# Integration and stochastic integration
in Gaussian multiplicative chaos

Isao Sauzedde  Isao Sauzedde – LPSM, Sorbonne Université, Paris
<EMAIL_ADDRESS>
###### Abstract.

We show that for $\gamma<\sqrt{4/3}$, it is possible to define the Lévy area
of a planar Brownian motion with the Liouville measure of intermittency
parameter $\gamma$ as the underlying area measure. We also consider the case
of smoother curves, and study some properties of the integration map thus
defined.

###### Key words and phrases:

Green’s formula, Planar Brownian motion, Lévy area, Liouville measure

###### Contents

  1. 1 Strategy of the proof of Theorem 1
  2. 2 Three estimates about the Liouville measures
  3. 3 Comparing $\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$
  4. 4 Estimating $\mathsf{M}_{N}$
  5. 5 Bootstrapping the bounds
  6. 6 The case of smoother curves
  7. 7 Weak Chen’s relation
  8. 8 From weak Chen’s relation to pathwise Chen’s relation
  9. 9 A Kolmogorov type criterion
  10. 10 Uniform estimates in $L^{q}$

## Introduction

In a previous paper [13], we showed a Green formula for $\alpha$-Hölder
continuous planar curves (with $\alpha>\frac{1}{2}$) and for the planar
Brownian motion. It allows us to compute the Young or Stratonovich integral of
a smooth differential $1$-form $\eta$ along the curve $X$ as the surface
integral of the winding function $\theta_{X}$ against the $2$-form ${\rm
d}\eta$. In the Brownian case, the surface integral has be interpreted through
a principal value method.

In the present paper, we look at the same integrals but with the smooth
$2$-form ${\rm d}\eta$ replaced with a random measure, the Liouville measure
of parameter $\gamma$. For a planar Brownian motion independent of the
Liouville measure, we show that, for $\gamma<\sqrt{4/3}$, the principal value
is still well defined and lies in $L^{2}$ of the product probability space. We
also prove continuity, Hölder continuity, and additivity (‘Chen’s relation’)
of the integral thus defined. For the case of $\alpha$-Hölder continuous
curves, we show similar results, provided $\gamma$ is small enough and
$\alpha$ is large enough.

For a continuous curve $X:[s,t]\to\mathbb{R}^{2}$, we denote by $\bar{X}$ the
concatenation of $X$ with the straight line segment from $X_{t}$ to $X_{s}$.
This is an oriented loop, and for all point
$z\in\mathbb{R}^{2}\setminus\operatorname{Range}(\bar{X})$, we denote by
$\theta_{X}(z)\in\mathbb{Z}$ the winding of this oriented loop $\bar{X}$
around the point $z$. Provided $X$ is a smooth curve, Green’s theorem is
easily seen to implies that for all smooth $1$-form $\eta$,

$\int_{\bar{X}}\eta=\int_{\mathbb{R}^{2}}\theta_{X}\;{\rm d}\eta.$ (1)

In [13], we showed that this equality extends to all $\alpha$-Hölder
continuous curves $X$ with $\alpha>\frac{1}{2}$. For this, the left-hand side
must be interpreted as a Young integral. We further proved that a similar
equality also holds for the Brownian motion (in the particular case
$\eta=x\;{\rm d}y$, but we can deduce the general case with a few extra work),
provided we interpret the left-hand side as a Stratonovich integral and the
right-hand side as a kind of principal value. To be more specific, and using
superscripts to indicates the coordinates in $\mathbb{R}^{2}$, and denoting
the Lebesgue measure by $\lambda$, one has almost surely

$\displaystyle\int_{0}^{1}X^{1}\circ{\rm
d}X^{2}+\int_{0}^{1}((1-r)X_{1}+rX_{0})^{1}(X_{0}-X_{1})^{2}\;{\rm d}r$
$\displaystyle=\lim_{K\to+\infty}\int_{\mathbb{R}^{2}}\mathbbm{1}_{\\{|\theta_{X}(z)|\leq
K\\}}\,\theta_{X}(z)\;{\rm d}\lambda(z)$
$\displaystyle=\lim_{K\to+\infty}\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X}(z),K))\;{\rm
d}\lambda(z).$ (2)

The goal of this paper is to study the right-hand sides of (1) and (2), but
with $\\!\;{\rm d}\eta$ (resp. ${\rm d}z$) replaced with a Gaussian
multiplicative chaos, that we denote by $\mathsf{M}$.

In this paper, $\mathsf{M}$ will be a random measure heuristically described
by the formula

$\forall
A\in\mathcal{B}(\mathbb{R}^{2}),\qquad\mathsf{M}(A)=\int_{A}\exp\big{(}\gamma\Phi_{z}-\tfrac{\gamma^{2}}{2}\mathbb{E}[\Phi_{z}^{2}]\big{)}\;{\rm
d}\lambda(z),$ (3)

where $\Phi$ is a centered Gaussian field, and $\gamma\in[0,2)$. It is assumed
that the covariance kernel
$K:\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}_{+}\cup\\{\infty\\}$ of
$\Phi$ takes the form

$K(z,w)=\log_{+}(|z-w|^{-1})+g(z,w),$

where $\log_{+}$ is the positive part of the logarithm and
$g:\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$ is a bounded and
$\mathcal{C}^{2}$-function with bounded derivatives up to order $2$. The
kernel $K$ is well defined, as well as the random field $\Phi$, even though
the logarithmic divergence of $K$ makes $\Phi$ a random distribution rather
than a random function. However, precisely because $\Phi$ is a random
distribution, (3) does not make sense, and the construction of $\mathsf{M}$
cannot rely on this formula. The first construction of a random measure that
is a reasonable candidate to be a mathematical incarnation of (3) was given
using the theory of multiplicative chaos by Kahane [9]. For an introduction to
Gaussian multiplicative chaos, see also [11] and [2]. In general, the larger
$\gamma$, the more irregular the measure. Most of the time in this paper, we
will be working in the case where $\gamma<\sqrt{2}$. We will thus be in the
so-called ‘$L^{2}$-phase’, in which it is relatively easy to define and to
study $\mathsf{M}$, using martingale methods and Hilbertian techniques.

For the reader more familiar with multiplicative chaos, let us mention that
the condition $g\in\mathcal{C}^{2}$ is not necessary to define the measure
$\mathsf{M}$, nor to study the integral along Hölder paths, but it is a
practical one when we look at the integral along Brownian paths.

We will denote by
$(\Omega^{\mathsf{M}},\mathcal{F}^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})$ the
probability space on which $\mathsf{M}$ is defined. We also let
$(\Omega^{X},\mathcal{F}^{X},\mathbb{P}^{X})$ be a second probability space,
on which a planar Brownian motion $X=(X_{t})_{t\in[0,1]}$ is defined. Finally,
we denote by $(\Omega,\mathcal{F},\mathbb{P})$ the product probability space.

The first part of this paper (Sections 1 to 5) is dedicated to the proof of
the following result. Let us emphasize that we stop the Brownian motion at
time $1$.

######  Theorem 1.

Assume $\gamma<\sqrt{4/3}$. Then, $\mathbb{P}$-almost surely, the integral

$\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X}(z),K))\;{\rm d}\mathsf{M}(z)$

admits a limit $\mathbb{A}^{X}$ as $K\to+\infty$.

Moreover, for all $p\in[2,\frac{4}{\gamma^{2}})$, this limit lies in
$L^{p}(\Omega^{X},L^{2}(\Omega^{\mathsf{M}}))$.111 The reader may find
surprising, as we do, that the integrability with respect to $\mathbb{P}^{X}$
depends on $\gamma$, and not the integrability with respect to
$\mathbb{P}^{\mathsf{M}}$. However, the result as it is written is what we
mean.

For $(s,t)\in\Delta=\\{(s,t)\in[0,1]^{2}:s\leq t\\}$, we denote by
$\mathbb{A}_{s,t}^{X}$ the almost surely defined limit,

$\mathbb{A}_{s,t}^{X}=\lim_{K\to+\infty}\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X_{|[s,t]}}(z),K))\;{\rm
d}\mathsf{M}(z).$ (4)

The second part of the paper (Section 6 to 9) is devoted to the study of the
map $(s,t)\mapsto\mathbb{A}_{s,t}^{X}$, and also, for a curve $Y$
deterministic and more regular than a Brownian motion, of the map
$\mathbb{A}_{s,t}^{Y}$ given by

$\mathbb{A}_{s,t}^{Y}=\int_{\mathbb{R}^{2}}\theta_{Y_{|[s,t]}}\;{\rm
d}\mathsf{M}.$ (5)

For a function $\mathbb{A}:\Delta\to\mathbb{R}$ to deserve the name of
‘algebraic $\mathsf{M}$-area enclosed by the curve $Z$’, it should satisfy a
relation that we will call a Chen relation. To write this relation, let us
denote, for all $s\leq u\leq t$, by $T_{s,u,t}$ the triangle with vertices
$Z_{s}$, $Z_{u}$ and $Z_{t}$. Let us also set $\varepsilon_{s,u,t}=1$ if
$Z_{s}$, $Z_{u}$ and $Z_{t}$ are found in this cyclic order along the
positively oriented boundary of $T_{s,u,t}$, and $\varepsilon_{s,u,t}=-1$
otherwise. We say that $\mathbb{A}$ satisfies the Chen relation (relative to
$Z$) if for all $s\leq u\leq t$,

${\mathbb{A}}_{s,t}={\mathbb{A}}_{s,u}+{\mathbb{A}}_{u,t}+\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t}).$

In a nutshell, the results of this second part can be summarized as follows
(some additional properties are given below).

######  Theorem 2.

Let $\gamma<2(\sqrt{2}-1)$, and $Z:[0,1]\to\mathbb{R}^{2}$ be either a planar
Brownian motion or an $\alpha$-Hölder continuous function with
$\alpha>\big{(}2(1-\frac{\gamma^{2}}{4})\big{)}^{-1}$. Then, for all
$(s,t)\in\Delta$, almost surely, $\mathbb{A}_{s,t}^{Z}$ is well-defined. It
admits a modification which is continuous, Hölder continuous, and satisfies
the Chen relation.

Let us also say that $\mathbb{A}$ is $\beta$-regular if there exists a
constant $C$ such that for all $(s,t)\in\Delta$,

$\mathbb{A}_{s,t}\leq C(t-s)^{\beta}.$ (6)

From now on, we set $\nu=2(1-\frac{\gamma^{2}}{4})$.333 This number $\nu$ is
the Hausdorff dimension of a Borel set of full $\mathsf{M}$-measure. However,
the reason why it matters to us is that it is equal to
$\frac{\xi(q)}{q}|_{q=2}=\frac{\xi(2)}{2}$, where $\xi$ the so-called
structure exponent of $\mathsf{M}$. This exponent is defined by the fact that
for $r>0$, the expectation of $\mathsf{M}(B(0,r))^{q}$ is of the order of
$r^{\xi(q)}$.

######  Theorem 3.

Let $\gamma<\sqrt{4/3}$. Then, $(\mathbb{A}_{s,t}^{X})_{(s,t)\in\Delta}$
admits a modification which satisfies the Chen relation.

Let $\gamma<2(\sqrt{2}-1)\simeq 0.81$. Then,
$(\mathbb{A}_{s,t}^{X})_{(s,t)\in\Delta}$ admits a modification which
satisfies the Chen relation, is $\beta$-regular for all
$\beta<\min(\frac{1}{2}-\frac{\gamma^{2}}{4},1+\frac{\gamma^{2}}{4}-\sqrt{2}\gamma)$,
and $\beta$-Hölder continuous for all
$\beta<\frac{1}{2}(1+\frac{\gamma^{2}}{4}-\sqrt{2}\gamma)$.

######  Theorem 4.

1\. For all $\gamma<2$ and $\alpha>\frac{1}{2}$, for all $\alpha$-Hölder
continuous function $Y:[0,1]\to\mathbb{R}^{2}$, for all $s<t$,
$\mathbb{P}^{\mathsf{M}}$-almost surely, the integral (5) is well-defined. The
function $\mathbb{A}^{Y}$ admits a modification which satisfies the Chen
relation,

2.1. For all $\gamma<\sqrt{2}\simeq 1.4$ and $\alpha>\frac{1}{\nu}$, the
function $\mathbb{A}^{Y}$ admits a modification which satisfies the Chen
relation, is almost surely $\beta$-regular for all

$\beta<\beta_{0}=\left\\{\begin{array}[]{ll}\alpha\nu-1&\mbox{if
}\alpha\geq\gamma^{-2},\\\
2\alpha(1+\frac{\gamma^{2}}{4})-2\gamma\sqrt{\alpha}&\mbox{if
}\alpha\in[\frac{3-2\sqrt{2}}{2}\gamma^{-2},\gamma^{-2}]\\\
\alpha\nu-\frac{1}{2}&\mbox{if
}\alpha\leq\frac{3-2\sqrt{2}}{2}\gamma^{-2}.\end{array}\right.$

2.2. For all $\gamma<2(\sqrt{2}-1)\simeq 0.81$ and $\alpha>\frac{1}{\nu}$,
this modification is $\beta^{\prime}$-Hölder continuous for
$\beta^{\prime}=\min((1-\sqrt{2}\gamma+\frac{\gamma^{2}}{4})\alpha,\beta)$,
for all $\beta<\beta_{0}$.

Figure 1 below represents the different ranges that appear in these results.

Figure 1.  Top left: Above the red line, $\mathbb{A}_{s,t}$ is well-defined
for given $s$ and $t$, and satisfies the Chen relation (up to modification).
Above the black curve, it admits a continuous modification. On the left of the
red point, $\mathbb{A}_{s,t}$ is well-defined for a Brownian motion, and
satisfies the Chen relation. On the left of the black point, it admits a
continuous modification. Remark the black dot is below the black curve. Top
right, bottom left, bottom right: Lower bound on the exponent
$\beta=\beta(\alpha,\gamma)$ of regularity. The black curves on the bottom
figures correspond to the values $\alpha=\gamma^{-2}$ for which the expression
of our lower bounds changes. The green curve in the bottom right figure
corresponds to the values $\alpha=\frac{3-2\sqrt{2}}{2\gamma^{2}}$, for which
the expression of our lower bounds changes a second time. The red line on the
bottom left figure is the correct value of $\beta$ in the classical case
$\mathsf{M}=\lambda$ (‘$\gamma=0$’)

During the proof, we will also show that $\mathbb{A}^{Y}_{s,t}$ is
‘stochastically $2\alpha$-regular’, in the following sense.

######  Theorem 5.

Assume that either $\gamma\in[0,2)$, $\alpha>\frac{1}{2}$ and $\rho=1$, or
that $\gamma<\sqrt{2}$, $\alpha>\frac{1}{\nu}$ and $\rho\in[1,\alpha\nu)$.
Then, for all $(s,t)\in\Delta$, $\mathbb{A}^{Y}_{s,t}$ lies in
$L^{\rho}(\Omega^{\mathsf{M}})$, and there exists a constant $C$ which does
not depend on $(s,t)$ and such that
$\|\mathbb{A}^{Y}_{s,t}\|_{L^{\rho}(\Omega^{\mathsf{M}})}\leq
C|t-s|^{2\alpha}$.

The two parts of the paper (Sections 1 to 5, Sections 6 to 10) are mostly
independent, but reading Proposition 1.1 and Lemmas 2.1 and 2.2 is necessary
to understand the second part.

##  1\. Strategy of the proof of Theorem 1

Let us recall that $X:[0,1]\to\mathbb{R}^{2}$ is a planar Brownian motion
defined on $(\Omega^{X},\mathcal{F}^{X},\mathbb{P}^{X})$, and $\theta_{X}$ is
the winding function associated with $X$. The random measure $\mathsf{M}$ is
defined on
$(\Omega^{\mathsf{M}},\mathcal{F}^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})$, and
is formally given by
$\mathsf{M}=e^{\gamma\phi-\frac{\gamma^{2}}{2}\mathbb{E}[\phi^{2}]}\lambda$,
where $\lambda$ is the Lebesgue measure and $\phi$ is a centered Gaussian
field on $\mathbb{R}^{2}$ with covariance kernel
$K:(z,w)\mapsto\log_{+}(|z-w|)+g(z,w)$, for a bounded $\mathcal{C}^{2}$
function $g$ with bounded derivatives up to order $2$. From now on and until
Section 5 included, we will assume that $\gamma<\sqrt{4/3}$. We are looking at
the existence of a limit, as $K\to+\infty$, of the integral with ‘cut-off’

$\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X}(z),K))\;{\rm d}z.$

Since $\theta_{X}$ takes its values in $\mathbb{Z}$, we can rewrite this
integral as

$\sum_{n=-\infty}^{+\infty}\max(-K,\min(n,K))\mathsf{M}(\\{z\in\mathbb{R}^{2}:\theta_{X}(z)=n\\}),$

which is more conveniently written

$\sum_{n=1}^{+\infty}\min(n,K)\big{(}\mathsf{M}(\\{z\in\mathbb{R}^{2}:\theta_{X}(z)=n\\})-\mathsf{M}(\\{z\in\mathbb{R}^{2}:\theta_{X}(z)=-n\\})\big{)}.$

Performing a summation by parts, this is finally equal to

$\sum_{N=1}^{K}\big{(}\mathsf{M}(\\{z\in\mathbb{R}^{2}:\theta_{X}(z)\geq
N\\})-\mathsf{M}(\\{z\in\mathbb{R}^{2}:\theta_{X}(z)\leq-N\\})\big{)}.$

This is the expression we will study. We will show that the general term goes
to zero sufficiently fast for the sum to be convergent. Consequently, for
$N\geq 1$, we define the following sets:

$\mathcal{D}_{N}=\\{z\in\mathbb{R}^{2}\setminus{\rm
Range}(\bar{X}):\theta_{X}(z)\geq N\\},\ \
\mathcal{D}_{-N}=\\{z\in\mathbb{R}^{2}\setminus{\rm
Range}(\bar{X}):\theta_{X}(z)\leq-N\\}.$ (7)

For a random variable $Z$ on $\Omega$, we set

$\|Z\|_{p,q}=\|Z\|_{L^{p}(\Omega^{X},L^{q}(\Omega^{\mathsf{M}}))}=\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[Z^{q}]^{\frac{p}{q}}]^{\frac{1}{p}}.$

Theorem 1 is then a direct consequence of the following estimation.

