39]. Let us
introduce the following modulus of continuity for fixed
$T>0,\delta>0,\gamma\in C([0,\infty[,M)$:

$w^{T}_{\delta}(\gamma)=\sup_{|s-t|<\delta,0\leq s,t\leq
T}d_{g}(\gamma(t),\gamma(s))\,.$ (5.6)

We then have

######  Lemma 8.

A sequence $\\{P^{k}\\}_{k\in\mathbb{N}}$ of probability measures on
$C([0,\infty[,M)$ is tight if for every $\varepsilon>0$ small enough and every
$T>0$ it holds

$\lim_{\delta\downarrow 0}\sup_{k\geq
1}P^{k}\left(w^{T}_{\delta}\geq\varepsilon\right)=0\,.$ (5.7)

The proof of the preceding lemma is a simple modification of the argument for
the case $M=\mathbb{R}$ given in Theorems 2.4.9 and 2.4.10 in [39]. We now
prove Lemma 6 by showing that (5.7) holds for the sequence $\\{P^{k,x}\\}$ of
laws of the intrinsic Brownian motions $X^{k,x}$. Since the metric $g_{k}$ is
smooth and the corresponding SDE (1.3) has unique strong solutions given
arbirtary initial values, we can assume that the processes $X^{k,x}$ are
defined on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$ and
are adapted with respect to a common filtration $\mathcal{F}_{t}$. Unwinding
the definitions, it is clear that to show (5.7) we need an estimate on

$\mathbb{P}(d_{g}(X^{k,x}_{t},X^{k,x}_{s})\geq\varepsilon)$

for arbitrary $|s-t|<\delta$ small. To estimate the latter quantity we follow
[44] (Section 4.3) and introduce, for arbitrary $x\in
M,k\in\mathbb{N},\delta>0$, the exit times

$\tau_{\varepsilon}^{k,x}=\inf\\{t>0:d_{g}(X^{k,x}_{t},x)>\varepsilon\\}\,.$
(5.8)

We claim that

######  Lemma 9.

There exists $\varepsilon_{M}>0$ only depending on $(M,g)$ such that for any
$0<\varepsilon<\varepsilon_{M}$ there exists a constant $C_{\varepsilon}>0$
(depending only on $\varepsilon,M$ and $g$) such that

$\sup_{x\in M}\mathbb{P}(\tau^{k,x}_{\varepsilon}\leq t)\leq C_{\varepsilon}t$
(5.9)

for any $t\geq 0$ and $k\in\mathbb{N}$ large enough.

We postpone the proof of the preceding lemma and show how to infer (5.7).
Observe that, since $X^{k,x}$ is Brownian motion on $M$, the coordinate
process $\eta=\\{\eta_{t}\\}_{t\geq 0}$ on $(C([0,\infty[,M),\mathcal{B})$
together with the family of laws $\\{P^{k,x}\\}_{x\in M}$ is a Markov family.
Fixing $T>0$, $\delta>0$, $0\leq s\leq t\leq T$ with $t-s<\delta$, and
$\varepsilon<\varepsilon_{M}$, we can then estimate, using the Markov property

$\displaystyle\mathbb{P}\left(d_{g}(X^{k,x}_{t},X^{k,x}_{s})\geq\varepsilon\right)$
$\displaystyle=P^{k,x}\left(d_{g}(\eta_{t},\eta_{s})\geq\varepsilon\right)$
$\displaystyle=\int_{M}P^{k,x}\left(d_{g}(\eta_{t},\eta_{s})\geq\varepsilon|\eta_{s}=y\right)\,d(\eta_{s})_{*}P^{k,x}(y)$
$\displaystyle=\int_{M}P^{k,y}\left(d_{g}(\eta_{t-s},y)\geq\varepsilon\right)\,d(\eta_{s})_{*}P^{k,x}(y)$

But due to (5.9) we have

$\displaystyle\sup_{y\in
M}P^{k,y}\left(d_{g}(\eta_{t-s},y)\geq\varepsilon\right)$
$\displaystyle=\sup_{y\in
M}\mathbb{P}\left(d_{g}(X^{k,y}_{t-s},y)\geq\varepsilon\right)\leq\sup_{y\in
M}\mathbb{P}\left(\tau^{k,y}_{\varepsilon}\leq t-s\right)$ $\displaystyle\leq
C_{\varepsilon}(t-s)$ $\displaystyle\leq C_{\varepsilon}\delta\,,$

which implies (5.7) for the sequence $\\{P^{k,x}\\}_{k\in\mathbb{N}}$ and
hence also the statement of Lemma 6. We are therefore left to show Lemma 9.