######  Proposition 1.1.

For all $p\in[2,\frac{4}{\gamma^{2}})$, there exists $\varepsilon>0$ and $C$
such that for all $N$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{-1-\varepsilon}.$

Let us remark that this is only possible thanks to the compensation between
$\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}(\mathcal{D}_{-N})$. Indeed, each
of these sets has a Lebesgue measure which is equivalent as $N$ tends ton
infinity (in $L^{q}$ for all $q$, and in the almost sure sense) to
$\frac{1}{2\pi N}$, so that even $\|\mathsf{M}(\mathcal{D}_{N})\|_{p,1}$ is
two large for the sum to be convergent. The fact that a compensation occurs
can be informally understood as a consequence of the fact that the sets
$\mathcal{D}_{N}$ and $\mathcal{D}_{-N}$ are extremely similar when $N$ is
large. For a better understanding of the asymptotic behaviour of
$\mathcal{D}_{N}$, we refer to our article [14].

The strategy for the proof of Proposition 1.1 is similar to the one we used in
[13] (Sections 3.2 and 3.3), and then in [14]. In short, we will decompose the
quantity $\mathsf{M}(\mathcal{D}_{N})$ into a sum of smaller quantities,
corresponding to contributions of pieces of the Brownian trajectory.

To this end, we fix some $t>0$, which should think of as being very small. For
a positive integer $N$, we set $T=\lfloor N^{t}\rfloor$. For
$i\in\\{1,\dots,T\\}$, we define $X^{i}$ to be the restriction of $X$ to
$[(i-1)T^{-1},iT^{-1}]$, $\bar{X}^{i}$ the concatenation of $X^{i}$ with the
segment between its endpoints, and $\theta^{i}(z)$ the number of times
$\bar{X}^{i}$ winds around $z$. We set

$\mathcal{D}^{i}_{N}=\\{z\in\mathbb{R}^{2}\setminus\operatorname{Range}(\bar{X}^{i}):\theta^{i}(z)\geq
N\\},\ \
\mathcal{D}^{i}_{-N}=\\{z\in\mathbb{R}^{2}\setminus\operatorname{Range}(\bar{X}^{i}):\theta^{i}(z)\leq-N\\}.$
(8)

Finally, we define

$\mathsf{M}_{N}=\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N}).$ (9)

This random variable should be thought of as a proxy for
$\mathsf{M}(\mathcal{D}_{N})$, when $N$ is large. To control the difference
between $\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$ will be the subject
of Section 3, in which we will prove the following bound.

######  Proposition (3.7).

Let $p\in(1,+\infty)$ and $\varepsilon>0$. Then, for $t$ small enough, there
exists $C$ such that for all $N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}\leq
CN^{-\frac{3\nu}{4}+2t+\varepsilon}.$

In the case where the random measure $\mathsf{M}$ has a law invariant by
translation, the variables $\mathsf{M}(\mathcal{D}^{i}_{N})$ are identically
distributed. If $\mathsf{M}$ also has a nice behaviour under scaling, then
these variables are distributed as a scaled version of
$\mathsf{M}(\mathcal{D}_{N})$, and we end up with the informal relation

$\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\simeq\sum_{i=1}^{T}(\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N}))=\sum_{i=1}^{T}f(T)Z_{i},$

where the $Z_{i}$ are random variables distributed as
$\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})$. Let us assume that
we can bound nicely the scaling function $f(T)$, and show that the variables
$Z_{i}$ are not too strongly correlated. Then, inserting a bound on
$\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})$ on the right-hand
side might lead to a _better_ bound on
$\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})$. We refer to this
as the _bootstrap_. The control on the correlation takes the following form.

Notation. The notation $f(N)\leq N^{a+o(1)}$ means that for all $h>0$, there
exists $C$ such that for all $N\geq 1$, $f(N)\leq CN^{a+h}$.

######  Lemma (4.2).

For $t,\varepsilon>0$, and $i,j\in\\{1,\dots,T\\}$, let
$F_{i,j}=\\{|X_{(i+1)T^{-1}}-X_{jT^{-1}}|\geq
T^{-\frac{1}{2}+\varepsilon}\\}$. For all
$p\in\big{[}2,\frac{4}{\gamma^{2}})$, there exists a constant $C$ such that
for all $N\geq 1$,

$\mathbb{E}^{X}\big{[}\big{|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j=1}^{T}\mathbbm{1}_{F_{i,j}}(\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N})(\mathsf{M}(\mathcal{D}^{j}_{N})-\mathsf{M}(\mathcal{D}^{j}_{-N}))\big{]}\big{|}^{\frac{p}{2}}\big{]}^{\frac{1}{p}}\\\
\leq
C\log(N+1)^{\frac{1}{p}}T^{\frac{\varepsilon}{2}}N^{-1}\big{(}N^{-\frac{1}{4}+o(1)}+T^{\frac{\gamma^{2}}{4}-\frac{1}{p}}\big{)}.$
(10)

The bootstrap then takes the following form. In the statement, there appears a
set $\mathcal{T}(\mathsf{M})$ of random measures that will be defined
precisely in Section 2. For the moment, suffice it to say that all the
elements of $\mathcal{T}(\mathsf{M})$ look like $\mathsf{M}$, in the sense
that they are deduced from $\mathsf{M}$ by (possibly random) translations and
symmetries. In many cases of interest, namely when $g$ is invariant by
translation, $\mathcal{O}(\mathsf{M})=\\{\mathsf{M}\\}$.

######  Lemma (5.1).

Consider $t>0,p\in[2,\frac{4}{\nu}]$ and $\zeta\in\mathbb{R}$. Assume that
there exists a constant $C$ such that for all
$\mathsf{M}^{\prime}\in\mathcal{T}(\mathsf{M})$ and all $N\geq 1$,

$\|\mathsf{M}^{\prime}(\mathcal{D}_{N})-\mathsf{M}^{\prime}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{\zeta}.$

Then, for all $\varepsilon>0$, there exists a constant $C^{\prime}$ such that
for all $N\geq 1$,

$\mathbb{E}^{X}\big{[}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{1\leq i\leq j\leq
T}\mathbbm{1}_{F_{i,j}^{c}}|(\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N}))(\mathsf{M}(\mathcal{D}^{j}_{N})-\mathsf{M}(\mathcal{D}^{j}_{-N}))|\big{]}^{\frac{p}{2}}\big{]}^{\frac{1}{p}}\leq
C^{\prime}\log(T+1)^{\frac{1}{p}}T^{\frac{\gamma^{2}}{4}-\frac{1}{p}+\varepsilon}N^{\zeta}.$

These propositions, properly combined, will allow us to prove the theorem.
During the proof, we will use the following estimation that we proved in [13,
Theorem 1.5]. For a measurable set $A$, the notation $|A|$ always denotes the
Lebesgue measure of $A$.

######  Lemma 1.2.

For all $p\in[2,+\infty)$ and all $\delta<\frac{1}{2}$, there exists a
constant $C$ such that for all $N\in\mathbb{N}$,

$\mathbb{E}\Big{[}\big{|}N|\mathcal{D}_{N}|-\frac{1}{2\pi}\big{|}^{p}\Big{]}^{\frac{1}{p}}\leq
CN^{-\delta}.$

The next section contains the material that will allow us to estimate the
$\mathbb{P}^{\mathsf{M}}$-expectation of some functionals of $\mathsf{M}$.

##  2\. Three estimates about the Liouville measures

In this section, we assume that $\gamma<\sqrt{2}$. Recall that $\nu$ is the
positive real number

$\nu=2-\frac{\gamma^{2}}{2}.$

One of the advantages of working in the $L^{2}$-phase (that is, under the
assumption $\gamma<\sqrt{2}$) is that one has the following explicit formula.

######  Lemma 2.1.

Let $A,B$ be two Borel subsets of $\mathbb{R}^{2}$ with finite Lebesgue
measure. Then,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)\mathsf{M}(B)]=\int_{A\times
B}\exp(\gamma^{2}K(z,w))\;{\rm d}z\;{\rm d}w.$

###### Proof.

We first do an informal computation, which is entirely valid when the kernel
$K$ is replaced with a continuous kernel $\tilde{K}$ (in which case the
centered Gaussian field $\phi$ with kernel $\tilde{K}$ is defined pointwise):

$\displaystyle\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)\mathsf{M}(B)]$
$\displaystyle=\mathbb{E}^{\mathsf{M}}\Big{[}\int_{A\times
B}e^{\gamma\phi_{z}-\frac{\gamma^{2}}{2}\mathbb{E}^{\mathsf{M}}[\phi_{z}^{2}]}e^{\gamma\phi_{w}-\frac{\gamma^{2}}{2}\mathbb{E}^{\mathsf{M}}[\phi_{w}^{2}]}\;{\rm
d}z\;{\rm d}w\Big{]}$ $\displaystyle=\int_{A\times
B}\mathbb{E}^{\mathsf{M}}\big{[}e^{\gamma(\phi_{z}+\phi_{w})-\frac{\gamma^{2}}{2}\mathbb{E}^{\mathsf{M}}[(\phi_{z}+\phi_{w})^{2}]}\big{]}e^{\gamma^{2}\tilde{K}(z,w)}\;{\rm
d}z\;{\rm d}w$ $\displaystyle=\int_{A\times
B}e^{\gamma^{2}\tilde{K}(z,w)}\;{\rm d}z\;{\rm d}w.$

For a smooth mollifier $\theta$ and a centered Gaussian field $\Phi$ with
kernel $K$, we define $\theta_{\varepsilon}$ as the function
$\varepsilon^{-2}\theta(\tfrac{\cdot}{\varepsilon})$, and $\Phi_{\varepsilon}$
as the convolution of $\Phi$ with $\theta_{\varepsilon}$. For a Borel
measurable set $A$ with finite Lebesgue measure, set also

$\mathsf{M}_{\varepsilon}(A)=\int_{A}e^{\gamma\Phi_{\varepsilon}(z)-\frac{\gamma^{2}}{2}\mathbb{E}[(\Phi_{\varepsilon}(z))^{2}]}\;{\rm
d}\lambda(z).$

Then, when $\varepsilon\to 0$, $\mathsf{M}_{\varepsilon}(A)$ converges in
$L^{2}$ toward $\mathsf{M}(A)$ (see for example [12, Theorem 2.3]444 The cited
theorem states convergence in probability, without assuming $\gamma<\sqrt{2}$,
but the proof is about the $L^{2}$ convergence provided $\gamma<\sqrt{2}$.
Remark that it is taken as a definition for the measure $\mathsf{M}$ (up to
some argument to make it not only a collection of random variables, but a
measure). The original construction of Kahane is rather different.).

It follows that for any two Lebesgue measurable sets $A$ and $B$ with finite
Lebesgue measures,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)\mathsf{M}(B)]=\lim_{\varepsilon\to
0}\mathbb{E}^{\mathsf{M}}[\mathsf{M}_{\varepsilon}(A)\mathsf{M}_{\varepsilon}(B)].$

On the other hand, the previous computation shows that

$\displaystyle\mathbb{E}^{\mathsf{M}}[\mathsf{M}_{\varepsilon}(A)\mathsf{M}_{\varepsilon}(B)]$
$\displaystyle=\int_{A\times
B}e^{\gamma^{2}\mathbb{E}^{\mathsf{M}}[X_{\varepsilon}(z)X_{\varepsilon}(w)]}\;{\rm
d}z\;{\rm d}w$ $\displaystyle\underset{\varepsilon\to
0}{\longrightarrow}\int_{A\times B}e^{\gamma^{2}K(z,w)}\;{\rm d}z\;{\rm d}w.$

The last convergence follows from pointwise convergence of
$\mathbb{E}[X_{\varepsilon}(z)X_{\varepsilon}(w)]$ toward $K(z,w)$, and
dominated convergence theorem (see [12, Theorem 2.3] again). ∎

According to , the scaling properties of the measure $\mathsf{M}$ are such
that, in the case where $K(z,w)=\log_{+}(\ |z+w|^{-1})$, for a set $A\subset
B(0,\frac{1}{2})$ and $r<1$, we have

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(rA)^{2}]=r^{2\nu}\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}],$

where $\nu$ is the constant defined at the beginning of this section. We
deduce that for each such set $A$, there exists a constant $C$ such that for
all $r\leq 1$,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(rA)^{2}]\leq C|rA|^{\nu}.$

We can actually choose $C$ such that the two terms are equal. The following
lemma states that the constant $C$ can be chosen to be uniform over all
measurable sets $A$.

######  Lemma 2.2.

There exists $C$ such that for all measurable set $A\subset\mathbb{R}^{2}$,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}]\leq C(|A|^{\nu}+|A|^{2}).$

###### Proof.

From the previous lemma, we know that

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}]=\int_{A^{2}}\exp(\gamma^{2}K(z,w))\;{\rm
d}z\;{\rm d}w.$

Since $K(z,w)\leq\log_{+}(|z-w|^{-1})+c$ for some constant $c$,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}]\leq
e^{\gamma^{2}c}\int_{A^{2}}\exp(\gamma^{2}\log_{+}(|z-w|^{-1}))\;{\rm
d}z\;{\rm d}w=e^{\gamma^{2}c}\int_{A^{2}}\max(|z-w|^{-\gamma^{2}},1)\;{\rm
d}z\;{\rm d}w.$

Let $r=|A|^{\frac{1}{2}}\pi^{-\frac{1}{2}}$ be the radius of a disk of the
same area as $A$, and

$S_{r}=\sup_{A^{\prime}:|A^{\prime}|=\pi
r^{2}}\int_{A^{\prime}}\max(|z|^{-\gamma^{2}},1)\;{\rm d}z.$

The supremum is achieved by the ball $B(0,r)$. By considering separately the
cases $r\leq 1$ and $r\geq 1$, we find that
$S_{r}\leq\frac{2\pi}{2-\gamma^{2}}r^{2-\gamma^{2}}+\pi r^{2}$.

By invariance under translations, $S_{r}$ is also equal to

$\sup_{w\in\mathbb{R}^{2}}\sup_{A^{\prime}:|A^{\prime}|=\pi
r^{2}}\int_{A^{\prime}}\max(|z-w|^{-\gamma^{2}},1)\;{\rm d}z.$

Thus,

$\displaystyle\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}]$ $\displaystyle\leq
e^{\gamma^{2}c}\int_{A^{2}}\max(|z-w|^{-\gamma^{2}},1)\;{\rm d}z\;{\rm d}w$
$\displaystyle\leq e^{\gamma^{2}c}|A|S_{r}\leq
e^{\gamma^{2}c}\Big{(}\frac{2\pi^{\frac{\gamma^{2}}{2}}}{2-\gamma^{2}}|A|^{2-\frac{\gamma^{2}}{2}}+|A|^{2}\Big{)}.$

This concludes the proof. ∎

The next estimate, should be understood as a quantitative version of the
following idea. Let first $A_{1}$, $A_{2}$ and $B$ be three sets such that
$A_{1}$ and $A_{2}$ are very close from each other, have comparable Lebesgue
measures, and are far from $B$; then,
$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A_{1})\mathsf{M}(B)]$ should be very close
to $\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A_{2})\mathsf{M}(B)]$. Let now $A_{1}$,
$A_{2}$, $B_{1}$, and $B_{2}$ be four sets such that

  * $\scriptstyle\diamond$

$A_{1}$ and $A_{2}$ are very close from each other, and have comparable
Lebesgue measures,

  * $\scriptstyle\diamond$

$B_{1}$ and $B_{2}$ are very close from each other, and have comparable
Lebesgue measures,

  * $\scriptstyle\diamond$

The $A_{i}$ are far from the $B_{i}$.

Then, the expectation
$\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(A_{2})-\mathsf{M}(A_{1}))(\mathsf{M}(B_{2})-\mathsf{M}(B_{1}))]$
should be even smaller than
$\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(A_{2})-\mathsf{M}(A_{1}))\mathsf{M}(B_{i})]$.
Remark that the control on the ‘four terms’ expectation should depend on the
distance between the $A_{i}$, the distance between the $B_{i}$, the distance
between $A_{i}$ and $B_{i}$, the Lebesgue measure of these four sets, and
finally the difference between their Lebesgue measure. Hence, we do not expect
a very short bound to appear.

For two disjoint sets $A$ and $B$, let us define

$\displaystyle d_{\sup}(A,B)$ $\displaystyle=\sup\\{d(z,z^{\prime}):z\in
A,z^{\prime}\in B\\},$ $\displaystyle d_{\inf}(A,B)$
$\displaystyle=\min(\inf\\{d(z,z^{\prime}):z\in A,z^{\prime}\in B\\},1),$

as illustrated by Figure 2 below.

Figure 2.  A typical position for the sets in Lemma 2.3, and some of the
‘distances’ between them. A typical choice for these sets would be
$A_{1}=\mathcal{D}^{i}_{N}$, $A_{2}=\mathcal{D}^{i}_{-N}$,
$B_{1}=\mathcal{D}^{j}_{N}$, and $B_{2}=\mathcal{D}^{j}_{-N}$.

######  Lemma 2.3.

There exists $C$ such that for all bounded Borel measurable sets
$A_{1},A_{2},B_{1},B_{2}$ with
$4\max(d_{\sup}(A_{1},A_{2}),d_{\sup}(B_{1},B_{2}))\leq d_{\inf}(A_{1},B_{1})$
and $|A_{1}|,|A_{2}|,|B_{1}|,|B_{2}|\leq 1$,

$|\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(A_{1})-\mathsf{M}(A_{2}))(\mathsf{M}(B_{1})-\mathsf{M}(B_{2})]|\leq
C\Big{(}d_{\inf}(A_{1},B_{1})^{-\gamma^{2}}\big{|}|B_{1}|-|B_{2}|\big{|}\big{|}|A_{1}|-|A_{2}|\big{|}\\\
+\big{(}d_{\sup}(A_{1},A_{2})+d_{\sup}(B_{1},B_{2})\big{)}d_{\inf}(A_{1},B_{1})^{-1-\gamma^{2}}\big{(}|A_{1}|\big{|}|B_{1}|-|B_{2}|\big{|}+\big{|}|A_{2}|-|A_{1}|\big{|}|B_{1}|\big{)}\\\
+d_{\inf}(A_{1},B_{1})^{-2-\gamma^{2}}d_{\sup}(A_{1},A_{2})d_{\sup}(B_{1},B_{2})|A_{1}||B_{1}|\Big{)}.$
(11)

Let us remark that the exponent $\gamma$, which is carried by $|A|$ when
looking at $\mathbb{E}^{\mathsf{M}}[\mathsf{M}(A)^{2}]$, is here carried
instead by $d_{\inf}(A_{1},B_{1})$. Remark also that all these terms but the
last vanish if $|A_{1}|=|A_{2}|$ and $|B_{1}|=|B_{2}|$.

###### Proof.

It is a simple computation. Using the explicit formula given by Lemma 2.1, we
have

$\displaystyle\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(A_{1})-\mathsf{M}(A_{2}))(\mathsf{M}(B_{1})-\mathsf{M}(B_{2})]$
$\displaystyle=\int_{A_{1}\times B_{1}}e^{\gamma K(z_{1},w_{1})}\;{\rm
d}z_{1}\;{\rm d}w_{1}-\int_{A_{1}\times B_{2}}e^{\gamma K(z_{1},w_{2})}\;{\rm
d}z_{1}\;{\rm d}w_{2}$ $\displaystyle\hskip 28.45274pt-\int_{A_{2}\times
B_{1}}e^{\gamma K(z_{2},w_{1})}\;{\rm d}z_{2}\;{\rm d}w_{1}+\int_{A_{2}\times
B_{2}}e^{\gamma K(z_{2},w_{2})}\;{\rm d}z_{2}\;{\rm d}w_{2}$

Writing $Q=A_{1}\times A_{2}\times B_{1}\times B_{2}$, the right-hand side of
this equation is equal to

$\frac{1}{|Q|}\int_{Q}\Big{(}|A_{1}||B_{1}|(e^{\gamma
K(z_{1},w_{1})}-e^{\gamma K(z_{1},w_{2})}-e^{\gamma K(z_{2},w_{1})}+e^{\gamma
K(z_{2},w_{2})})\\\ +|A_{1}|(|B_{2}|-|B_{1}|)(e^{\gamma
K(z_{2},w_{2})}-e^{\gamma K(z_{1},w_{2})})+(|A_{2}|-|A_{1}|)|B_{1}|(e^{\gamma
K(z_{2},w_{2})}-e^{\gamma K(z_{2},w_{1})})\\\
+(|A_{2}|-|A_{1}|)(|B_{2}|-|B_{1}|)e^{\gamma K(z_{2},w_{2})}\Big{)}\;{\rm
d}z_{1}\;{\rm d}z_{2}\;{\rm d}w_{1}\;{\rm d}w_{2}.$

This, in absolute value, is less than

$\displaystyle\sup_{(z_{1},z_{2},w_{1},w_{2})\in Q}\Big{(}$
$\displaystyle|A_{1}||B_{1}||e^{\gamma K(z_{1},w_{1})}-e^{\gamma
K(z_{1},w_{2})}-e^{\gamma K(z_{2},w_{1})}+e^{\gamma K(z_{2},w_{2})}|$
$\displaystyle+|A_{1}|\big{|}|B_{2}|-|B_{1}|\big{|}|e^{\gamma
K(z_{2},w_{2})}-e^{\gamma
K(z_{1},w_{2})}|+\big{|}|A_{2}|-|A_{1}|\big{|}|B_{1}||e^{\gamma
K(z_{2},w_{2})}-e^{\gamma K(z_{2},w_{1})}|$ $\displaystyle\hskip
170.71652pt+\big{|}|A_{2}|-|A_{1}|\big{|}\big{|}|B_{2}|-|B_{1}|\big{|}e^{\gamma
K(z_{2},w_{2})}\Big{)}.$

We set $D_{A}$ a ball of diameter $d_{\sup{}}(A_{1},A_{2})$ containing $A_{1}$
and $A_{2}$, and $D_{B}$ a ball of diameter $d_{\sup{}}(B_{1},B_{2})$
containing $B_{1}$ and $B_{2}$, so tat the last expression is less than

$|A_{1}||B_{1}|d_{\sup{}}(A_{1},A_{2})d_{\sup{}}(B_{1},B_{2})\sup_{z\in
D_{A},w\in D_{B}}\|\nabla^{2}e^{\gamma K(z,w)}\|\\\
+|A_{1}|\big{|}|B_{2}|-|B_{1}|\big{|}d_{\sup{}}(A_{1},A_{2})\sup_{z\in
D_{A},w_{2}\in B_{2}}\|\nabla e^{\gamma K(z,w_{2})}\|\\\
+\big{|}|A_{2}|-|A_{1}|\big{|}|B_{1}|d_{\sup{}}(B_{1},B_{2})\sup_{z_{2}\in
A_{2},w\in D_{B}}\|\nabla e^{\gamma K(z_{2},w)}\|\\\
+\big{|}\\!|A_{2}|-|A_{1}|\\!\big{|}\big{|}\\!|B_{2}|-|B_{1}|\\!\big{|}\sup_{z_{2}\in
A_{2},w_{2}\in B_{2}}e^{\gamma K(z_{2},w_{2})}.$

The three suprema appearing in this expression are respectively less than
$Cd_{\inf{}}(D_{A},D_{B})^{-2-\gamma^{2}}$,
$Cd_{\inf{}}(D_{A},B_{2})^{-1-\gamma^{2}}$ and
$Cd_{\inf{}}(A_{2},B_{2})^{-\gamma^{2}}$, where the constant $C$ depends on
the exact expression of the kernel $K$ but not on the four sets
$A_{1}$,$A_{2}$,$B_{1}$,$B_{2}$.

Since $d_{\inf{}}(D_{A},D_{B})\geq
d_{\inf{}}(A_{1},B_{1})-d_{\sup{}}(A_{1},A_{2})-d_{\sup{}}(B_{1},B_{2})\geq\frac{1}{2}d_{\inf{}}(A_{1},B_{1})$,
we deduce the announced inequality.

∎

For reasons that will appear only at the end of the proof, we need to consider
not only the random measure $\mathsf{M}$, but also measures obtained by
possibly random (but independent from $\mathsf{M}$) translation of
$\mathsf{M}$.

For $z\in\mathbb{R}^{2}$, we define the random measure $\tau_{z}(\mathsf{M})$
by setting, for all Borel measurable set $A$,

$~\tau_{z}(\mathsf{M})(A)=\mathsf{M}(A+z).$

For a probability law $p$ on $\mathbb{R}^{2}$, we also set
$\tau_{p}(\mathsf{M})=\int_{\mathbb{R}^{2}}\tau_{z}(\mathsf{M})\;{\rm d}p(z)$.
Finally, we define

$\mathcal{T}(\mathsf{M})=\\{\tau_{p}(\mathsf{M}):p\mbox{ is a probability law
on }\mathbb{R}^{2}\\}.$

For a random measure $\tilde{\mathsf{M}}$, let us define

$\displaystyle|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{0}$
$\displaystyle=\inf\big{\\{}C>0:\forall A\text{ Borel set with }|A|\leq 1,\
\mathbb{E}^{\mathsf{M}}[\tilde{\mathsf{M}}(A)^{2}]\leq
C|A|^{\nu}\big{\\}}^{\frac{1}{2}}$ $\displaystyle=\inf\big{\\{}C>0:\text{
Lemma \ref{le:firstBound} holds}\\}^{\frac{1}{2}}$

and, similarly555We chose this notation because it is related to the second
derivatives of the kernel $K$.,

$|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{2}=\inf\big{\\{}C>0:\text{ Lemma
\ref{le:4points} holds}\\}^{\frac{1}{2}}.$

It is easily shown that, for all
$\tilde{\mathsf{M}}\in\mathcal{T}(\mathsf{M})$,
$|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{0}\leq|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}$
and
$|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{2}\leq|\\!|\\!|\mathsf{M}|\\!|\\!|_{2}$.

In the following two sections, all the results stated with $\mathsf{M}$ also
hold for any measure $\tilde{\mathsf{M}}\in\mathcal{T}(\mathsf{M})$, with the
same constants. Actually, they hold for all random measure
$\tilde{\mathsf{M}}$ with $|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{0}<+\infty$
and $|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{2}<+\infty$. Furthermore, they hold
uniformly on $\mathcal{T}(\mathsf{M})$ provided the constants $C$ are replaced
with
$C(|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{0}+|\\!|\\!|\tilde{\mathsf{M}}|\\!|\\!|_{2})$.

##  3\. Comparing $\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$

Let us recall the notation that we introduced in Section 1. We fix $t>0$, a
small positive real. We also fix a (large) positive integer $N$, and define
$T=\lfloor N^{t}\rfloor$. In the estimations that follow, $t$ is going to be
fixed, and $N$ is going to tend to infinity.

For each $i\in\\{1,\dots,T\\}$, we define

  * $\scriptstyle\diamond$

$X^{i}$ to be the restriction of $X$ to $[(i-1)T^{-1},iT^{-1}]$,

  * $\scriptstyle\diamond$

$\bar{X}^{i}$ the concatenation of $X^{i}$ with the segment between its
endpoints,

  * $\scriptstyle\diamond$

$\theta^{i}(z)$ the number of times $\bar{X}^{i}$ winds around $z$,

  * $\scriptstyle\diamond$

$\mathcal{D}^{i}_{N}=\\{z\in\mathbb{R}^{2}\setminus\operatorname{Range}(\bar{X}^{i}):\theta^{i}(z)\geq
N\\}$.

We then define

$\mathsf{M}_{N}=\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N}).$

Let us stress the fact that the law of the random sets $\mathcal{D}^{i}_{N}$
can be deduced from that of $\mathcal{D}_{N}$ by a scaling and a random
translation. This self-similar behaviour will come into play in a crucial way
in Section 5.

As explained in Section 1, one of our main objects of interest is
$\mathsf{M}(\mathcal{D}_{N})$, the $\mathsf{M}$-measure of the set of points
around which the Brownian motion winds at least $N$ times (see (7)), and our
strategy to study it is to compare it with $\mathsf{M}_{N}$.

The goal of this section is to prove that $\mathsf{M}(\mathcal{D}_{N})$ and
$\mathsf{M}_{N}$ are close, in the sense of Proposition 3.7. This result will
allow us, in Section 5, to transfer to $\mathsf{M}(\mathcal{D}_{N})$ the
information about $\mathsf{M}_{N}$ that we will gather in Section 4.

To compare $\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$, we introduce
several sets. To start with, let us denote by $\mathcal{E}$ the union of the
range of $\bar{X}$ and the $T$ segments joining the endpoints of $X^{i}$ for
$i\in\\{1,\ldots,T\\}$. It is an important fact for us that this set
$\mathcal{E}$ is Lebesgue-negligible, and this implies that it is
$\mathbb{P}^{\mathsf{M}}$-almost surely $\mathsf{M}$-negligible. Then, given
integers $N,M_{1},M_{2}$, indices $i,j\in\\{1,\dots,T\\}$, and a multi-index
$\mathbf{i}\in\mathbf{I}_{k}=\\{(i_{1},\dots,i_{k})\in\\{1,\dots,T\\}^{k}:i_{1}<\dots<i_{k}\\}$,
we define

$\displaystyle\mathcal{D}^{i,j}_{N,M_{1}}$
$\displaystyle=\\{z\in\mathbb{R}^{2}\setminus\mathcal{E}:|\theta^{i}(z)|\geq
N,|\theta^{j}(z)|\geq M_{1}\\},$
$\displaystyle\mathcal{D}^{\mathbf{i}}_{M_{2}}$
$\displaystyle=\\{z\in\mathbb{R}^{2}\setminus\mathcal{E}:\forall
l\in\\{1,\dots,k\\},|\theta^{i_{l}}(z)|\geq M_{2}\\}.$

The integer $k$ that we will use will only depend on $\gamma$, and can be
though of as being fixed.

The following lemma gives us a relation between the set $\mathcal{D}_{N}$ and
the sets that we just defined.

######  Lemma 3.1.

Assume that $TM_{2}\leq\frac{N}{k}-T$ and $kM_{1}+(M_{2}+1)T<N$. Then,

$\bigcup_{i=1}^{T}\mathcal{D}^{i}_{N+TM_{1}}\setminus\bigcup_{i\neq
j}\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}}\ \subseteq\
\mathcal{D}_{N}\setminus\mathcal{E}\ \subseteq\
\bigcup_{i=1}^{T}\mathcal{D}^{i}_{N-kM_{1}-(M_{2}+1)T}\cup\bigcup_{i\neq
j}\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}}\cup\bigcup_{\mathbf{i}\in\mathbf{I}_{k}}\mathcal{D}^{\mathbf{i}}_{M_{2}}.$

These inclusions are purely deterministic and the statement remains true if
replace the Brownian curve with any other curve. The proof is based on a
discussion of the highest values taken by the winding functions of the pieces
of our curve. It would probably be best done by the reader for himself, but we
offer a detailed argument for the second inclusion, which is the less simple
one.

###### Proof.

For a given point $z\in\mathbb{R}^{2}\setminus\mathcal{E}$, let us sort the
values $(|\theta^{i}(z)|)_{i\in\\{1,\dots,T\\}}$ in non-increasing order and
denote them by $\eta_{1}\geq\dots\geq\eta_{T}$. Let us also denote
$\tilde{\eta}_{1}$ one of the values $(\theta^{i}(z))_{i\in\\{1,\dots,T\\}}$
such that $|\tilde{\eta}_{1}|=\eta_{1}$. We have the following implications:

$\displaystyle\textstyle
z\notin\bigcup_{i=1}^{T}\mathcal{D}^{i}_{N-kM_{1}-({M_{2}}+1)T}$
$\displaystyle\ \Longrightarrow\ \tilde{\eta}_{1}<N-kM_{1}-({M_{2}}+1)T,$
$\displaystyle\textstyle
z\notin\bigcup_{\mathbf{i}\in\mathbf{I}_{k}}\mathcal{D}^{\mathbf{i}}_{M_{2}}$
$\displaystyle\ \Longrightarrow\ \eta_{k}<{M_{2}},$ $\displaystyle\textstyle
z\notin\bigcup_{i\neq j}\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}}$ $\displaystyle\
\Longrightarrow\ \big{(}\eta_{1}<\tfrac{N}{k}\mbox{ or
}\eta_{2}<M_{1}\big{)}.$

If $z$ is in none of the sets appearing on the left of these implications, we
are in one of two cases, depending on which of the two assertions on the right
of the third implication holds.

If $\eta_{1}<\frac{N}{k}$, then $\eta_{2},\ldots,\eta_{k-1}<\frac{N}{k}$ and

$\sum_{i=1}^{T}\theta^{i}(z)\leq\sum_{i=1}^{T}\eta_{i}<(k-1)\frac{N}{k}+(T-(k-1)){M_{2}}\leq
N-T.$

If $\eta_{2}<M_{1}$, then $\eta_{3},\dots,\eta_{k-1}<M_{1}$ and

$\sum_{i=1}^{T}\theta^{i}(z)\leq\tilde{\eta}_{1}+\sum_{i=2}^{T}\eta_{i}<(N-kM_{1}-({M_{2}}+1)T)+(k-2)M_{1}+(T-(k-1)){M_{2}}\leq
N-T.$

In both cases, we conclude that $\sum_{i=1}^{T}\theta^{i}(z)\leq N-T$. The
difference between this sum and $\theta_{X}(z)$ is the winding at $z$ of a
piecewise linear curve with $T+1$ pieces, which cannot exceed $T-1$. Thus
$\theta_{X}(z)<N$, so that $z\notin\mathcal{D}_{N}$. ∎

In order to compare $\mathsf{M}(\mathcal{D}_{N})$ with $\mathsf{M}_{N}$, we
are going to take the $\mathsf{M}$-measures of the sets of which we just
proved the inclusion. The $\mathsf{M}$-measures of the first unions appearing
in the leftmost and rightmost terms of Lemma 3.1 will be close, but not
exactly equal, to $\mathsf{M}_{N}$.

A first difference is that we are taking the measure of a union instead of the
sum of the measures. This problem turns out not to be a serious one, and will
be treated in the proof of Proposition 3.7.

A second difference is that instead of $\mathsf{M}_{N}$, there seems to appear
$\mathsf{M}_{N^{\prime}}$ for two integers $N^{\prime}$ close to $N$. To go
around this difficulty, we will in fact apply Lemma 3.1 to several well-chosen
values of $N$, and use Lemma 3.4 to connect the various estimations that we
obtain in this way.

A third difference is that there are correction terms appearing on both sides,
and which we need to control: this will be done by the following Lemma 3.2 and
Corollary 3.3.

The first estimation in the next statement is a mild reformulation of the
Lemma 2.4 that we obtained in [13]. The second one is a slight improvement of
the Lemma 2.5 in the same paper, which corresponds to the case $k=3$. The
extension from $k=3$ to general $k$ is obtain by following the same proof. It
is a long, but elementary, computation that involves a decomposition of
$(\mathbb{R}^{2})^{k-1}$ into a family of products of balls and complementary
of balls in $\mathbb{R}^{2}$.

######  Lemma 3.2.

For all positive integer $k$ and all $r\in(0,+\infty)$, there exists $C$ such
that for all positive integers $N,M_{1},M_{2}$ and $T$, the following holds.