###### Proof of Lemma 9.

We want to use the property (2.3) for $X^{k,x}$ to gain information about the
exit-time of the process $X^{k,x}$. In order to do so we need to construct a
suitable testfunction $f$. Since the manifold $(M,g)$ is closed, the
injectivity radius is bounded from below by a constant $\varepsilon_{M}>0$. In
particular, for any fixed $x\in M$, the Riemannian distance function
$d_{g}(x,\cdot)$ is $C^{2}$ on the open set
$B_{\varepsilon}(x)\setminus\\{x\\}$, where $B_{\varepsilon}(x)$ is the open
geodesic ball with radius $\varepsilon$ (this follows for example from the
formula (4.2) in [66]). Given $0<\varepsilon<\varepsilon_{M}$ we let
$\theta\in C^{\infty}_{c}(\mathbb{R})$ be a cutoff function with
$0\leq\theta\leq 1$ on $\mathbb{R}$, $\theta_{\varepsilon}\equiv 1$ on
$]-\frac{\varepsilon}{4},\frac{\varepsilon}{4}[$ and
$\theta_{\varepsilon}\equiv 0$ outside
$]-\frac{\varepsilon}{2},\frac{\varepsilon}{2}[$. Then the function

$f^{x}_{\varepsilon}:=\theta_{\varepsilon}\circ d_{g}(x,\cdot)\,$

is a $C^{2}$-function on $M$ with support contained in $B_{\varepsilon}(x)$.
As in (5.5), we can estimate for $k$ large enough

$\|\Delta_{g_{k}}f^{x}_{\varepsilon}\|_{0}\leq
C\|f^{x}_{\varepsilon}\|_{2}\leq C_{\varepsilon}$

for a constant $C_{\varepsilon}$ only depending on $\varepsilon$ and $(M,g)$,
but not on $k$ or $x$. Now fix $t\geq 0$. We use that $X^{k,x}$ is the
intrinsic Browinan motion on $(M,g_{k})$ and the fact that $0\leq
f^{x}_{\varepsilon}\leq 1$ with $f^{x}_{\varepsilon}(x)=1$ and
$f^{x}_{\varepsilon}=0$ outside $B_{\frac{\varepsilon}{2}}(x)$ to find

$\displaystyle\mathbb{P}(\tau^{k,x}_{\varepsilon}\leq t)$
$\displaystyle=1-\mathbb{P}(\tau^{k,x}_{\varepsilon}>t)$ $\displaystyle\leq
1-\mathbb{E}\left(\chi_{\\{\tau^{k,x}_{\varepsilon}>t\\}}f^{x}_{\varepsilon}\left(X^{k,x}_{\tau^{k,x}_{\varepsilon}\wedge
t}\right)\right)$
$\displaystyle=1-\mathbb{E}\left(f^{x}_{\varepsilon}\left(X^{k,x}_{\tau^{k,x}_{\varepsilon}\wedge
t}\right)\right)+\mathbb{E}\left(\chi_{\\{\tau^{k,x}_{\varepsilon}\leq
t\\}}f^{x}_{\varepsilon}\left(X^{k,x}_{\tau^{k,x}_{\varepsilon}\wedge
t}\right)\right)$
$\displaystyle=f^{x}_{\varepsilon}(x)-\mathbb{E}\left(f^{x}_{\varepsilon}\left(X^{k,x}_{\tau^{k,x}_{\varepsilon}\wedge
t}\right)\right)$
$\displaystyle=-\mathbb{E}\int_{0}^{\tau^{k,x}_{\varepsilon}\wedge
t}\frac{1}{2}\Delta_{g^{k}}f^{x}_{\varepsilon}(X^{k,x}_{s})\,ds\leq
C_{\varepsilon}t$

for some constant $C_{\varepsilon}$ depending only on $M,g$ and $\varepsilon$.
This concludes the proof. ∎

##  6\. Proof of Theorem 4

Theorem 4 relies on the existence and short-time asymptotics for heat kernels
for $C^{0,\alpha}$ metrics. We review the necessary background before turning
to the proof of Theorem 4.

###  6.1. Preliminaries

####  6.1.1. Heat equation on Riemannian manifolds

Let $h$ be a continuous metric on $M$. Then the Laplace-Beltrami operator
$-\Delta_{h}$ is welldefined as an unbounded operator on $L^{2}(M)$. Recall
that for a $v\in W^{1,2}(M)$, the action on $-\Delta_{h}v$ on a test function
$\varphi\in W^{1,2}(M)$ is given by

$\langle-\Delta_{h}v,\varphi\rangle=\int_{M}h(\nabla_{h}v,\nabla_{h}\varphi)d\text{vol}_{h}\,.$

Classical theory (see e.g. Theorem 4.1 Chapter III in [43] and [63]) asserts
that for any $f\in L^{2}(M)$, the Cauchy problem for the heat equation
$\partial_{t}v-\Delta_{h}v=0$, $v(0)=f$ has a unique (weak) solution $v\in
L^{2}(]0,\infty[,W^{1,2})$ and that it can be written as

$v(t,x)=\int_{M}k_{h}(t,x,y)f(y)d\text{vol}_{h}(y)\,,$

where $k_{h}$ is called the _heat kernel_ of $h$. It is a smooth function of
$(t,x,y)$ when $h$ is smooth. In the proof of Theorem 4 below we will consider
$h=u^{\sharp}e$, which is only $C^{0,\alpha}$. In particular, the classical
formula (3.10) does not hold a priori. We therefore need the following lemma.