  * $\scriptstyle\diamond$

For all $i,j\in\\{1,\dots,T\\},i\neq j$,

$\mathbb{E}^{X}[|\mathcal{D}^{i,j}_{N,M_{1}}|^{r}]\leq
C\log(TNM_{1}+1)^{3r+1}\frac{(TNM_{1})^{-r}}{|j-i|+1}.\hskip 42.67912pt$

  * $\scriptstyle\diamond$

For all $\mathbf{i}\in\mathbf{I}_{k}$,

$\mathbb{E}^{X}[|\mathcal{D}^{\mathbf{i}}_{M_{2}}|^{r}]\leq
C\log(M_{2}T+1)^{(k+1)(r+1)-2}\frac{T^{-r}M_{2}^{-kr}}{\prod_{j=1}^{k-1}(i_{j+1}-i_{j}+1)}.$

From these estimations, Lemma 2.2 allows us to deduce corresponding
estimations in the Liouville case. We then sum over $i\neq j$ or over
$\mathbf{i}$, and we obtain the following result.

######  Corollary 3.3.

For all positive integer $k$, for all $p\in[1,+\infty)$, there exists a
constant $C$ such that for all positive integers $N,M_{1},M_{2},T$,

$\displaystyle\sum_{i\neq j}\|\mathsf{M}(\mathcal{D}^{i,j}_{N,M_{1}})\|_{p,2}$
$\displaystyle\leq
C\log(TNM_{1}+1)^{\frac{3\nu}{2}+\frac{1}{p}}T^{2-\frac{1}{p}-\frac{\nu}{2}}N^{-\frac{\nu}{2}}M_{1}^{-\frac{\nu}{2}},$
$\displaystyle\sum_{\mathbf{i}\in\mathbf{I}_{k}}\|\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{M_{2}})\|_{p,2}$
$\displaystyle\leq
C\log(M_{2}T+1)^{(k+1)(\frac{\nu}{2}+\frac{1}{p})-2}T^{k-\frac{k}{p}-\frac{\nu}{2}}M_{2}^{-k\frac{\nu}{2}}.$

###### Proof.

We prove the first inequality, the second proof is identical. For $i\neq j$,
by Lemma 2.2,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(\mathcal{D}^{i,j}_{N,M_{1}})^{2}]^{\frac{1}{2}}\leq
C(|\mathcal{D}^{i,j}_{N,M_{1}}|^{\frac{\nu}{2}}+|\mathcal{D}^{i,j}_{N,M_{1}}|).$

Applying Lemma 3.2 with $r=\frac{p\nu}{2}$ and with $r=p$, we obtain

$\displaystyle\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[\mathsf{M}(\mathcal{D}^{i,j}_{N,M_{1}})^{2}]^{\frac{p}{2}}]^{\frac{1}{p}}$
$\displaystyle\leq
C^{\prime}\mathbb{E}^{X}[|\mathcal{D}^{i,j}_{N,M_{1}}|^{\frac{p\nu}{2}}+|\mathcal{D}^{i,j}_{N,M_{1}}|^{p}]^{\frac{1}{p}}$
$\displaystyle\leq
C^{\prime\prime}\log(TNM_{1}+1)^{\frac{3\nu}{2}+\frac{1}{p}}\frac{(TNM_{1})^{-\frac{\nu}{2}}}{(|j-i|+1)^{\frac{1}{p}}}.$

We then sum over $i\neq j$ to get the announced bound. ∎

The next lemma compares the measures of $\mathcal{D}_{N}$ and
$\mathcal{D}_{N^{\prime}}$ when $N$ and $N^{\prime}$ are close.

######  Lemma 3.4.

Let $M=M(N)$ be an integer-valued function of $N$ such that $M\to\infty$ and
$\frac{M}{N}\to 0$ as $N$ tends to infinity. Then, for all $r\in(0,+\infty)$,

$\mathbb{E}^{X}\big{[}\big{|}|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|\big{|}^{r}\big{]}^{\frac{1}{r}}\leq
O(MN^{-2})+M^{\frac{2}{r}}N^{-\frac{3}{2}-\frac{1}{r}+o(1)}.$

If additionally $\liminf\frac{M^{q}}{N}>0$ for some $q<2$, then for all
$r\in(0,+\infty)$, there exists $C$ such that for all $N\geq 1$,

$\mathbb{E}^{X}\big{[}\big{|}|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|\big{|}^{r}\big{]}^{\frac{1}{r}}\leq
CMN^{-2}.$

###### Proof.

We use the following convergence, which is the main result of [15].

$\mathbb{E}^{X}\big{[}(n^{2}|\\{z\in\mathbb{R}^{2}:\theta(z)=n\\}|-\tfrac{1}{2\pi})^{2}\big{]}\underset{n\to\infty}{\longrightarrow}0.$
(12)

In particular, there exists $C$ such that for all $n\geq 1$,
$\mathbb{E}^{X}[|\\{z\in\mathbb{R}^{2}:\theta(z)=n\\}|^{2}]^{\frac{1}{2}}\leq
Cn^{-2}$. Summing from $n=N$ to $N+M-1$, we deduce that

$\mathbb{E}^{X}[(|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|)^{2}]^{\frac{1}{2}}\leq
CMN^{-2}.$ (13)

This is sufficient to conclude in the case $r=2$. The case $r<2$ follows from
Hölder inequality. For $r>2$, we fix some $p>r$. From the triangle inequality,

$\mathbb{E}^{X}\big{[}\big{|}|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|\big{|}^{p}\big{]}^{\frac{1}{p}}\leq\mathbb{E}^{X}\big{[}\big{|}|\mathcal{D}_{N}|-\tfrac{1}{2\pi
N}\big{|}^{p}\big{]}^{\frac{1}{p}}+\mathbb{E}^{X}\big{[}\big{|}|\mathcal{D}_{N+M}|-\tfrac{1}{2\pi(N+M)}\big{|}^{p}\big{]}^{\frac{1}{p}}\tfrac{M}{2\pi
N(N+M)}$

By Lemma 1.2, there is thus $C$ such that for all $N$ and $M$,

$\mathbb{E}^{X}[(|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|)^{p}]^{\frac{1}{p}}\leq
C(N^{-\frac{3}{2}+o(1)}+MN^{-2}).$ (14)

We now interpolate between the inequalities (13) and (14). Setting
$\theta=\frac{r^{-1}-p^{-1}}{2^{-1}-p^{-1}}$, the Hölder inequality is written

$\|\ {\raisebox{1.42271pt}{$\scriptscriptstyle\bullet$}}\ \|_{L^{r}}\leq\|\
{\raisebox{1.42271pt}{$\scriptscriptstyle\bullet$}}\ \|_{L^{2}}^{\theta}\|\
{\raisebox{1.42271pt}{$\scriptscriptstyle\bullet$}}\ \|_{L^{p}}^{1-\theta}.$

Hence,

$\mathbb{E}^{X}[(|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|)^{r}]^{\frac{1}{r}}\leq
C\big{(}MN^{-2}+\big{(}MN^{-2}\big{)}^{\theta}(N^{-\frac{3}{2}-o(1)})^{1-\theta}\big{)}.$

As $p$ goes to infinity, $\theta$ goes to $\frac{2}{r}$, and we end up with

$\mathbb{E}^{X}[(|\mathcal{D}_{N}|-|\mathcal{D}_{N+M}|)^{r}]^{\frac{1}{r}}\leq
C\big{(}MN^{-2}+M^{\frac{2}{r}}N^{-\frac{3}{2}-\frac{1}{r}+o(1)}\big{)}.$

This is the first announced bound.

For the last bound, it suffices to remark that, for any $r>2$, for
$\varepsilon$ small enough,
$M^{\frac{2}{r}}N^{-\frac{3}{2}-\frac{1}{r}+\varepsilon}$ is negligible
compared to $MN^{-2}$. ∎

######  Remark 3.5.

In [15], it is stated that the convergence (12) can be extended to higher
moments, and the proof is sketched. We are convinced that this sketch can
indeed, to the price of a lot of effort, be turned into a proof, but to the
best of our knowledge, this has not been done. If this statement is true, as
we think it is, the proof of Lemma 3.4 becomes almost trivial, and the
additional assumption becomes superflous.

######  Corollary 3.6.

Let $M=M(N)$ be an integer-valued function of $N$ such that
$M^{q}N^{-1}\to\infty$ for some $q<2$, and $MN^{-1}\to 0$ as $N$ tends to
infinity. Then, for all $p\in[1,+\infty)$, there exists $C$ such that for all
$N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{N+M})\|_{p,2}\leq
CN^{-\nu}M^{\frac{\nu}{2}}.$

###### Proof.

Remark that $\mathcal{D}_{N+M}\subseteq\mathcal{D}_{N}$. We have

$\displaystyle\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{N+M})\|_{p,2}$
$\displaystyle=\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(\mathcal{D}_{N}\setminus\mathcal{D}_{N+M}))^{2}]^{\frac{p}{2}}]^{\frac{1}{p}}$
$\displaystyle\leq
C\mathbb{E}^{X}[(|\mathcal{D}_{N}\setminus\mathcal{D}_{N+M}|^{\nu}+|\mathcal{D}_{N}\setminus\mathcal{D}_{N+M}|^{2})^{\frac{p}{2}}]^{\frac{1}{p}}\
\ \mbox{(using Lemma \ref{le:firstBound})}.$

We apply Lemma 3.4 with $r=\frac{\nu p}{2}$ and with $r=p$, and we obtain

$\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[(\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{N+M}))^{2}]^{\frac{p}{2}}]^{\frac{1}{p}}\leq
C_{2}\big{(}(MN^{-2})^{\frac{\nu}{2}}+MN^{-2}\big{)}\leq
2C_{2}M^{\frac{\nu}{2}}N^{-\nu},$

which we square to get the result. ∎

We are now finally ready for the comparison between
$\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$.

######  Proposition 3.7.

Let $p\in(1,+\infty)$. Then, for all
$t<\min(\tfrac{\nu}{8}\tfrac{p}{p-1},\tfrac{1}{2})$ there exists $C$ such that
for all $N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}\leq
CN^{-\frac{3\nu}{4}+2t}.$

###### Proof.

Let us set $M_{1}=\lfloor N^{\frac{1}{2}}\rfloor$ and ${M_{2}}=\lfloor
N^{\frac{1}{4}}\rfloor$. The exponents $\tfrac{1}{2}$ and $\tfrac{1}{4}$ here
are chosen in order to optimize some bound later on, and we suggest the reader
should think of $M_{1}$ and $M_{2}$ as ‘some powers of $N$, satisfying $1\ll
M_{2}\ll M_{1}$ and $M_{1}T\ll N$’ during the proof. Actually, the value
$\frac{1}{4}$ can be replaced with any value strictly between $0$ and
$\frac{1}{2}$, which even allows to extend the result of the lemma to
$t<\min(\tfrac{\nu}{4}\tfrac{p}{p-1},\tfrac{1}{2})$. This, however, is useless
for us.

For any fixed integer $k$, the relations $T{M_{2}}\leq\frac{N}{k}-T$ and
$kM_{1}+({M_{2}}+1)T<N$ holds as soon as $N$ is large enough. We can then
apply the Lemma 3.1. Since
$\mathcal{D}^{i}_{N+TM_{1}}\cap\mathcal{D}^{j}_{N+TM_{1}}\subseteq\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}}$,
we deduce the following inequalities, which holds $\mathbb{P}$-almost surely.

$\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N+TM_{1}})-\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}})\leq\mathsf{M}(\mathcal{D}_{N})\leq\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N-2TM_{1}})+\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}})+\sum_{\mathbf{i}\in\mathbf{I}_{k}}\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{{M_{2}}}).$
(15)

Remark that we have replaced the sets $\mathcal{D}^{i}_{N-kM_{1}-(M_{2}+1)T}$
that appears in Lemma 3.1 with the larger set $\mathcal{D}^{i}_{N-2TM_{1}}$.
This is possible because $kM_{1}+(M_{2}+1)T\leq 2TM_{1}$.

If we try to compare directly the first and last expressions of (15), we are
lead to compare $\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N-2TM_{1}})$ with
$\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N+TM_{1}})$, which is not very
convenient. To circumvent the difficulty, we apply (15) with $N$ replaced by
$\tilde{N}=N-3TM_{1}$, and also with $N$ replaced by $\tilde{N}=N+3TM_{1}$.

We then obtain the following inequalities, $\mathbb{P}$-almost surely.

$\mathsf{M}(\mathcal{D}_{N+3TM_{1}})-2\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}})-\sum_{\mathbf{i}}\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{{M_{2}}})\leq\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N+TM_{1}})-\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}})\\\
\leq\mathsf{M}(\mathcal{D}_{N})\leq\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N-2TM_{1}})+\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N}{k},M_{1}})+\sum_{\mathbf{i}}\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{{M_{2}}})\\\
\leq\mathsf{M}(\mathcal{D}_{N-3TM_{1}})+2\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N-TM_{1}}{k},M_{1}})+\sum_{\mathbf{i}}\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{{M_{2}}}).$
(16)

The exact same inequalities also hold if the middle term
$\mathsf{M}(\mathcal{D}_{N})$ is replaced by
$\mathsf{M}_{N}=\sum_{i=1}^{T}\mathsf{M}(\mathcal{D}^{i}_{N})$. It follows
that the difference between $\mathsf{M}(\mathcal{D}_{N})$ and $\mathsf{M}_{N}$
is less than the difference between the left-most and right-most terms of
(16):

$\big{|}\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\big{|}\leq\big{(}\mathsf{M}(\mathcal{D}_{N-3TM_{1}})-\mathsf{M}(\mathcal{D}_{N+3TM_{1}})\big{)}+4\sum_{i\neq
j}\mathsf{M}(\mathcal{D}^{i,j}_{\frac{N-TM_{1}}{k},M_{1}})+2\sum_{\mathbf{i}}\mathsf{M}(\mathcal{D}^{\mathbf{i}}_{{M_{2}}}).$
(17)

The three terms on the right-hand side are the ones that appears in Corollary
3.3 and Lemma 3.6 (applied with $M=TM_{1}$). Applying these lemmas, we obtain

$\displaystyle\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}$
$\displaystyle\leq
C\big{(}(TM_{1})^{\frac{\nu}{2}}N^{-\nu}+\log(TNM_{1}+1)^{\frac{3\nu}{2}+\frac{1}{p}}T^{2-\frac{1}{p}-\frac{\nu}{2}}N^{-\frac{\nu}{2}}M_{1}^{-\frac{\nu}{2}}$
$\displaystyle\hskip
85.35826pt+\log(M_{2}T+1)^{(k+1)(\frac{\nu}{2}+\frac{1}{p})-2}T^{k-\frac{k}{p}-\frac{\nu}{2}}M_{2}^{-k\frac{\nu}{2}}\big{)}$
$\displaystyle\leq
C^{\prime}\big{(}N^{-\frac{3\nu}{4}+t\frac{\nu}{2}}+\log(N+1)^{\frac{3\nu}{2}+\frac{1}{p}}N^{-\frac{3\nu}{4}+t(2-\frac{1}{p}-\frac{\nu}{2})}$
$\displaystyle\hskip
85.35826pt+\log(N+1)^{(k+1)(\frac{\nu}{2}+\frac{1}{p})-2}N^{-t\frac{\nu}{2}+k(t\frac{p-1}{p}-\frac{\nu}{8})}\big{)}.$

The bound on $t$ is such that the exponent factorized by $k$ is strictly
negative. It follows that for $k$ sufficiently large, the last term is
negligible compared to the others. We end up with, for some $C,C^{\prime},d$,
for all $N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}\leq
C\log(N+1)^{d}N^{-\frac{3\nu}{4}+t\max(\frac{\nu}{2},2-\frac{1}{p}-\frac{\nu}{2})}\leq
C^{\prime}N^{-\frac{3\nu}{4}+2t}.$

This concludes the proof. ∎

######  Remark 3.8.

Let us recall that our ultimate goal is to show that
$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}$ is
asymptotically less than $N^{-1-\varepsilon}$, for some $\varepsilon>0$. The
condition $\gamma<\sqrt{4/3}$ that we imposed in Theorem 1 is equivalent to
the condition $\frac{3\nu}{4}>1$, for which we just showed that
$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}$ is asymptotically less
than $N^{-1-\varepsilon}$, provided $t$ and $\varepsilon$ are small enough. If
we were to try to improve our proof to larger $\gamma$, the more important
step would be to improve the bound $\frac{3\nu}{4}$ in Proposition 3.7.

##  4\. Estimating $\mathsf{M}_{N}$

We now take the next step towards the proof of Theorem 1, according to the
strategy presented in Section 1 and recalled at the beginning of the previous
section. We use the notation introduced in Section 3, as well as that of (8).

Our next goal is thus to estimate $\mathsf{M}_{N}$, and more precisely

$(\mathsf{M}_{N}-\mathsf{M}_{-N})^{2}=\sum_{i,j=1}^{T}(\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N}))(\mathsf{M}(\mathcal{D}^{j}_{N})-\mathsf{M}(\mathcal{D}^{j}_{-N})).$

This task will, in fact, only be completed in Section 5, because of some ‘bad’
couples $(i,j)$ in this sum. Basically, we expect the expression
$(\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N}))(\mathsf{M}(\mathcal{D}^{j}_{N})-\mathsf{M}(\mathcal{D}^{j}_{-N}))$
to have a $\mathbb{P}^{\mathsf{M}}$-expectation very close to $0$, provided
that $X^{i}$ and $X^{j}$ are far from each other: these are the good couples
$(i,j)$. When these Brownian pieces go close to each other (bad couples), it
is difficult to have a good bound on the
$\mathbb{P}^{\mathsf{M}}$-expectation. To prove that these bad couples do not
contribute too much to $\mathsf{M}_{N}$ is a substantial problem on its own,
and we will address it in the next section. In the present section, we bound
the sum over the good couples only.666The proofs presented in this section can
be substantially simplified in the case where $p=2$, using the symmetry
properties of the Brownian motion. The corresponding arguments are less robust
but much simpler than the ones that we give below, and we will include them in
a forthcoming version of this paper.

Let us introduce a notation which will make many expressions much shorter than
they would otherwise be: we set, for all $i\in\\{1,\ldots,T\\}$,

$R_{i}=\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N}).$ (18)

We fix some $\varepsilon>0$ and set
$\beta=\tfrac{1}{2}-\tfrac{\varepsilon}{3}$. For $j>i+1$, we define the
following events of large probability in $\Omega^{X}$:

$E=\\{\|X\|_{\mathcal{C}^{\beta}}\leq\tfrac{1}{4}T^{\frac{\varepsilon}{3}}\\},\
\ F_{i,j}=\\{\|X_{(i+1)T^{-1}}-X_{jT^{-1}}\|\geq
T^{-\frac{1}{2}+\varepsilon}\\}.$

For $j\in\\{i,i+1\\}$, we simply set $F_{i,j}=\varnothing$ in order to
harmonize some results. The complement of an event $G$ in $\Omega^{X}$ is
denoted by $G^{c}$. We first get rid of the event $E^{c}$.

######  Lemma 4.1.

For all $p\in[1,+\infty)$, and all $r>0$, there exists $C$ such that for all
$N\geq 1$,

$\|\mathbbm{1}_{E^{c}}(\mathsf{M}_{N}-\mathsf{M}_{-N})\|_{p,2}\leq CN^{-r}.$

###### Proof.

From the triangle inequality in $L^{p}(\Omega^{X},L^{2}(\Omega^{\mathsf{M}}))$
and Cauchy–Schwarz inequality between $\mathbbm{1}_{E^{c}}$ and
$\mathbb{E}^{\mathsf{M}}[R_{i}^{2}]^{\frac{p}{2}}$ in $\Omega^{X}$,

$\displaystyle\|\mathbbm{1}_{E^{c}}(\mathsf{M}_{N}-\mathsf{M}_{-N})\|_{p,2}\leq\sum_{i=1}^{T}\|\mathbbm{1}_{E^{c}}R_{i}\|_{p,2}$
$\displaystyle=\sum_{i=1}^{T}\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[\mathbbm{1}_{E^{c}}R_{i}^{2}]^{\frac{p}{2}}]^{\frac{1}{p}}\leq\sum_{i=1}^{T}\mathbb{P}^{X}(E^{c})^{\frac{1}{2p}}\mathbb{E}^{X}[\mathbb{E}^{\mathsf{M}}[R_{i}^{2}]^{p}]^{\frac{1}{2p}}.$

By Lemma 2.2, for some $C$, for all $N$,

$\mathbb{E}^{\mathsf{M}}[\mathsf{M}(\mathcal{D}^{i}_{N})^{2}]\leq
C(|\mathcal{D}^{i}_{N}|^{\nu}+|\mathcal{D}^{i}_{N}|^{2}).$

By Lemma 1.2 and a scaling argument, for some $C^{\prime}$, for all $N$,

$\mathbb{E}^{X}[|\mathcal{D}^{i}_{N}|^{\nu p}+|\mathcal{D}^{i}_{N}|^{2p}]\leq
C^{\prime}(TN)^{-\nu p}\leq C^{\prime}.$

The same bounds hold for $N$ replaced with $-N$, and we deduce that for some
$C^{(2)}$, for all $N$,

$\mathbb{E}^{X}[\mathbbm{1}_{E^{c}}\mathbb{E}^{\mathsf{M}}[(\mathsf{M}_{N}-\mathsf{M}_{-N})^{2}]^{\frac{p}{2}}]^{\frac{1}{p}}\leq
C^{(2)}T\mathbb{P}^{X}(E^{c})^{\frac{1}{2p}}.$

From Kolmogorov continuity theorem, for all $\beta<\frac{1}{2}$,
$\|X\|_{\mathcal{C}^{\beta}}$ admits moments of all orders. From Markov’s
inequality, the tail probability $\mathbb{P}^{X}(E^{c})$ decreases more
quickly than any polynomial in $T$ (hence in $N$). This concludes the proof. ∎

######  Lemma 4.2.

For all $p\in\big{[}2,\frac{4}{\gamma^{2}})$, for all $\varepsilon>0$ and
$t>0$,

$\mathbb{E}^{X}\big{[}\big{|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j=1}^{T}\mathbbm{1}_{E\cap
F_{i,j}}R_{i}R_{j}\big{]}\big{|}^{\frac{p}{2}}\big{]}^{\frac{1}{p}}\leq
CT^{\frac{\varepsilon}{2}}N^{-1+o(1)}\big{(}N^{-\frac{1}{4}}+T^{\frac{\gamma^{2}}{4}-\frac{1}{p}}\big{)}.$

###### Proof.

Let $i,j\in\\{1,\dots,T\\}$ be such that $j>i+1$. During this proof, in order
to try and maintain the length of expressions within reasonable bounds, we
will write

$\displaystyle D^{ij}\hskip 8.5359pt$
$\displaystyle=|\mathcal{D}^{i}_{N}|+|\mathcal{D}^{i}_{-N}|+|\mathcal{D}^{j}_{N}|+|\mathcal{D}^{j}_{-N}|,$
$\displaystyle\triangle D^{ij}$
$\displaystyle=\big{|}|\mathcal{D}^{i}_{N}|-|\mathcal{D}^{i}_{-N}|\big{|}+\big{|}|\mathcal{D}^{j}_{N}|-|\mathcal{D}^{j}_{-N}|\big{|},$
$\displaystyle X_{i,j}\hskip 8.5359pt$
$\displaystyle=|X_{(i+1)T^{-1}}-X_{jT^{-1}}|.$

On $F_{i,j}\cap E$, both $4d_{\sup}(\mathcal{D}^{i}_{N},\mathcal{D}^{i}_{-N})$
and $4d_{\sup}(\mathcal{D}^{j}_{N},\mathcal{D}^{j}_{-N})$ are less than
$d_{\inf}(\mathcal{D}^{i}_{N},\mathcal{D}^{j}_{N})$, so that we can apply
Lemma 2.3. Since $d_{\inf}(\mathcal{D}^{i}_{N},\mathcal{D}^{j}_{N})\leq
X_{i,j}$, we obtain, for some constant $C$,

$\big{|}\mathbb{E}^{\mathsf{M}}[R_{i}R_{j}]\big{|}\leq
C\big{(}X_{i,j}^{-\gamma^{2}}(\triangle
D^{ij})^{2}+T^{-\beta}\|X\|_{\mathcal{C}^{\beta}}X_{i,j}^{-1-\gamma^{2}}D^{ij}\triangle
D^{ij}+T^{-2\beta}\|X\|_{\mathcal{C}^{\beta}}^{2}X_{i,j}^{-2-\gamma^{2}}(D^{ij})^{2}\big{)}.$

We raise to the power $\frac{p}{2}$, multiply by $\mathbbm{1}_{E\cap
F_{i,j}}$, and take the $\mathbb{P}^{X}$-expectation. We obtain

$\displaystyle\mathbb{E}^{X}$ $\displaystyle\Big{[}\mathbbm{1}_{E\cap
F_{i,j}}\big{|}\mathbb{E}^{\mathsf{M}}[R_{i}R_{j}]\big{|}^{\frac{p}{2}}\Big{]}\leq
C^{\prime}\big{(}\mathbb{E}^{X}\big{[}\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}\gamma^{2}}(\triangle
D^{ij})^{p}\big{]}$ (19)
$\displaystyle+T^{-\frac{p\beta}{2}}\mathbb{E}^{X}\big{[}\mathbbm{1}_{F_{i,j}}\|X\|_{\mathcal{C}^{\beta}}^{\frac{p}{2}}X_{i,j}^{-\frac{p}{2}(1+\gamma^{2})}(D^{ij})^{\frac{p}{2}}(\triangle
D^{ij})^{\frac{p}{2}}\big{]}+T^{-p\beta}\mathbb{E}^{X}\big{[}\mathbbm{1}_{F_{i,j}}\|X\|_{\mathcal{C}^{\beta}}^{p}X_{i,j}^{-\frac{p}{2}(2+\gamma^{2})}(D^{ij})^{p}\big{]}\big{)}.$

The variables $D^{ij}$ and $\triangle D^{ij}$ are measurable with respect to
the $\sigma$-algebra generated by $(X_{s})_{s\leq(i+1)T^{-1}}$ and
$(X_{s}-X_{jT^{-1}})_{s\geq jT^{-1}}$, and hence jointly independent from
$X_{i,j}$. Thus,

$\displaystyle\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}\gamma^{2}}\hskip
53.49132pt(\triangle D^{ij})^{p}\hskip 1.9919pt]$
$\displaystyle=\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}\gamma^{2}}]\hskip
21.25403pt\mathbb{E}^{X}[(\triangle D^{ij})^{p}],$
$\displaystyle\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}(2+\gamma^{2})}(D^{ij})^{p}\hskip
47.23167pt]$
$\displaystyle=\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}(2+\gamma^{2})}]\
\mathbb{E}^{X}[(D^{ij})^{p}],$ (20)
$\displaystyle\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}(1+\gamma^{2})}(D^{ij})^{\frac{p}{2}}(\triangle
D^{ij})^{\frac{p}{2}}]$
$\displaystyle=\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-\frac{p}{2}(1+\gamma^{2})}]\
\mathbb{E}^{X}[(D^{ij})^{\frac{p}{2}}(\triangle D^{ij})^{\frac{p}{2}}].$

With Lemma 1.2 and the scaling properties of the Brownian motion, we obtain
the following bounds. For all $r\in[1,+\infty)$, there exists $C$ such that
for all $N$,

$\mathbb{E}^{X}[(D^{ij})^{r}]^{\frac{1}{r}}\leq CT^{-1}N^{-1},\ \
\mathbb{E}^{X}[(\triangle D^{ij})^{r}]^{\frac{1}{r}}\leq
T^{-1}N^{-\frac{3}{2}+o(1)}.$ (21)

We also need to control the expectations that depends on $X_{i,j}$. This
variable is distributed according to $2\pi p_{(j-i-1)T^{-1}}(u)u\;{\rm d}u$,
where $\;{\rm d}u$ denotes the Lebesgue measure on $\mathbb{R}^{+}$ and
$p_{t}(u)=(2\pi t)^{-1}\exp(-\frac{u^{2}}{2t})$.

With elementary computations, we obtain, for any $r>0$,

$\displaystyle\mathbb{E}^{X}[\mathbbm{1}_{F_{i,j}}X_{i,j}^{-r}]$
$\displaystyle=2\pi\int_{T^{-\frac{1}{2}+\varepsilon}}^{+\infty}u^{1-r}p_{(j-i-1)T^{-1}}(u)\;{\rm
d}u$
$\displaystyle=(T^{\frac{1}{2}}(j-i-1)^{-\frac{1}{2}})^{r}\int_{(j-i-1)^{-\frac{1}{2}}T^{\varepsilon}}^{+\infty}\rho^{1-r}p_{1}(\rho)\;{\rm
d}\rho$
$\displaystyle\leq\left\\{\begin{array}[]{ll}C\phantom{\log(T+1)}\hskip
1.42271ptT^{\frac{r}{2}}(j-i-1)^{-\frac{r}{2}}&\mbox{if }1-r>-1\\\
C\log(T+1)T^{\frac{r}{2}}(j-i-1)^{-\frac{r}{2}}&\mbox{if }1-r=-1\\\
C\phantom{\log(T+1)}\hskip 1.42271ptT^{\frac{r}{2}}(j-i-1)^{-1}&\mbox{if
}1-r<-1\end{array}\right.$ (25) $\displaystyle\leq
C\log(T+1)T^{\frac{r}{2}}(j-i-1)^{-\min(\frac{r}{2},1)}.$ (26)

Combining (19), (20), (21) and (26) leads to

$\displaystyle\mathbb{E}^{X}$ $\displaystyle\big{[}\mathbbm{1}_{E\cap
F_{i,j}}\big{|}\mathbb{E}^{\mathsf{M}}[R_{i}R_{j}]\big{|}^{\frac{p}{2}}\big{]}\leq
C\log(T+1)\big{(}T^{\frac{p\gamma^{2}}{4}}(j-i-1)^{-\min(\frac{p\gamma^{2}}{4},1)}T^{-p}N^{-\frac{3}{2}p+o(1)}$
$\displaystyle\hskip 71.13188pt+T^{-\frac{p}{4}+\frac{\varepsilon
p}{3}}T^{\frac{p}{4}(1+\gamma^{2})}(j-i-1)^{-\min(\frac{p}{4}(1+\gamma^{2}),1)}T^{-p}N^{-\frac{5p}{4}+o(1)}$
$\displaystyle\hskip 71.13188pt+T^{-\frac{2p}{4}+\frac{\varepsilon
p}{2}}T^{\frac{p}{4}(2+\gamma^{2})}(j-i-1)^{-\min(\frac{p}{4}(2+\gamma^{2}),1)}T^{-p}N^{-p}\big{)}$
$\displaystyle\leq C^{\prime}\log(T+1)T^{\frac{\varepsilon
p}{2}}N^{-p}T^{p(\frac{\gamma^{2}}{4}-1)}\big{(}(j-i-1)^{-\frac{p\gamma^{2}}{4}}N^{-\frac{p}{4}+o(1)}+(j-i-1)^{-1}\big{)}.$

From the triangle inequality in $L^{\frac{p}{2}}(\Omega^{X})$,

$\displaystyle\mathbb{E}^{X}\big{[}\big{|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j=1}^{T}\mathbbm{1}_{E\cap
F_{i,j}}R_{i}R_{j}\big{]}\big{|}^{\frac{p}{2}}\big{]}^{\frac{1}{p}}$
$\displaystyle\leq\Big{(}\sum_{i,j=1}^{T}\mathbb{E}^{X}\big{[}\big{|}\mathbb{E}^{\mathsf{M}}\big{[}\mathbbm{1}_{E\cap
F_{i,j}}R_{i}R_{j}\big{]}\big{|}^{\frac{p}{2}}\big{]}^{\frac{2}{p}}\
\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\sum_{i,j=1}^{T}\big{(}C\log(T+1)T^{\frac{\varepsilon
p}{2}}N^{-p}T^{p(\frac{\gamma^{2}}{4}-1)}$
$\displaystyle\big{(}(j-i-1)^{-\min(\frac{p\gamma^{2}}{4},1)}N^{-\frac{p}{2}+o(1)}+(j-i-1)^{-\min(\frac{p}{4}(1+\gamma^{2}),1)}N^{-\frac{p}{4}+o(1)}+(j-i-1)^{-1}\big{)}\big{)}^{\frac{2}{p}}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq
C^{\prime}\log(T+1)^{\frac{1}{p}}T^{\frac{\varepsilon}{2}}N^{-1}T^{\frac{\gamma^{2}}{4}}\big{(}N^{-\frac{1}{4}+o(1)}T^{-\frac{\gamma^{2}}{4}}+T^{-\frac{1}{p}}\big{)}.$

Since $\log(T+1)\leq N^{o(1)}$, This concludes the proof. ∎

##  5\. Bootstrapping the bounds

Let us summarize what we did up to here. Our goal is to bound
$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}$. In
Section 3, we showed that, in this estimation, $\mathsf{M}(\mathcal{D}_{N})$
can safely be replaced with $\mathsf{M}_{N}$, provided $t$ is small and
$\gamma<\sqrt{4/3}$. Then, we split $(\mathsf{M}_{N}-\mathsf{M}_{-N})^{2}$
into a sum of two terms. Let us call them $\mathsf{M}_{\rm good}$ and
$\mathsf{M}_{\rm bad}$. The first one is a sum over ‘good terms’, and we have
been able in Section 4 to show that this term is small, provided
$p<\frac{4}{\gamma^{2}}$.

What remains to be done is thus to control $\mathsf{M}_{\rm bad}$. Here is the
place where the bootstrap really starts. In a perfect world, our dearest wish
would be to have $\|\mathsf{M}_{\rm bad}\|_{p,2}<CN^{-1-h}$, for some $C$ and
$h>0$. This however is not what we will obtain in the first place. We will
first give a bound on $\mathsf{M}_{\rm bad}$ which is something like

$\|\mathsf{M}_{\rm
bad}\|_{p,2}<CN^{\xi}\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}.$
(27)

It might seem at first that this approach is doomed to fail, because it seems
that the problem is now to control
$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}$, which
actually was the problem we started with. The crucial point is to obtain a
_negative_ exponent $\xi$: indeed, in that case, the relation

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{-1-\varepsilon}+CN^{\zeta}\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}$

does imply

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
C^{\prime}N^{-1-\varepsilon},$

for a new constant $C^{\prime}$.

We will show a relation of the form (27), with
$\xi=t(\frac{\gamma^{2}}{4}-\frac{1}{p})$. Let us recall that
$\mathcal{T}(\mathsf{M})$ is the set of random measures obtained from
$\mathsf{M}$ by translation by a random variable independent from
$\mathsf{M}$, and possibly a symmetry with respect to the horizontal axis.

Let us also recall that the parameter $\varepsilon>0$ appears in the
definition of the event $F_{i,j}$.