######  Lemma 10.

Let $\\{h_{\varepsilon}\\}$ be a family of smooth Riemannian metrics on $M$
and $\\{k_{h_{\varepsilon}}\\}$ the corresponding heat kernels. Assume that
$\|h_{\varepsilon}-h\|_{0}\to 0$ for $\varepsilon\to 0$. Then for any $f\in
C^{2}(M)$ and any $t>0$, $x\in M$ it holds

$\lim_{\varepsilon\to
0}\int_{M}k_{h_{\varepsilon}}(t,x,y)f(y)d\mathrm{vol}_{h_{\varepsilon}}(y)=\int_{M}k_{h}(t,x,y)f(y)d\mathrm{vol}_{h}(y)\,.$
(6.1)

###### Proof.

The function
$v_{\varepsilon}(t,x)=\int_{M}k_{h_{\varepsilon}}(t,x,y)f(y)d\mathrm{vol}_{h_{\varepsilon}}(y)$
is the unique smooth solution of the heat equation
$\partial_{t}v_{\varepsilon}-\Delta_{h_{\varepsilon}}v_{\varepsilon}=0$ on $M$
with $v_{\varepsilon}(0,\cdot)=f$. Testing the equation with $v_{\varepsilon}$
and integrating yields a uniform $L^{\infty}((0,T),L^{2}(M))\cap
L^{2}((0,T),W^{1,2}(M))$ bound for $\\{v_{\varepsilon}\\}$ and therefore a
weakly in $L^{2}((0,T),W^{1,2}(M))$ converging subsequence. Passing to the
limit in the weak formulation of the heat equation, we find (using the
uniqueness of weak solutions) that the limit is $v(t,\cdot)$. On the other
hand, the coordinate expression of $v_{\varepsilon}$ in some fixed chart is a
smooth solution to

$\partial_{t}\left(\sqrt{|h_{\varepsilon}|}v_{\varepsilon}\right)-\partial_{i}\left(\sqrt{|h_{\varepsilon}|}h_{\varepsilon}^{ij}\partial_{j}v_{\varepsilon}\right)=0$

in some ball $B_{r}(0)\subset\mathbb{R}^{n}$ (recall that $|h|=\det h$). Due
to the convergence assumption and the compactness of $M$ there exist
$0<\lambda<\Lambda<+\infty$ such that for all $\varepsilon>0$ small enough it
holds

$\lambda\leq\sqrt{|h_{\varepsilon}|}\leq\Lambda$

and

$\lambda Id\leq\sqrt{|h_{\varepsilon}|}h_{\varepsilon}^{ij}\ \leq\Lambda
Id\,.$

The De Giorgi–Nash–Moser theorem (see Theorem 18 in [65] for our setting) then
implies a uniform (in $\varepsilon$)
$C^{0,\alpha}([s,T]\times\bar{B}_{\frac{r}{2}})$ \- bound for the functions
$v_{\varepsilon}$ for all $s>0$. By the theorem of Arzelà-Ascoli, a
subsequence of $v_{\varepsilon}$ therefore converges uniformly in $(t,x)$. By
the above weak convergence, the limit must be $v$, which shows the claim. ∎

####  6.1.2. Recovering Riemannian metrics from their distance function

In the proof of Theorem 1 we used that one can recover the Riemannian metric
from its distance function by differentiation. This proof relies on the
regularity assumption that the metric is twice differentiable. In the proof of
Theorem 4 we only know a priori that the metric $u^{\sharp}e$ is
(Hölder-)continuous. The following Lemma is necessary to close this gap.

######  Lemma 11.

Let $g\in C^{2}$ and $h\in C^{0}$ be two Riemannian metrics on $M$. Assume
that for any $x\in M$ it holds

$d_{g}(x,y)=d_{h}(x,y)$ (6.2)

for all $y$ in a $g$-geodesic neighborhood of $x$. Then $g=h$.

As usual, the distance function $d_{h}(x,y)$ is defined by

$d_{h}(x,y)=\inf_{\gamma}\int_{0}^{1}|\gamma^{\prime}|_{h}\,dt=:\inf_{\gamma}L_{h}(\gamma)\,,$

where the infimum is taken over all piecewise smooth curves
$\gamma:[0,1]\to\mathcal{M}$ with $\gamma(0)=x,\gamma(1)=y$.