######  Lemma 5.1.

Set $t>0$, $p\in[2,\frac{4}{\gamma^{2}})$, and $\zeta\in\mathbb{R}$. Assume
that there exists $C$ such that for all $N\geq 1$,

$\|\mathsf{M}^{\prime}(\mathcal{D}_{N})-\mathsf{M}^{\prime}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{\zeta}.$

Then, for all $\varepsilon>0$, there exists a constant $C^{\prime}$ such that
for all $N\geq 1$,

$\mathbb{E}^{X}\Big{[}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{1\leq i\leq j\leq
T}\mathbbm{1}_{F_{i,j}^{c}}|R_{i}R_{j}|\big{]}^{\frac{p}{2}}\Big{]}^{\frac{1}{p}}\leq
C^{\prime}\log(T+1)^{\frac{1}{p}}T^{\frac{\gamma^{2}}{4}-\frac{1}{p}+\varepsilon}N^{\zeta}.$

###### Proof.

Before we dive into the proof, let us look at the behaviour of
$|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}$ under scaling of $\mathsf{M}$. For all
$z\in\mathbb{R}^{2}$ and $\lambda\geq 1$, let $\mathsf{M}_{z,\lambda}$ be the
measure defined by setting

$\mathsf{M}_{z,\lambda}(A)=\mathsf{M}(\lambda^{-\frac{1}{2}}A+z).$

In particular, $\mathsf{M}_{z,1}=\tau_{z}(\mathsf{M})$ with the definition of
Section 2 For $A$ with $|A|\leq 1$, we have $|\lambda^{-\frac{1}{2}}A+z|\leq
1$, so that

$\displaystyle\mathbb{E}[\mathsf{M}_{z,\lambda}(A)^{2}]^{\frac{1}{2}}$
$\displaystyle\leq|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}|\lambda^{-\frac{1}{2}}A+z|^{\frac{\nu}{2}}$
$\displaystyle=|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}\lambda^{-\frac{\nu}{2}}|A|^{\frac{\nu}{2}}.$

Thus,
$|\\!|\\!|\mathsf{M}_{z,\lambda}|\\!|\\!|_{0}\leq\lambda^{-\frac{\nu}{2}}|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}$.
Similarly, we have
$|\\!|\\!|\mathsf{M}_{z,\lambda}|\\!|\\!|_{2}\leq\lambda^{-\frac{\nu}{2}}|\\!|\\!|\mathsf{M}|\\!|\\!|_{2}$.
Besides, the assumption of the lemma extends automatically to all
$\mathsf{M}^{\prime}\in\mathcal{T}(M)$: for all $M^{\prime}\in\mathcal{T}(M)$,
for all $N\geq 1$,

$\|\mathsf{M}^{\prime}(\mathcal{D}_{N})-\mathsf{M}^{\prime}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{\zeta}.$

That being said, let us start the proof. Let us fix two indices
$i,j\in\\{1,\dots,T\\}$ with $i\leq j$. We bound $|R_{i}R_{j}|$ by
$\frac{1}{2}(R_{i}^{2}+R_{j}^{2})$, and we will treat separately the term with
$i$ from the term with $j$. The reason why we do not apply the same treatment
to these two terms is that the event $F^{c}_{i,j}$ is independent of $R_{i}$
but _not_ of $R_{j}$.

For $i\in\\{1,\dots,T\\}$, let us denote by $J_{i}$ the random variable

$J_{i}=\sum_{j\geq i}\mathbbm{1}_{F_{i,j}^{c}},$

which is independent of $R_{i}$. Then, using the triangle inequality in
$L^{\frac{p}{2}}(\Omega^{X})$ (recall that $p\geq 2$), we have

$\displaystyle\mathbb{E}^{X}\Big{[}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{j\geq
i}\mathbbm{1}_{F_{i,j}^{c}}R_{i}^{2}\big{]}^{\frac{p}{2}}\Big{]}^{\frac{1}{p}}=\mathbb{E}^{X}\Big{[}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i=1}^{T}J_{i}R_{i}^{2}\big{]}^{\frac{p}{2}}\Big{]}^{\frac{1}{p}}$
$\displaystyle\leq\Big{(}\sum_{i=1}^{T}\mathbb{E}^{X}\big{[}\mathbb{E}^{\mathsf{M}}\big{[}J_{i}R_{i}^{2}\big{]}^{\frac{p}{2}}\big{]}^{\frac{2}{p}}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\sum_{i=1}^{T}\mathbb{E}^{X}[J_{i}^{\frac{p}{2}}]^{\frac{2}{p}}\|R_{i}\|_{p,2}^{2}\Big{)}^{\frac{1}{2}}.$

For $j>i+1$,

$\displaystyle\mathbb{P}^{X}(F_{i,j}^{c})$
$\displaystyle=\mathbb{P}^{X}(\|X_{(i+1)T^{-1}}-X_{jT^{-1}}\|\leq
T^{-\frac{1}{2}+\varepsilon})=\int_{0}^{T^{-\frac{1}{2}+\varepsilon}}\frac{\exp(-\frac{r^{2}}{2(j-i-1)T^{-1}})}{2\pi(j-i-1)T^{-1}}r\;{\rm
d}r$ $\displaystyle\leq\frac{T^{2\varepsilon}}{2(j-i-1)}.$ (28)

It follows that

$\mathbb{E}^{X}[J_{i}^{\frac{p}{2}}]\leq
T^{\frac{p}{2}-1}\sum_{j=i}^{T}\mathbb{P}^{X}(F_{i,j}^{c})\leq
CT^{\frac{p}{2}-1+2\varepsilon}\log(T+1).$

Let $\tilde{\mathcal{D}}_{N}$ be a random set which is equal in distribution
to $\mathcal{D}_{N}$ under $\mathbb{P}^{X}$, but which is independent from
$X$. Then, the random set $\mathcal{D}^{i}_{N}$ is equal in distribution to
$T^{-\frac{1}{2}}\tilde{\mathcal{D}}_{N}+X_{i}$. It follows that

$\displaystyle\|R_{i}\|_{p,2}=\|\mathsf{M}(\mathcal{D}^{i}_{N})-\mathsf{M}(\mathcal{D}^{i}_{-N})\|_{p,2}$
$\displaystyle\leq\sup_{z\in\mathbb{R}^{2}}\|\mathsf{M}(T^{-\frac{1}{2}}\tilde{\mathcal{D}}_{N}+z)-\mathsf{M}(T^{-\frac{1}{2}}\tilde{\mathcal{D}}_{N}+z)\|_{p,2}$
$\displaystyle=\sup_{z\in\mathbb{R}^{2}}\|\mathsf{M}_{z,T}(\mathcal{D}_{N})-\mathsf{M}_{z,T}(\mathcal{D}_{-N})\|_{p,2}$
$\displaystyle\leq
C(|\\!|\\!|\mathsf{M}_{z,T}|\\!|\\!|_{0}+|\\!|\\!|\mathsf{M}_{z,T}|\\!|\\!|_{2})N^{\zeta}$
$\displaystyle\leq
CT^{-\frac{\nu}{2}}(|\\!|\\!|\mathsf{M}|\\!|\\!|_{0}+|\\!|\\!|\mathsf{M}|\\!|\\!|_{2})N^{\zeta}.$

Putting all together,

$\displaystyle\mathbb{E}^{X}\Big{[}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{1\leq
i\leq j\leq
T}\mathbbm{1}_{F_{i,j}^{c}}R_{i}^{2}\big{]}^{\frac{p}{2}}\Big{]}^{\frac{1}{p}}$
$\displaystyle\leq
CT^{\frac{-\nu}{2}}N^{\zeta}\big{(}\sum_{i=1}^{T}C^{\prime}\log(T+1)^{\frac{2}{p}}T^{1-\frac{2}{p}-\frac{4\varepsilon}{p}}\big{)}^{\frac{1}{2}}$
$\displaystyle\leq
C\log(T+1)^{\frac{1}{p}}T^{1-\frac{1}{p}-\frac{\nu}{2}+\frac{2\varepsilon}{p}}N^{\zeta}.$

Having dealt with the terms $i\leq j$, we now have to deal with the ones for
which $j<i$. For this, we replace the Brownian motion $X$ with the time-
reversed Brownian motion $\tilde{X}:t\mapsto X_{1-t}-X_{1}$, which is
independent from $(X_{1},\mathsf{M})$. We remark that
$\mathsf{M}_{X_{1},1}(\tilde{\mathcal{D}}_{N}^{i})={\mathsf{M}}(\mathcal{D}_{-N}^{T-i-1})$,
where $\tilde{\mathcal{D}}_{N}^{i}$ is the set defined as
$\mathcal{D}_{N}^{i}$, but with $\tilde{X}$ replacing $X$. We are now back in
the situation ‘$i\leq j$’ since $i^{\prime}=T-i-1<T-j-1=j^{\prime}$. We can
then use the same bounds as in the case $i\leq j$, and we end up with the same
bound, which concludes the proof. ∎

We are now ready to prove Proposition 1.1. We first recall it.

######  Proposition (1.1).

Assume that $\gamma\leq\sqrt{4/3}$ and $p\in[2,\frac{4}{\gamma^{2}})$. Then,
there exists $\delta>0$ and $C$ such that for all $N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{-1-\delta}.$

###### Proof.

Set $t\in\big{(}0,\frac{-3\nu}{8}\big{)}$, so that, by Proposition 3.7, there
exists $\delta_{1}>0,C_{1}$ such that for all $N\geq 1$,

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}\leq
C_{1}N^{-1-\delta_{1}}.$

Set also
$\varepsilon\in\big{(}0,\min\big{(}\frac{1}{p}-\frac{\gamma^{2}}{4},\frac{1}{2}\big{)}\big{)}$,
and
$\delta_{2}\in\big{(}0,t(\frac{1}{p}-\frac{\gamma^{2}}{4}-\varepsilon\big{)}$,
so that by Lemma 4.2, there exists $C_{2}$ such that for all $N\geq 1$,

$\big{\|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j}\mathbbm{1}_{E\cap
F_{i,j}}R_{i}R_{j}\big{]}^{\frac{1}{2}}\big{\|}_{p}\leq
C_{2}N^{-1-\delta_{2}}.$

We inductively show that for all $k\in\mathbb{N}$, there exists a constant $C$
such that

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
C(N^{-1-\min(\delta_{1},\delta_{2})}+N^{-\frac{\nu}{2}-k\delta_{2}}).$

At rank $k=0$, it follows directly from Lemma 2.2.

The induction hypothesis, together with Lemma 5.1, ensures that there exists
$C$ such that for all $N\geq 1$,

$\big{\|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j}\mathbbm{1}_{E\cap
F_{i,j}^{c}}R_{i}R_{j}\big{]}^{\frac{1}{2}}\big{\|}_{p}\leq
CN^{\max(-1-\delta_{1},-1-\delta_{2},\xi-k\delta_{2})}.$

To go from rank $k$ to rank $k+1$, let us decompose
$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}$ as
follows:

$\displaystyle\|\mathsf{M}^{\prime}(\mathcal{D}_{N})-\mathsf{M}^{\prime}(\mathcal{D}_{-N})\|_{p,2}$
$\displaystyle\leq\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}_{N}\|_{p,2}+\|\mathsf{M}(\mathcal{D}_{-N})-\mathsf{M}_{-N}\|_{p,2}+\|\mathsf{M}_{N}-\mathsf{M}_{-N}\|_{p,2}$
$\displaystyle\leq
2C_{1}N^{-1-\delta_{1}}+\big{\|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j}\mathbbm{1}_{E^{c}}R_{i}R_{j}\big{]}^{\frac{1}{2}}\big{\|}_{p}+\big{\|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j}\mathbbm{1}_{E\cap
F_{i,j}}R_{i}R_{j}\big{]}^{\frac{1}{2}}\big{\|}_{p}$ $\displaystyle\hskip
170.71652pt+\big{\|}\mathbb{E}^{\mathsf{M}}\big{[}\sum_{i,j}\mathbbm{1}_{E\cap
F_{i,j}^{c}}R_{i}R_{j}\big{]}^{\frac{1}{2}}\big{\|}_{p}$ $\displaystyle\leq
2C_{1}N^{-1-\delta_{1}}+CN^{-r}+C_{2}N^{-1-\delta_{2}}+CN^{\max(-1-\delta_{1},-1-\delta_{2},\xi-k\delta_{2})},$

which implies the induction hypothesis at rank $k+1$. This concludes the
induction. For $k$ large enough, we obtain

$\|\mathsf{M}(\mathcal{D}_{N})-\mathsf{M}(\mathcal{D}_{-N})\|_{p,2}\leq
CN^{-1-\min(\delta_{1},\delta_{2})},$

which concludes the proof of the proposition, hence also of Theorem 1. ∎

The work that we have done so far allowed us to define the ‘algebraic
Liouville area enclosed by the Brownian curve’ as the $\mathbb{P}$-almost sure
limit

$\mathbb{A}_{0,1}=\lim_{K\to+\infty}\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X_{|[0,1]}},K))\;{\rm
d}\mathsf{M}.$

Our reasoning extends without trouble if we replace $(0,1)$ with any couple
$(s,t)\in\Delta=\\{(s,t)\in[0,1]^{2}:s\leq t\\}$: for all such couple,
$\mathbb{P}$-almost surely, the limit

$\mathbb{A}_{s,t}=\lim_{K\to+\infty}\int_{\mathbb{R}^{2}}\max(-K,\min(\theta_{X_{|[s,t]}},K))\;{\rm
d}\mathsf{M}$ (29)

exists. The fact that the trajectory now starts from a random point does not
necessitate any additional work.

In the next section, we will define a similar quantity when the Brownian
motion is replaced with a deterministic smoother curve. A classical analogue,
when the measure $\mathsf{M}$ is replaced with the Lebesgue measure, would be
to say that after defining the Lévy area, we are now constructing the Young
integral.

The reason with why do this before pushing further the analysis of the
Brownian case is twofold. First, there are some results that we will apply to
both the Brownian and the smoother case. In order to understand these results
fully when we state them, it is better to have both definitions in mind. The
second reason is that our proof that Chen’s relation holds in the Brownian
case is rather involved, and we think that it is more easily understood once
we have proved the Chen relation for smoother curves.

##  6\. The case of smoother curves

The results that we present now are mostly independent from the first part of
the paper. In this section, we replace the planar Brownian motion $X$ with a
function $Y:[0,1]\to\mathbb{R}^{2}$ which is $\alpha$-Hölder continuous. We
will say that we work under the _relaxed assumptions_ if we only assume that
$\gamma<2$ and $\alpha>\frac{1}{2}$, and under the _strengthened assumptions_
if we assume $\gamma<\sqrt{2}$ and
$\alpha>\frac{1}{2(1-\frac{\gamma^{2}}{4})}$.

For a real number $q\in[0,\frac{4}{\gamma^{2}})$, we define

$\xi(q)=q(1+\frac{\gamma^{2}}{4})-q^{2}\frac{\gamma^{2}}{4},$

the so-called _structure exponent_ of $\mathsf{M}$.

Under the relaxed assumptions, we will show that for all
$(s,t)\in\Delta=\\{(s,t)\in[0,1]^{2}:s\leq t\\}$, the $\mathsf{M}$-area
$\mathbb{A}^{Y}_{s,t}$ delimited by $Y_{|[s,t]}$ is almost surely defined as

$\mathbb{A}^{Y}_{s,t}=\int_{\mathbb{R}^{2}}\theta_{Y_{|[s,t]}}\;{\rm
d}\mathsf{M},$ (30)

and lies in $L^{1}(\Omega^{\mathsf{M}})$.

Under the strengthened assumptions, we show that it actually lies in
$L^{2}(\Omega^{\mathsf{M}})$, and that the map $\mathbb{A}^{Y}:\Delta\to
L^{2}(\Omega^{\mathsf{M}})$ admits some Hölder regularity. 777The strengthened
assumptions are necessary to work in the $L^{2}$ framework. Nonetheless, it
should be possible to extend some of the results of the next sections to the
relaxed assumptions, provided that we succeed to work in
$L^{1+\varepsilon}(\Omega^{\mathsf{M}})$.

To simplify notation, we will write $Y_{s,t}=Y_{|[s,t]}$. Let us recall that
$\lambda$ denotes the Lebesgue measure on the plane.

The following result gives us information about the winding function of a
Hölder continuous curve.

######  Lemma 6.1.

If $Y$ is $\alpha$-Hölder continuous with $\alpha>\frac{1}{2}$, then
$\theta_{Y}$ is defined Lebesgue-almost everywhere and lies in
$L^{r}(\mathbb{R}^{2},\lambda)$ for any $r\in[1,2\alpha)$.

For any $r\in[1,2\alpha)$, there exists a constant $C$ which depends on
$\alpha$ and $r$ but not on $Y$ and such that for all $s<t\in[0,1]$,

$\|\theta_{Y_{s,t}}\|_{L^{r}(\mathbb{R}^{2},\lambda)}\leq
C(t-s)^{\frac{2\alpha}{r}}\|Y\|_{\mathcal{C}^{\alpha}}^{\frac{2}{r}}.$

The first part of this lemma was proved in Theorem 0.2 of our previous work
[13] (but with Hölder norm instead of $p$-variation norm888Recall that
$\|Y_{s,t}\|_{p-{\rm var}}\leq(t-s)^{\alpha}\|Y\|_{\mathcal{C}^{\alpha}}$. ).
The second part is Lemma 7.8 in the same paper.

######  Corollary 6.2.

Under the relaxed assumptions, for all $(s,t)\in\Delta$, almost surely, the
function $\theta_{Y_{s,t}}$ lies in $L^{1}(\mathbb{R}^{2},\mathsf{M})$ and the
random variable $\int_{\mathbb{R}^{2}}|\theta_{Y_{s,t}}|\;{\rm d}\mathsf{M}$
lies in $L^{1}(\Omega^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})$.

###### Proof.

The first point follows from the previous lemma. For the second, it suffices
to remark that

$\mathbb{E}^{\mathsf{M}}\bigg{[}\int_{\mathbb{R}^{2}}|\theta_{Y}|\;{\rm
d}\mathsf{M}\bigg{]}=\int_{\mathbb{R}^{2}}|\theta_{Y}|\;{\rm
d}\lambda<+\infty.$

∎

We are now interested in higher moments of
$\int_{\mathbb{R}^{2}}\theta_{Y}\;{\rm d}\lambda$.

######  Lemma 6.3.

Let $r>(1-\tfrac{\gamma^{2}}{4})^{-1}$ and $f\in
L^{r}(\mathbb{R}^{2},\lambda)$ with support in the unit ball.

Then, $\mathbb{P}^{\mathsf{M}}$-almost surely, $f\in
L^{r}(\mathbb{R}^{2},\mathsf{M})$, and for any
$\rho\in[1,r(1-\tfrac{\gamma^{2}}{4}))$, the random variable
$\|f\|_{L^{\rho}(\mathbb{R}^{2},\mathsf{M})}$ lies in
$L^{2\rho}(\Omega^{\mathsf{M}})$.