###### Proof.

Let $x\in M$, $y$ in a $g$-geodesic neighborhood of $x$ and let
$\gamma:[0,L]\to M$ be the unique $g$-geodesic connecting $x$ and $y$. Using
the theorem of Arzelà-Ascoli we can find a piecewise smooth curve
$\bar{\gamma}:[0,1]\to M$ connecting $x$ and $y$ which minimizes the length
$L_{h}$ (see also Section 2.1 in [58] and references therein). We claim that
$\bar{\gamma}$ is a reparametrization of $\gamma$. Fix $s\in(0,1)$ and observe
that

$\displaystyle d_{g}(x,y)$ $\displaystyle\leq
d_{g}(x,\bar{\gamma}(s))+d_{g}(y,\bar{\gamma}(s))=d_{h}(x,\bar{\gamma}(s))+d_{h}(y,\bar{\gamma}(s))$
$\displaystyle\leq\int_{0}^{s}|\bar{\gamma}^{\prime}|_{h}\,dt+\int_{s}^{1}|\bar{\gamma}^{\prime}|_{h}\,dt=L_{h}(\bar{\gamma})=d_{h}(x,y)=d_{g}(x,y)\,,$

i.e., $d_{g}(x,y)=d_{g}(x,\bar{\gamma}(s))+d_{g}(y,\bar{\gamma}(s))$. But this
implies that $\bar{\gamma}(s)=\gamma(t(s))$ for some $t(s)\in[0,L]$. Indeed,
connecting $x$ and $\bar{\gamma}(s)$ by the unique $g$-geodesic and following
the unique $g$-geodesic from $\bar{\gamma}(s)$ to $y$ yields a
$d_{g}$-minimizing curve connecting $x,y$, implying that the curve is $\gamma$
by uniqueness of $d_{g}$-minimizing curves.

Now fix $x\in M$ and $X\in T_{x}M$ and let $\gamma:(-1,1)\to M$ be a
$g$-geodesic with $\gamma(0)=x$ and $\gamma^{\prime}(0)=X$. Fix $T<1$ such
that $\gamma(T)$ belongs to a geodesic neighborhood of $x$. Let then
$\bar{\gamma}:[0,1]\to M$ be a $L_{h}$-minimizing curve connecting $x$ and
$\gamma(T)$, and fix $t\in(0,T)$. By the above reasoning it holds
$\gamma(t)=\bar{\gamma}(s(t))$ for some $s(t)\in(0,1)$ and hence, using (6.2)
and the indepence of the length of a curve with respect to reparametrizations,

$\displaystyle\int_{0}^{s}|\gamma^{\prime}|_{g}\,dt$
$\displaystyle=d_{g}(x,\gamma(s))=d_{h}(x,\bar{\gamma}(t(s)))=L_{h}(\bar{\gamma}|_{[0,t(s)]})=L_{h}(\gamma|_{[0,s]})$
(6.3) $\displaystyle=\int_{0}^{s}|\gamma^{\prime}|_{h}\,dt\,.$ (6.4)

Dividing by $s$ and letting $s$ approach zero then yields $|X|_{g}=|X|_{h}$,
from which the claim follows. ∎

####  6.1.3. Mollifying embeddings

For $\varepsilon>0$ let $u^{\varepsilon}:M\to\mathbb{R}^{q}$ denote the
mollified map $u^{\varepsilon}=u*\varphi_{\varepsilon}$ as defined in
Subsection 5.1.4.

######  Lemma 12.

There exists $\varepsilon_{*}>0$ such that $u^{\varepsilon}$ is an embedding
for $0<\varepsilon<\varepsilon_{*}$.

###### Proof.

Fix a finite atlas for $M$. Since $u$ is an immersion and $M$ is compact,
there exists $\eta>0$ such that $|Du_{p}\xi|\geq 5\eta$ for all $p\in M$ and
$\xi\in\mathbb{S}^{n-1}$. We can estimate

$|Du^{\varepsilon}_{p}\xi|\geq 5\eta-\|u-u^{\varepsilon}\|_{1}\geq 4\eta\,,$
(6.5)

which shows that $u^{\varepsilon}$ is an immersion for $\varepsilon$ small
enough, since $\|u-u^{\varepsilon}\|_{1}\leq
C\varepsilon^{\alpha}\|u\|_{1,\alpha}$ by Lemma 4. It remains to show that
$u^{\varepsilon}$ is injective for $\varepsilon$ small enough. First of all,
since $u\in C^{1}$, there exists $\rho>0$ such that $|Du_{p}-Du_{q}|\leq\eta$
for all $d_{g}(p,q)<\rho$. Without loss of generality we assume
$\rho<\text{inj}(M)$. Since $u$ is an embedding, there is $\delta>0$ such that
$|u(p)-u(q)|\geq 3\delta$ for $d_{g}(p,q)\geq\rho$. For such $p,q$ we
therefore find

$|u^{\varepsilon}(p)-u^{\varepsilon}(q)|\geq
3\delta-2\|u-u^{\varepsilon}\|_{0}\geq\delta$

for $\varepsilon$ small enough. On the other hand, fix $d_{g}(p,q)<\rho$ and
let $x,y$ denote the coordinates of $p,q$. By the mean value theorem we find