Besides, there exists a constant $C$, which depends only on $r$, $\rho$ and
$K$, and such that for all $f\in L^{r}(\mathbb{R}^{2},\mathsf{M})$,

$\big{\|}\|f\|_{L^{\rho}(\mathbb{R}^{2},\mathsf{M})}\big{\|}_{L^{2\rho}(\Omega^{\mathsf{M}})}\leq
C\|f\|_{L^{r}(\mathbb{R}^{2},\lambda)}.$

###### Proof.

The fact that $f$ lies $\mathbb{P}^{\mathsf{M}}$-almost surely in
$L^{r}(\mathbb{R}^{2},\mathsf{M})$ follows directly from

$\mathbb{E}\bigg{[}\int_{\mathbb{R}^{2}}|f|^{r}\;{\rm
d}\mathsf{M}\bigg{]}=\int_{\mathbb{R}^{2}}|f|^{r}\;{\rm d}\lambda<+\infty.$

For the remaining part of the lemma, let $C$ be such that $K(z,w)\leq
C+\log(|z-w|^{-1})$ for all $(z,w)\in(\mathbb{R}^{2})^{2}$. We have

$\displaystyle\mathbb{E}\bigg{[}\Big{(}\int|f|^{\rho}\;{\rm
d}\mathsf{M}\Big{)}^{2}\bigg{]}$
$\displaystyle=e^{C\gamma^{2}}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}\frac{|f|^{\rho}(z)|f|^{\rho}(w)}{|z-w|^{\gamma^{2}}}\;{\rm
d}z\;{\rm d}w$
$\displaystyle=e^{C\gamma^{2}}\int_{B(0,1)}\frac{|f|^{\rho}(z)}{|z-w|^{\frac{\gamma^{2}}{2}}}\bigg{(}\int_{B(0,1)}\frac{|f|^{\rho}(w)}{|z-w|^{\frac{\gamma^{2}}{2}}}\;{\rm
d}w\bigg{)}\;{\rm d}z.$

By Hölder inequality applied with
$p=\frac{r}{\rho}>\frac{1}{1-\frac{\gamma^{2}}{4}}$ and
$q=\frac{r}{r-\rho}<\frac{4}{\gamma^{2}}$,

$\int_{B(0,1)}\frac{|f|^{\rho}(w)}{|z-w|^{\frac{\gamma^{2}}{2}}}\;{\rm
d}w\leq\|f\|^{\rho}_{L^{r}(\mathbb{R}^{2},\lambda)}\bigg{(}\int_{B(0,2)}\frac{1}{|w|^{q\frac{\gamma^{2}}{2}}}\;{\rm
d}w\bigg{)}^{\frac{1}{q}}.$

The latter integral is finite, and with another identical computation we end
up with

$\int_{\mathbb{R}^{2}}\frac{|f|^{\rho}(z)}{|z-w|^{\frac{\gamma^{2}}{2}}}\bigg{(}\int_{\mathbb{R}^{2}}\frac{|f|^{\rho}(w)}{|z-w|^{\frac{\gamma^{2}}{2}}}\;{\rm
d}w\bigg{)}\;{\rm d}z\leq
C^{\prime}\|f\|_{L^{r}(\mathbb{R}^{2},\lambda)}^{2\rho},$

hence

$\mathbb{E}\bigg{[}\Big{(}\int|f|^{\rho}\;{\rm
d}\mathsf{M}\Big{)}^{2}\bigg{]}^{\frac{1}{2\rho}}\leq
C^{\prime\prime}\|f\|_{L^{r}(\mathbb{R}^{2},\lambda)},$

which concludes the proof. ∎

Recall that $\nu=2(1-\frac{\gamma^{2}}{4})$.

######  Corollary 6.4.

Let $Y:[0,1]\to\mathbb{R}^{2}$ be an $\alpha$-Hölder continuous function for
some $\alpha>\frac{1}{\nu}$. Then, for all $(s,t)\in\Delta$,
$\mathbb{A}^{Y}_{s,t}$ lies in
$L^{2}(\Omega^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})$. Besides, for all
$\varepsilon>0$, there exists $C$ such that, for all $s<t$,

$\|\mathbb{A}^{Y}_{s,t}\|_{L^{2}(\Omega^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})}\leq
C(t-s)^{\alpha\nu-\varepsilon}.$

###### Proof.

Lemma 6.1 ensures that for all $r\in\big{(}\frac{2}{\nu},2\alpha\big{)}$,
there exists $C$ such that for all $s<t\in[0,1]$, $\theta_{Y_{s,t}}\in
L^{r}(\mathbb{R}^{2},\lambda)$ and
$\|\theta_{Y_{s,t}}\|_{L^{r}(\mathbb{R}^{2},\lambda)}\leq
C(t-s)^{\frac{2\alpha}{r}}$.

Lemma 6.3 applied with $\rho=1<\frac{\nu r}{2}$ then ensures that
$\mathbb{P}^{\mathsf{M}}$-almost surely, $\theta_{Y_{s,t}}\in
L^{1}(\mathbb{R}^{2},\mathsf{M})$, and that

$\|\mathbb{A}^{Y}_{s,t}\|_{L^{2}(\Omega^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})}\leq
C^{\prime}(t-s)^{\frac{2\alpha}{r}}.$

We conclude by taking $r$ arbitrarily close to $\frac{2}{\nu}$. ∎

######  Remark 6.5.

The bound $r<2\alpha$ in Lemma 6.1 is optimal. In Lemma 6.3, the bound
$\rho<\frac{\nu r}{2}$ is optimal in the sense that there exists $f\in
L^{r}(\mathbb{R}^{2},\lambda)$ such that
$\|\|f\|_{L^{\rho}(\mathbb{R}^{2},\mathsf{M})}\|_{L^{2\rho}(\Omega^{\mathsf{M}})}=+\infty$
for $\rho>\frac{\nu r}{2}$ (roughly speaking, $L^{r}(\mathbb{R}^{2},\lambda)$
is not included in the ‘fibered space’
$L^{2\rho}(\Omega^{\mathsf{M}},L^{\rho}(\mathbb{R}^{2},\mathsf{M}))$). We do
not know about the cases $r=2\alpha$ and $\rho=\frac{\nu r}{2}$.

The optimality in Lemma 6.1 is seen by considering the curve $Y$ which goes
once along each of the circles with center $(0,n^{-\alpha^{\prime}})$ and
radius $n^{-\alpha^{\prime}}$ (see Figure 3 below). This curve has finite
$p$-variation for all $p>\frac{1}{\alpha^{\prime}}$, hence it can be
parameterized as an $\alpha$-Hölder continuous function for all
$\alpha<\alpha^{\prime}$. Nonetheless, there exists $C$ such that for all
$N>0$, $|\\{z:\theta_{Y}(z)=N\\}|=CN^{-2\alpha^{\prime}-1}$. It follows that
the winding function $\theta_{Y}$ does not belong to
$L^{2\alpha^{\prime}}(\mathbb{R}^{2},\lambda)$.

Figure 3.  The function $Y$, here for $\beta=2/3$. Inspired from Figure 5,
p.11 in [10].

For the optimality in Lemma 6.3, one could look precisely at the function
$\theta_{Y}$, but it is slightly simpler to look at the function
$f:z\mapsto|z|^{-\alpha}\mathbbm{1}_{B(0,1)}(z)$ with $\alpha<\frac{2}{r}$.
This function is easily seen to lie in $L^{r}(\mathbb{R}^{2},\lambda)$, but
for any $\rho\geq\frac{\nu}{\alpha}$,

$\displaystyle\mathbb{E}^{\mathsf{M}}\bigg{[}\Big{(}\int_{\mathbb{R}^{2}}f(z)^{\rho}\;{\rm
d}z\Big{)}^{2}\bigg{]}$
$\displaystyle=\int_{B(0,1)^{2}}|z|^{-\alpha\rho}|w|^{-\alpha\rho}|z-w|^{-\gamma^{2}}\;{\rm
d}z\;{\rm d}w$
$\displaystyle\geq\int_{B(0,1)}\int_{B(0,|w|)}|w|^{-\alpha\rho}|w|^{-\alpha\rho}2^{-\gamma^{2}}|w|^{-\gamma^{2}}\;{\rm
d}z\;{\rm d}w$
$\displaystyle=2^{-\gamma^{2}}\pi\int_{0}^{1}r^{-2\alpha\rho-\gamma^{2}+3}\;{\rm
d}r=+\infty.$

##  7\. Weak Chen’s relation

We will need the following version of the Chen relation, where the quantifier
on $s$, $u$ and $t$ and the _almost sure_ has been exchanged. Let us recall
from the introduction that $T_{s,u,t}$ is the triangle delimited by $Z_{s}$,
$Z_{u}$, and $Z_{t}$ (the continuous function $Z:[0,1]\to\mathbb{R}^{2}$ is
assumed to be fixed), and that $\varepsilon_{s,u,t}\in\\{\pm 1\\}$ depends one
the cyclic order between these points on the boundary of $T_{s,u,t}$.

######  Definition 7.1.

Set $\Delta_{3}=\\{(s,t,u):0\leq s\leq u\leq t\leq 1\\}$. We say that a
collection $(\mathbb{A}_{s,t})_{(s,t)\in\Delta}$ of random variables satisfies
the _weak Chen relation_ (relative to the continuous function $Z$) if for all
$(s,u,t)\in\Delta_{3}$, $\mathbb{P}$-almost surely,

$\mathbb{A}_{s,t}=\mathbb{A}_{s,u}+\mathbb{A}_{u,t}+\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t}).$

Let us remark that, in contrast with the Chen relation, the weak Chen relation
is preserved by the replacement of the family $\mathbb{A}$ by one of its
modifications (see Definition 8.1).

######  Lemma 7.2.

Under the relaxed assumptions, that is, if $\alpha>\frac{1}{2}$ and
$\gamma<2$, the family of random variables $\mathbb{A}^{Y}$ defined by (30)
satisfies the weak Chen relation relative to $Y$.

###### Proof.

We give two proofs. The first is much simpler, but understanding the second
one will help us to understand the proof of the similar result for the
Brownian motion.

_First proof._ Let us fix $(s,u,t)\in\Delta_{3}$. The equality

$\theta_{s,t}=\theta_{s,u}+\theta_{u,t}+\varepsilon_{s,u,t}\mathbbm{1}_{T_{s,u,t}}$

holds $\lambda$-almost everywhere. Hence, $\mathbb{P}^{\mathsf{M}}$-almost
surely, this equality holds $\mathsf{M}$-almost everywhere. Since all of these
functions are $\mathbb{P}^{\mathsf{M}}$-almost surely $\mathsf{M}$-integrable
(Corollary 6.2), $\mathbb{P}^{\mathsf{M}}$-almost surely,

$\int_{\mathbb{R}^{2}}\theta_{s,t}\;{\rm
d}\mathsf{M}=\int_{\mathbb{R}^{2}}\theta_{s,u}\;{\rm
d}\mathsf{M}+\int_{\mathbb{R}^{2}}\theta_{u,t}\;{\rm
d}\mathsf{M}+\int_{\mathbb{R}^{2}}\varepsilon_{s,u,t}\mathbbm{1}_{T_{s,u,t}}\;{\rm
d}\mathsf{M}.$

This is exactly the announced equality.

_Second proof._ Let us choose $(s,u,t)\in\Delta_{3}$. We decompose the plane
according to the values of the two winding functions $\theta_{s,u}$ and
$\theta_{u,t}$. Unfortunately, we also have to take the triangle $T_{s,u,t}$
into account, which muddles the proof. We invite the reader to write down the
simplified version when $Y_{s}=Y_{u}=Y_{t}$.

For three relative integers $j$, $k$, and $n$, we define the following sets:

$\displaystyle\mathcal{A}_{j,k}$
$\displaystyle=\\{z\in\mathbb{R}^{2}:\theta_{s,u}(z)=j,\
\theta_{u,t}(z)=k\\},$ $\displaystyle\mathcal{A}_{n}$
$\displaystyle=\\{z\in\mathbb{R}^{2}:\theta_{s,t}(z)=n\\},$
$\displaystyle\mathcal{A}^{1}_{j}$
$\displaystyle=\\{z\in\mathbb{R}^{2}:\theta_{s,u}(z)=j\\},$
$\displaystyle\mathcal{A}^{2}_{k}$
$\displaystyle=\\{z\in\mathbb{R}^{2}:\theta_{u,t}(z)=k\\}.$

We also define $\mathcal{A}_{j,k}^{\wedge}$, $\mathcal{A}_{n}^{\wedge}$,
$\mathcal{A}^{1,\wedge}_{n}$ and $\mathcal{A}^{2,\wedge}_{n}$ the intersection
of the triangle $T_{s,u,t}$ with (respectively) $\mathcal{A}_{j,k}$,
$\mathcal{A}_{n}$, $\mathcal{A}^{1}_{n}$ and $\mathcal{A}^{2}_{n}$.

For $z\in\mathcal{A}^{\wedge}_{j,k}$,
$\theta_{s,t}(z)=j+k+\varepsilon_{s,u,t}$. Hence, for all integer $n$,

$\mathcal{A}^{\wedge}_{n}=\bigsqcup_{j,k:j+k+\varepsilon_{s,u,t}=n}\mathcal{A}^{\wedge}_{j,k}.$

Besides, for all $j\in\mathbb{N}$,

$\mathcal{A}^{1,\wedge}_{j}=\bigsqcup_{k\in\mathbb{N}}\mathcal{A}^{\wedge}_{j,k},$

and for all $k\in\mathbb{N}$,

$\mathcal{A}^{2,\wedge}_{k}=\bigsqcup_{j\in\mathbb{N}}\mathcal{A}^{\wedge}_{j,k}.$

It follows from these relations that

$\displaystyle\int_{T_{s,u,t}}\theta_{s,t}\;{\rm d}\mathsf{M}$
$\displaystyle=\sum_{j,k\in\mathbb{N}^{2}}(j+k+\varepsilon_{s,u,t})\mathsf{M}(\mathcal{A}^{\wedge}_{j,k})$
$\displaystyle=\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t})+\sum_{j\in\mathbb{N}}j\mathsf{M}(\mathcal{A}^{1,\wedge}_{j})+\sum_{k\in\mathbb{N}}k\mathsf{M}(\mathcal{A}^{2,\wedge}_{k})$
$\displaystyle=\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t})+\int_{T_{s,u,t}}\theta_{s,u}\;{\rm
d}\mathsf{M}+\int_{T_{s,u,t}}\theta_{u,t}\;{\rm d}\mathsf{M}.$

We then replace the sets $\mathcal{A}_{j,k}^{\wedge}$,
$\mathcal{A}_{n}^{\wedge}$, $\mathcal{A}^{1,\wedge}_{n}$ and
$\mathcal{A}^{2,\wedge}_{n}$ with the sets $\mathcal{A}_{j,k}^{\vee}$,
$\mathcal{A}_{n}^{\vee}$, $\mathcal{A}^{1,\vee}_{n}$ and
$\mathcal{A}^{2,\vee}_{n}$ defined as the intersection of
$\mathbb{R}^{2}\setminus T_{s,u,t}$ with (respectively) $\mathcal{A}_{j,k}$,
$\mathcal{A}_{n}$, $\mathcal{A}^{1}_{n}$ and $\mathcal{A}^{2}_{n}$. The same
computations hold, except that for $z\in\mathcal{A}^{\vee}_{j,k}$,
$\theta_{s,t}(z)$ is equal to $j+k$ instead of $j+k+\varepsilon_{s,u,t}$. We
end up with

$\int_{\mathbb{R}^{2}\setminus T_{s,u,t}}\theta_{s,t}\;{\rm
d}\mathsf{M}=\int_{\mathbb{R}^{2}\setminus T_{s,u,t}}\theta_{s,u}\;{\rm
d}\mathsf{M}+\int_{\mathbb{R}^{2}\setminus T_{s,u,t}}\theta_{u,t}\;{\rm
d}\mathsf{M},$

which allows to conclude the proof. ∎

We now prove the corresponding result for the Brownian motion.

######  Lemma 7.3.

If $\gamma<\sqrt{4/3}$, then $\mathbb{A}^{X}$ satisfies the weak Chen
relation.

###### Proof.

We invite the reader to skim through the (rather long) proof a first time, and
to convince herself or himself that it is merely a question of interchanging
the summation order in a double sum, hence of showing that some residual terms
are small.

We fix $(s,u,t)\in\Delta_{3}$. For simplicity, we assume that
$\varepsilon(s,u,t)=1$ (that is, the triangle has ‘positive orientation’).

For a positive integer $N$, we set

$\displaystyle\mathcal{D}_{N}$ $\displaystyle=\\{z:\theta_{X_{|[s,t]}}(z)\geq
N\\}$ $\displaystyle\mathcal{D}^{1}_{N}$
$\displaystyle=\\{z:\theta_{X_{|[s,u]}}(z)\geq N\\}$
$\displaystyle\mathcal{D}^{2}_{N}$
$\displaystyle=\\{z:\theta_{X_{|[u,t]}}(z)\geq N\\},$

and as usual the same notation with $N<0$ is used with the inequality
reversed.

For two positive integers $N,M$, we denote by $\mathcal{D}_{N,M}$ the set

$\mathcal{D}_{N,M}=\Big{(}\big{(}\mathcal{D}^{1}_{N}\cup\mathcal{D}^{1}_{-N}\big{)}\cap\big{(}\mathcal{D}^{2}_{M}\cup\mathcal{D}^{2}_{-M}\big{)}\Big{)}\cup\Big{(}\big{(}\mathcal{D}_{M}^{1}\cup\mathcal{D}^{1}_{-M}\big{)}\cap\big{(}\mathcal{D}^{2}_{N}\cup\mathcal{D}^{2}_{-N}\big{)}\Big{)}.$

This is the set of points $z$ such that among $|\theta_{s,u}(z)|$ and
$|\theta_{u,t}(z)|$, one is at least $N$ and the other is at least $M$. We
also use the notations $\mathcal{A}_{N}$
$\mathcal{A}_{k,j}$,$\mathcal{A}^{1}_{k}$ and $\mathcal{A}^{2}_{j}$, of the
previous proof.