$\displaystyle|u^{\varepsilon}(x)-u^{\varepsilon}(y)-Du^{\varepsilon}(x)(x-y)|$
$\displaystyle\leq|\left(Du^{\varepsilon}(x+\tau(y-x))-Du^{\varepsilon}(x)\right)||x-y|$
$\displaystyle\leq 3\eta|x-y|$

for $\varepsilon$ small enough. Therefore, using (6.5),

$|u^{\varepsilon}(x)-u^{\varepsilon}(y)|\geq|Du^{\varepsilon}(x)(x-y)|-3\eta|x-y|\geq\eta|x-y|$

for $\varepsilon$ small enough, which yields the claim. ∎

###  6.2. Proof of Theorem 4

Let $u$ be isometric. As in the proof of Theorem 3 we can assume that the
processes $X^{\varepsilon}_{t}$ are defined on a common probability space
$(\Omega,\mathcal{F},\mathbb{P})$, and hence also
$Y^{\varepsilon}_{t}=u^{\varepsilon}(X^{\varepsilon}_{t})$. The law of
$Y^{\varepsilon}$ is a probability measure on the path space
$C([0,\infty[,\mathbb{R}^{q})$ (which is equipped with the metric $\rho_{e}$
defined as in (5.1)). Fix therefore
$f:C([0,\infty[,\mathbb{R}^{q})\to\mathbb{R}$ continuous and bounded. Then

$\int_{C([0,\infty[,\mathbb{R}^{q})}f(\gamma)\,dY^{\varepsilon}_{*}\mathbb{P}(\gamma)=\int_{C([0,\infty[,M)}f\circ
u^{\varepsilon}(\gamma)\,dX^{\varepsilon}_{*}\mathbb{P}(\gamma)\,.$

Because $u$ is isometric and $\alpha>\frac{1}{2}$ it follows from Lemma 5 and
Theorem 3 that $X^{\varepsilon}_{t}$ converges in law to $X_{t}$ for
$\varepsilon\to 0$. Hence, if we show that $f\circ u^{\varepsilon}$ converges
uniformly on compact subsets of $C([0,\infty[,M)$ to $f\circ u$, it will
follow (see e.g. Problem 2.4.12 in [39])

$\displaystyle\lim_{\varepsilon\downarrow
0}\int_{C([0,\infty[,\mathbb{R}^{q})}f(\gamma)\,dY^{\varepsilon}_{*}\mathbb{P}(\gamma)$
$\displaystyle=\int_{C([0,\infty[,M)}f\circ
u(\gamma)\,dX_{*}\mathbb{P}(\gamma)$
$\displaystyle=\int_{C([0,\infty[,\mathbb{R}^{q})}f(\gamma)\,dY_{*}{\mathbb{P}}(\gamma)\,,$

as required. Thus, fix any compact subset $K\subset C([0,\infty[,M)$ and a
sequence $\varepsilon_{i}\downarrow 0$. Observe that $u(K)$ is compact in
$C([0,\infty[,\mathbb{R}^{q})$ (equipped with the usual topology of compact
convergence) thanks to the estimate

$|u(\gamma_{i}(t))-u(\gamma(t))|\leq C[u]_{1}d_{g}(\gamma_{i}(t),\gamma(t))$

if $d_{g}(\gamma_{i}(t),\gamma(t))$ is smaller than the injectivity radius of
$(M,g)$. The same holds for the set $u^{\varepsilon}(K)$ for any
$\varepsilon>0$. Since moreover

$|u^{\varepsilon}(\gamma_{i}(t))-u(\gamma(t))|\leq\|u^{\varepsilon}-u\|_{0}+C[u]_{1}d_{g}(\gamma_{i}(t),\gamma(t))$

it follows that also the set
$\tilde{K}=\bigcup_{i=1}^{\infty}u^{\varepsilon_{i}}(K)$ is compact. As a
continuous function, $f$ is uniformly continuous on $\tilde{K}$ and given
$\delta>0$ we find $\eta>0$ such that $|f(z)-f(w)|<\delta$ for any
$z,w\in\tilde{K}$ with $\rho_{e}(z,w)<\eta$. Since
$\|u^{\varepsilon}-u\|_{0}\leq C\varepsilon[u]_{1}$ by Lemma 4, we deduce that
for any $T>0$

$\sup_{\gamma\in K}\sup_{0\leq t\leq
T}|u_{\varepsilon_{i}}(\gamma(t))-u(\gamma(t))|\leq C\varepsilon[u]_{1}<\eta$

for any $i$ large enough, i.e., for such $i$ it holds
$\tilde{\rho}(u_{\varepsilon_{i}}\circ\gamma,u\circ\gamma)<\eta$ for any
$\gamma\in K$. This yields

$\sup_{\gamma\in K}|f\circ u_{\varepsilon_{i}}(\gamma)-f\circ
u(\gamma)|<\delta$

for large enough $i$,, which shows the claim.

Conversely, assume that $Z^{\varepsilon,x}_{t}$ converges in law to
$Y^{x}_{t}$ for any $x\in M$. As in the proof of Theorem 1 we consider the
processes $\bar{Z}^{\varepsilon,x}_{t}=u^{-1}(Z^{\varepsilon,x}_{t})$, which
are Brownian motions on $(M,(u^{\varepsilon})^{\sharp}e)$ starting at $x$.
Since the metric $h_{e}:=(u^{\varepsilon})^{\sharp}e$ is smooth it holds

$\mathbb{E}[f(\bar{Z}^{\varepsilon,x}_{t})]=\int_{M}k_{h_{\varepsilon}}(t,x,y)f(y)d\mathrm{vol}_{h_{\varepsilon}}(y)$

for any $f\in C^{\infty}(M)$. By assumption, $\bar{Z}^{\varepsilon,x}_{t}$
converges in law to $u^{-1}(Y_{t}^{x})=X^{x}_{t}$ which is Brownian motion on
$(M,g)$. Consequently,

$\lim_{\varepsilon\to
0}\int_{M}k_{h_{\varepsilon}}(t,x,y)f(y)d\mathrm{vol}_{h_{\varepsilon}}(y)=\int_{M}k_{g}(t,x,y)f(y)d\mathrm{vol}_{g}(y)\,.$

On the other hand, $h_{\varepsilon}\to u^{\sharp}e$ in $C^{0,\alpha}$. We can
therefore apply Lemma 10 to deduce that in fact

$\displaystyle\int_{M}k_{g}(t,x,y)f(y)d\mathrm{vol}_{g}(y)$
$\displaystyle=\lim_{\varepsilon\to
0}\int_{M}k_{h_{\varepsilon}}(t,x,y)f(y)d\mathrm{vol}_{h_{\varepsilon}}(y)$
$\displaystyle=\int_{M}k_{u^{\sharp}e}(t,x,y)f(y)d\mathrm{vol}_{u^{\sharp}e}(y)\,$

for any $f\in C^{\infty}(M)$. As in the proof of Theorem 1 we can localize and
invoke Varadhan’s lemma (more precisely the low-regularity version due to
[54]) to find $d_{g}(x,y)=d_{u^{\sharp}e}(x,y)$ for all $x,y\in M$ close
enough. We conclude by appealing to Lemma 11.

##  7\. Thermodynamics of isometric embedding

###  7.1. Isometric embeddings, artificial intelligence and turbulence

The work presented in this paper is part of a program to construct Gibbs
measures supported on isometric embeddings. Informally, the underlying
questions are: ‘how do we construct typical isometric embeddings’ and ‘what
are their universal properties’?

This program was first motivated by the embedding-turbulence analogy [20]. The
primary empirical reality in turbulence is the universality of the Kolmogorov
spectrum. The Kolmogorov spectrum is a feature of the statistical theory of
turbulence. By contrast, despite its origin, the Onsager conjecture (now
theorem) has no explicit probabilistic content. Our work began as a
randomization of Nash’s scheme in order to bridge this divide. In this
context, the construction of Gibbs measures allows critical exponents in PDE
theory to be seen as the mathematical counterpart of critical exponent
phenomena in condensed matter physics [26]. The isometric embedding problem
also appears in quantum field theory under the guise of the nonlinear sigma
model [24, 25]. The physical context of these problems provides a rich source
of inspiration. For example, the embedding-turbulence analogy allows us to
conjecture universality for isometric embeddings and to seek models and
numerical methods to test such conjectures.

The isometric embedding problem also arises in several mathematical approaches
to artificial intelligence. Diffusion maps, heat kernel embeddings, and
geometric deep learning are successful frameworks in machine learning that
implicitly use isometric embedding. This is a vast area; the papers [5, 11,
37, 59] provide an introduction to some of its scope. A foundational result in
the area, due to Berard, Besson and Gallot [6], is the construction of
‘almost’ isometric embeddings $(M,g)\to L^{2}(M,g;\mathbb{R})$ using heat
kernels. Unlike Nash’s embedding theorems, the heat kernel embeddings are
approximate, not exact, and they map $M$ into an infinite-dimensional space,
not $\mathbb{R}^{q}$. However, they are geometrically natural and admit robust
numerical implementations. Thus, the use of heat kernel embeddings has
stimulated new attempts to determine ‘canonical’ isometric embeddings in
$\mathbb{R}^{q}$ [56, 67], extensions of the heat kernel method to RCD(K,N)
spaces [2, 32], and computational relaxation schemes [45]. Despite the
extensive use of heat kernel embeddings, there appears to have been no attempt
to use probabilistic methods, especially stochastic calculus, to rethink
Nash’s theorems prior to our work.

Grenander’s pattern theory provides a Bayesian framework for many problems in
cognition (see [47, 50] for introductions). However, these Bayesian methods
are not competitive with deep learning in computer vision, since they rely on
the construction of geometric priors and fast optimization over parameters
that determine the priors. Neither of these steps has been adequately
resolved. Our construction of Gibbs measures for isometric embeddings is also
stimulated by a desire to bridge the divide between Bayesian methods and deep
learning.

The above applications emphasize the need to balance rigorous results with
fast algorithms, and to balance concept with technique, when re-examining
Nash’s work. The main conceptual insight formulated in [46] is that the use of
information theory provides unity between the many applications above. In the
context of physics, we follow Jaynes and approach Gibbs measures from an
information theoretic perspective [36]. We also augment the embedding-
turbulence analogy by developing an embedding-RMT (for random matrix theory)
analogy, so that the ties to mathematical physics are made explicit. The heart
of the matter, however, is an information theoretic interpretation of
embedding that goes roughly as follows.

We view embedding as a form of information transfer between a source and an
observer. The process of information transfer is complete when all
measurements of distances by the observer agree with those at the source. This
viewpoint is both Bayesian and information theoretic. It places the emphasis
not on the structure of the manifold, but on a more primitive aspect of the
problem, the measurement of length. In the Bayesian interpretation, the world
is random and both the source and the observer are stochastic processes with
well-defined parameters. The process of successive approximation implict in
Nash’s work can now be modeled as an optimal control strategy by an observer
tuning a model in response to measurements of signals from the source. Thus,
embedding is simply ‘replication’ and the process of replication is complete
when all measurements by the observer and the source agree on a common set of
questions (here it is the question: ‘what is the distance between points $x$
and $y$ ?’).

The approach in this paper has been chosen to implement this information
theoretic viewpoint in its simplest form.

###  7.2. A gedanken experiment for the measurement of distance

The power of the isometric embedding problem lies in its minimalism and
generality. It is minimal because equation (1.1) captures the equality of
infinitesimal lengths in $(M,g)$ and on $u(M)\subset(\mathbb{R}^{q},e)$. It is
general because Nash’s embedding theorems hold for all Riemannian manifolds
$(M,g)$ under mild regularity and toplogical assumptions. The construction of
Gibbs measures in mathematical physics is typically guided by an underlying
dynamical system. Thus, in order to construct Gibbs measures supported on
solutions to (1.1) it is necessary to use a dynamical system that respects the
minimalism of the problem.

The gedanken experiment that underlies Theorem 4 is this: How do we model the
measurement of length for an embedding of $M$? The ideal model must be
consistent with the rigorous analysis of equation (1.1), as well as its
scientific applications.

Theorem 4 is formulated to capture the minimal thermodynamics of embedding in
the following sense. It formalizes the idea that $u$ is isometric if and only
if intrinsic and extrinsic observers measure exactly the same length between
each pair of points. Of course, this is what the PDE (1.1) means implicitly.
What is new is the idea that the use of Brownian motion on $(M,g)$ provides an
explicit model for the measurement of distance by distinct explorers of the
space $u(M)$.

More precisely, the equivalence in law of the two stochastic processes $Y_{t}$
and $Z_{t}$, corresponds conceptually to the fact that intrinsic and extrinsic
Brownian observers explore the space $\Sigma=u(M)$ in an equivalent manner.
The reason we distinguish between $W_{t}\in\mathbb{R}^{n}$ and
$B_{t}\in\mathbb{R}^{q}$ in the SDE’s defining $U_{t}$ and $Z_{t}$ is that the
use of distinct probability spaces to construct Brownian motion formalizes the
notion of measurement of distance in different frames of reference. The heat
kernel $k_{\tau}^{Y}(x,y)$ for the process $Y_{t}$ provides the best guess of
the distance $d_{g}(x,y)$ between $x$ and $y$ as measured by an intrinsic
observer at temperature $\tau$, with $d_{g}(x,y)^{2}\approx-2\tau\log
k_{\tau}^{Y}(x,y)$. Similarly, the heat kernel $k_{\tau}^{Z}$ provides an
approximation to measurements by an extrinsic observer. Thus, these two
constructions of Brownian motion, along with the use of Varadhan’s lemma,
provide a minimal model for the measurement of distances by each observer.

###  7.3. Curvature, RLE and stochastic mechanics

Theorem 2 may also be interpreted along the lines of Nelson’s stochastic
mechanics [22, 53]. The correction term $\tfrac{1}{2}H$ in equation (2.2) may
be interpreted as a constraint that ensures that the process $Z_{t}$ does not
leave $\Sigma$. It is the analog in stochastic mechanics of the centripetal
acceleration in Newtonian mechanics. The interpretation of (minus a half) mean
curvature as a constraint force is an analog of Maxwell’s derivation of the
pressure on the boundary of a domain containing a hard-sphere gas using the
time average of momentum transfer during collision. Unlike Maxwell, however,
we do not postulate the existence of a hard sphere gas, only the existence of
Wiener measure. Theorem 2 and (conjectural) equation (2.5) provide a rigorous
microscopic explanation for the pressure, even in the low-regularity regime.

An imprecise, but useful, caricature of this effect is as follows: subtract
the drift from the Stratonovich equation (1.7) to obtain the formal Itô SDE

$d\tilde{Z}_{t}=\sum_{a=1}^{q}P_{a}\left|{}_{\tilde{Z}_{t}}\right.\circ
dB_{t}^{a}-\frac{1}{2}H(\tilde{Z}_{t})\,dt\stackrel{{\scriptstyle\mathrm{formally}}}{{=}}\sum_{a=1}^{q}P_{a}\left|{}_{\tilde{Z}_{t}}\right.dB_{t}^{a}.$
(7.1)

The formal Itô equation is suggestive because it tells us that the
fluctuations are tangential. However, Theorem 2 tells us that $\tilde{Z}_{t}$
does not lie in $u(M)$: it is pushed outwards normally by the Itô correction.

We have used this insight to provide a geometric construction of Dyson
Brownian motion [31], to construct an analog of Dyson Brownian motion in the
Siegel half-space [49], to find Gibbs sampling algorithms for low-rank psd
matrices [68, 69], and to shed new light on deep learning and optimization
[48]. When applied to the embedding problem itself, we see that we may replace
Nash’s discrete scheme outlined in Section 2.3 by a geometric stochastic flow
of a short embedding that uses only tangential noise to ‘push normally
outwards’. The more subtle issue, however, is to determine the most
fundamental evolution equation of this nature. Here we may use random matrix
theory as a guide.

In each of the above examples, symmetries simplify the analysis, allowing us
to formulate a unifying Riemannian Langevin equation (RLE) of the form
111Equation (7.2) is formal because, like equation (7.1), it is written in the
Itô form, not the Stratonovich form. We have chosen this form so that the
correspondence with the (Euclidean) Langevin equation is clear.

$\displaystyle dX_{t}=\mathrm{grad}_{g}S(X_{t})+dB^{g,\beta}_{t}\,.$ (7.2)

Here $X_{t}$ is a stochastic process on a Riemannian manifold $(M,g)$,
$B^{g,\beta}_{t}$ is intrinsic Brownian motion on $(M,g)$ at inverse
temperature $\beta$, and $S=\log W$ is a Boltzmann entropy such that the
‘number of microstates’ $W$ is the volume of a group orbit. An additional
potential may be included so that the entropy $S$ is replaced by minus a free
energy; however, it is the understanding of $S$ that is most fundamental in
each of these problems.

The stochastic evolution used to establish the existence of Gibbs measures for
the isometric embedding problem and turbulence must adhere strictly to
thermodynamic principles. The above examples reveal clearly that the RLE for
stochastic gradient ascent of entropy is the most fundamental such model. This
leads us to seek natural infinite-dimensional group orbits corresponding to
the isometric embedding problem, defining their entropy $S$ by replacing
volume with a Fredholm determinant. The theory of stochastic flows provides a
powerful tool for this purpose. But as in this paper, this viewpoint leads us
back to classical questions in stochastic flows and random matrix theory,
allowing us to see them in a new light.

The fundamental problem within this class is as follows. Given $(M,g)$ our
task is to determine the low-regularity renormalization of Brownian motion in
the diffeomorphism group $\mathrm{Diff}(M)$, extending work of Baxendale and
Kunita [4, 41]. The renormalization involves resolving a classical degeneracy:
there are many Gaussian sections of $T\mathcal{M}$ whose stochastic flows have
one-point marginals that are Brownian motion [41, Ch.4]. That is, there are
many intrinsic constructions of Brownian motion on $(\mathcal{M},g)$. The
fundamental new idea that is suggested by the embedding-RMT analogy is that a
principled renormalization is not the Eells, Elworthy, Malliavin construction.
Instead, we approach Brownian motion in $\mathrm{Diff}(M)$ using stochastic
gradient flows given by Stratonovich SDE

$dX_{t}=\mathrm{grad}_{g}(\circ\,d\psi_{t}),$ (7.3)

for a stationary random field $\psi_{t}:\mathcal{M}\to\mathbb{R}$. The law of
$\psi$ is determined by a stochastic Nash lemma, replacing [51, Lemma 1] with
an infinite-dimensional matrix-completion problem.

In this way, we see a subtle persistence of Nash’s ideas, even when the
isometric embedding problem is studied with entirely probabilistic techniques.
These RLE will be explained at greater length in forthcoming work by the
authors.

Taken together, our theorems and gedanken experiment, express the idea that
the true nature of the isometric embedding problem lies in understanding the
implicit role of measurement, and gauge invariance, in the character of
physical law.

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