We will use the following bounds: for all $\varepsilon>0$, there exists $C$
such that for all $N,M\geq 1$,

$\mathbb{E}^{X}[|\mathcal{D}_{N,M}|]\leq C(NM)^{-1+\varepsilon}.$ (31)

This is an easy consequence of Lemma 2.4 in [13].

3.4

For a point
$z\in\mathbb{R}^{2}\setminus(T_{s,u,t}\cup\operatorname{Range}(X))$, it is
easily seen that $z\in\mathcal{A}_{N}$ if and only if there exists
$k\in\mathbb{Z}$ such that $z\in\mathcal{A}^{1}_{k}\cap\mathcal{A}^{2}_{N-k}$,
in which case this $k$ is unique. We let $\mathsf{M}^{\vee}$ be the (random)
measure defined by $\mathsf{M}^{\vee}(A)=\mathsf{M}(A\setminus T_{s,u,t})$.

For all $N$, we compute

$\displaystyle\sum_{k=1}^{N}\mathsf{M}^{\vee}(\mathcal{D}_{k})$
$\displaystyle=\sum_{k=1}^{N}\sum_{l=k}^{+\infty}\mathsf{M}^{\vee}(\mathcal{A}_{l})=\sum_{k=1}^{N}\sum_{l=k}^{+\infty}\sum_{j=-\infty}^{+\infty}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{l-j})=\sum_{k=1}^{N}\sum_{j=-\infty}^{+\infty}\sum_{m=k-j}^{+\infty}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m})$
$\displaystyle=\sum_{j=-\infty}^{+\infty}\sum_{m=1-j}^{+\infty}\min(N,m+j)\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m})$
$\displaystyle=\sum_{j=-\infty}^{+\infty}\sum_{m=-\infty}^{+\infty}\max(0,\min(N,m+j))\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m}).$

This computation remains true if we replace each set $\mathcal{A}^{*}_{\ell}$
by $\mathcal{A}^{*}_{-\ell}$ and leave everything else unchanged. Doing this
susbstitution, subtracting the resulting equality from the one that we just
obtained, and using the notation $[j]_{n}=\max(-n,\min(n,j))$, we find

$\sum_{k=1}^{N}\big{(}\mathsf{M}^{\vee}(\mathcal{D}_{k})-\mathsf{M}^{\vee}(\mathcal{D}_{-k})\big{)}=\sum_{j=-\infty}^{+\infty}\sum_{m=-\infty}^{+\infty}[m+j]_{N}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m}).$
(32)

On the other hand, using the fact that
$(\mathcal{A}^{2}_{m})_{m\in\mathbb{Z}}$ is a partition of $\mathbb{R}^{2}$,
we find, by a superficially identical, but in fact different computation,

$\displaystyle\sum_{k=1}^{N}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{k})$
$\displaystyle=\sum_{k=1}^{N}\sum_{j=k}^{+\infty}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j})=\sum_{k=1}^{N}\sum_{j=k}^{+\infty}\sum_{m=-\infty}^{+\infty}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m})$
$\displaystyle=\sum_{j=1}^{+\infty}\sum_{m=-\infty}^{+\infty}\min(N,j)\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m})$
$\displaystyle=\sum_{j=-\infty}^{+\infty}\sum_{m=-\infty}^{+\infty}\max(0,\min(N,j))\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m}).$

Replacing $k$ by $-k$ as we did before and combining the two results, we
obtain

$\sum_{k=1}^{N}\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{k})-\mathsf{M}^{\vee}(\mathcal{D}^{1}_{-k})\big{)}=\sum_{j=-\infty}^{+\infty}\sum_{m=-\infty}^{+\infty}[j]_{N}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{m}).$
(33)

The same equation holds after exchanging the superscript $1$ and $2$, so that

$\displaystyle\sum_{k=1}^{N}\Big{(}\big{(}\mathsf{M}^{\vee}(\mathcal{D}_{k})-\mathsf{M}^{\vee}(\mathcal{D}_{-k})\big{)}-\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{k})-\mathsf{M}^{\vee}(\mathcal{D}^{1}_{-k})+\mathsf{M}^{\vee}(\mathcal{D}^{2}_{k})-\mathsf{M}^{\vee}(\mathcal{D}^{2}_{-k})\big{)}\Big{)}$
$\displaystyle\hskip
142.26378pt=\sum_{k,j=-\infty}^{+\infty}\big{(}[k+j]_{N}-[j]_{N}-[k]_{N}\big{)}\mathsf{M}^{\vee}(\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{k}).$
(34)

Our goal is now to show that this sums goes to $0$ as $N$ goes to infinity. To
this end, we decompose $\mathbb{N}^{2}$ as follows. We fix a parameter
$m\in(0,1)$ and set $M=\lfloor N^{m}\rfloor$. We then partition
$\mathbb{N}^{2}$ into five subsets $E_{1},E_{2},E_{3},E_{4},E_{5}$,
illustrated on Figure 4.

$\displaystyle E_{1}=[0,\tfrac{N}{2})^{2},$ $\displaystyle
E_{2}=([M,+\infty)\times[\tfrac{N}{2},+\infty))\cup([\tfrac{N}{2},+\infty)\times[M,+\infty)),$
$\displaystyle E_{3}=([0,M)\times[N-M,N+M))\cup([N-M,N+M)\times[0,M)),$
$\displaystyle
E_{4}=([0,M)\times[N+M,+\infty))\cup([N+M,+\infty)\times[0,M)),$
$\displaystyle
E_{5}=([0,M)\times[\tfrac{N}{2},N-M))\cup([\tfrac{N}{2},N-M)\times[0,M)).$

Figure 4.  On the left: decomposition of $\mathbb{N}^{2}$. On the right:
approximate bounds on the corresponding sum.

We now partition $\mathbb{Z}^{2}$ into the sets
$F_{i}=\\{(k,j)\in\mathbb{Z}^{2}:(|k|,|j|)\in E_{i}\\}$, for
$i\in\\{1,\ldots,5\\}$. We decompose the sum (34) accordingly into five sums
$S_{1}^{\vee},\dots,S_{5}^{\vee}$.

$\bullet$ It is easily seen that $S_{1}^{\vee}=S_{5}^{\vee}=0$.

$\bullet$ For $(k,j)\in F_{2}$, the inequality
$|(k+j)_{N}-(j)_{N}-(k)_{N}|\leq N$ holds. Moreover, the
$\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{k}$ are disjoint subsets of
$\mathcal{D}_{N,M}$. Hence,

$|S_{2}^{\vee}|\leq N\mathsf{M}^{\vee}(\mathcal{D}_{N,M}).$

$\bullet$ For $(k,j)\in F_{3}$, $|(k+j)_{N}-(j)_{N}-(k)_{N}|\leq N$, and the
$\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{k}$ are disjoint subsets of
$(\mathcal{D}^{1}_{N-M}\setminus\mathcal{D}^{1}_{N+M+1})\cup(\mathcal{D}^{1}_{N-M}\setminus\mathcal{D}^{2}_{N+M+1})$.
Hence,

$|S_{3}^{\vee}|\leq
N\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{N-M}\setminus\mathcal{D}^{1}_{N+M+1})+\mathsf{M}^{\vee}(\mathcal{D}^{2}_{N-M}\setminus\mathcal{D}^{2}_{N+M+1})\big{)}.$

$\bullet$ For $(k,j)\in F_{4}$, $|(k+j)_{N}-(j)_{N}-(k)_{N}|\leq 2M$, and the
$\mathcal{A}^{1}_{j}\cap\mathcal{A}^{2}_{k}$ are disjoint subsets of
$\mathcal{D}^{1}_{N}\cup\mathcal{D}^{1}_{-N}\cup\mathcal{D}^{2}_{N}\cup\mathcal{D}^{2}_{-N}$.
Hence,

$|S_{4}^{\vee}|\leq
2M\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{N})+\mathsf{M}^{\vee}(\mathcal{D}^{2}_{N})\big{)}.$

Altogether, we have

$|S_{1}^{\vee}+S_{2}^{\vee}+S_{3}^{\vee}+S_{4}^{\vee}+S_{5}^{\vee}|\\\ \leq
N\big{(}\mathsf{M}^{\vee}(\mathcal{D}_{N,M})+\mathsf{M}^{\vee}(\mathcal{D}^{1}_{N-M}\setminus\mathcal{D}^{1}_{N+M+1})+\mathsf{M}^{\vee}(\mathcal{D}^{2}_{N-M}\setminus\mathcal{D}^{2}_{N+M+1})\big{)}+\\\
2M\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{N})+\mathsf{M}^{\vee}(\mathcal{D}^{2}_{N})\big{)}.$
(35)

We take the expectation under $\mathbb{P}^{\mathsf{M}}$ on both sides. Using
the fact that the intensity of the random measure $\mathsf{M}$ is the Lebesgue
measure, we obtain

$\mathbb{E}^{\mathsf{M}}\big{[}|S_{1}+S_{2}+S_{3}+S_{4}+S_{5}|\big{]}\leq
N\big{(}|\mathcal{D}_{N,M}|+|\mathcal{D}^{1}_{N-M}\setminus\mathcal{D}^{1}_{N+M+1}|)+|\mathcal{D}^{2}_{N-M}\setminus\mathcal{D}^{2}_{N+M+1}|\big{)}+\\\
2M\big{(}|\mathcal{D}^{1}_{N}|+|\mathcal{D}^{2}_{N}|)\big{)}.$

We take the expectation under $\mathbb{P}^{X}$ on both sides. Using Equation
(31), as well as the fact that $|\mathcal{A}_{N}|$ is equivalent in $L^{2}$
(hence in $L^{1}$) to $\frac{|t-s|}{2\pi N^{2}}$ (recall Equation 12), we
obtain

$\mathbb{E}\big{[}|S_{1}+S_{2}+S_{3}+S_{4}+S_{5}|\big{]}\leq
C^{\prime}N((NM)^{-1+\varepsilon}+MN^{-2})+MN^{-1}\underset{N\to+\infty}{\longrightarrow}0,$

where the last convergence holds for an arbitrary choice of $m\in(0,1)$ and
$\varepsilon\in(0,\frac{1}{2})$.

From this long discussion, it follows that, $\mathbb{P}$-almost surely,

$\sum_{k=1}^{+\infty}\big{(}\mathsf{M}^{\vee}(\mathcal{D}_{k})-\mathsf{M}^{\vee}(\mathcal{D}_{-k})\big{)}=\sum_{k=1}^{+\infty}\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{1}_{k})-\mathsf{M}^{\vee}(\mathcal{D}^{1}_{-k})\big{)}+\sum_{k=1}^{+\infty}\big{(}\mathsf{M}^{\vee}(\mathcal{D}^{2}_{k})-\mathsf{M}^{\vee}(\mathcal{D}^{2}_{-k})\big{)}.$

For a point $z$ in the interior of $T_{s,u,t}$ (and outside the range of $X$),
the relation between $\mathcal{A}_{N}$ and the
$\mathcal{A}^{1}_{k},\mathcal{A}^{2}_{j}$ has to be shifted by $1$:

$z\in\mathcal{A}^{1}_{N}\Longleftrightarrow\exists
k\in\mathbb{Z}:z\in\mathcal{A}^{1}_{k}\cap\mathcal{A}^{2}_{N-k-1}.$

This explains the apparition of the additional term $\mathsf{M}(T_{s,u,t})$ in
the Chen relation. Computations similar to the previous ones lead to the
equality

$\sum_{k=1}^{+\infty}\big{(}\mathsf{M}(\mathcal{D}_{k}\cap
T_{s,u,t})-\mathsf{M}(\mathcal{D}_{-k}\cap
T_{s,u,t})\big{)}=\sum_{k=1}^{+\infty}\big{(}\mathsf{M}(\mathcal{D}^{1}_{k}\cap
T_{s,u,t})-\mathsf{M}(\mathcal{D}^{1}_{-k}\cap T_{s,u,t})\big{)}\\\
+\sum_{k=1}^{+\infty}\big{(}\mathsf{M}(\mathcal{D}^{2}_{k}\cap
T_{s,u,t})-\mathsf{M}(\mathcal{D}^{2}_{-k}\cap
T_{s,u,t})\big{)}+\mathsf{M}(T_{s,u,t}).$ (36)

Combining the two equalities gives the desired result. ∎

##  8\. From weak Chen’s relation to pathwise Chen’s relation

The goal in this section is to show that, up to modification, a map
$\mathbb{A}$ that satisfies the weak Chen relation does satisfies the Chen
relation. For this, we need the $\mathsf{M}$-measure of a triangle to be a
continuous function of its vertices, and we are able to do this only under the
assumption that $\gamma<2(\sqrt{2}-1)$.

Let us first recall some terminology for functions of two parameters. For
functions of three parameters, we use the same definitions with $\Delta$
replaced by $\Delta_{3}=\\{(s,u,t)\in[0,1]^{3}:s\leq u\leq t\\}$.

######  Definition 8.1.

Let $X$ and $\tilde{X}$ be two collections of random variables on the same
probability space, both indexed by $\Delta$. We say that they are
_modifications_ of each other, or that one is a modification of the other, if
for all $w\in\Delta$, almost surely, $X_{w}=\tilde{X}_{w}$.

For instance, the collections $\mathbb{A}$ defined by (29) and (30) are
defined only up to modification.

######  Definition 8.2.

A collection $X$ of random variables indexed by $\Delta$ and with values in
$\mathbb{R}^{d}$ is said to be _separable_ (with respect to the class of
closed sets) if there exist a countable set $I$ (called the separability set)
and a negligible event $\mathcal{N}$ such that for all open set $U$ of
$\Delta$ and all closed set $F$ of $\mathbb{R}^{d}$, the following inclusion
holds:

$\\{\forall w\in U\cap I:X_{w}\in F\\}\setminus\\{\forall w\in U:X_{w}\in
F\\}\subseteq\mathcal{N}.$

Our impression is that the terminology of separability, which might have been
very commonly used in the past, has gone lost with time. It seems to us that
modern introductions to stochastic processes tend to forget about it, to the
profit of stronger properties such as continuity or càdlàg property. We will
use some results which can be found in [5] and [8]. A discussion of these
questions can also be found in [4, Chapter IV, 24-30]. In the first cited
text, things are stated for functions from $\mathbb{R}$ to $\mathbb{R}$, but
the results that we use extend without any technical complications to our
situation. On the contrary, the framework is much more general in the second
cited text. We state these results in the form which is adapted to our
framework. The reason why we use separability is that, in order to prove some
regularity result, we need to _first_ prove the pathwise Chen relation. This
relation, since it can rewritten as

$\forall(s,u,t)\in\Delta_{3},\
\delta\mathbb{A}_{s,u,t}=\mathbb{A}_{s,t}-\mathbb{A}_{s,u}-\mathbb{A}_{u,t}-\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t})\in\\{0\\},$

clearly follows from the weak one, _provided the family $\delta\mathbb{A}$ is
separable_. The following result, due to J.L. Doob, states that any family has
a separable modification. Though we will use this result, it is not directly
sufficient to us: what we want is not a modification of $\delta\mathbb{A}$
equal to $0$, but a modification $\tilde{\mathbb{A}}$ of $\mathbb{A}$ such
that the corresponding family $\delta\tilde{\mathbb{A}}$ is separable. To show
the existence of such a family is the main purpose of this section.

######  Lemma 8.3 ([5, Theorem 2.4], [8, Theorem 1 in Section III.2]).

Let $X$ be a collection of random variables in the same probability space,
indexed by $\Delta$ (resp. $\Delta_{3}$) and with values in $\mathbb{R}^{d}$.
Then, there exists a separable modification of $X$.

We will also need the following characterization of the separability
condition.

######  Lemma 8.4 ([8, Lemma 1 in Section III.2]).

A collection $(X_{\mathbf{t}})_{\mathbf{t}\in\Delta_{3}}$ is separable if and
only if there exists a negligible set $\mathcal{N}$ and a countable set
$I\subseteq\Delta_{3}$ such that for all $\omega\in\Omega\setminus\mathcal{N}$
and $\mathbf{t}\in\Delta_{3}$, the value $X_{\mathbf{t}}(\omega)$ lies in

$\bigcap_{U}\overline{\\{X_{s}(\omega):s\in U\cap I\\}},$

where $U$ ranges over the open sets in $\Delta_{3}$ containing $\mathbf{t}$.

This characterization allows us to prove the following result, which we were
unable to find in the literature.

######  Corollary 8.5.

Assume that $(X_{\mathbf{t}})_{\mathbf{t}\in\Delta_{3}}$ and
$(Y_{\mathbf{t}})_{\mathbf{t}\in\Delta_{3}}$ are separable. Then,
$(X_{\mathbf{t}},Y_{\mathbf{t}})_{\mathbf{t}\in\Delta_{3}}$ and
$(X_{\mathbf{t}}+Y_{\mathbf{t}})_{\mathbf{t}\in\Delta_{3}}$ are separable.

###### Proof.

Let $I_{1},\mathcal{N}_{1}$ (resp. $I_{2},\mathcal{N}_{2}$) be the sets that
appear in the characterization of the separability of $X$ (resp. $Y$). Let
$I=I_{1}\cup I_{2}$, and $\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}$. For
all $\mathbf{t}\in\Delta_{3}$, and $U$ open set in $\Delta_{3}$ containing
$\mathbf{t}$, for all $\omega\in\Omega\setminus\mathcal{N}$, we know that
$X_{\mathbf{t}}(\omega)$ lies in
$\overline{\\{X_{\mathbf{s}}(\omega):\mathbf{s}\in U\cap I\\}}$ and that
$Y_{\mathbf{t}}(\omega)$ lies in
$\overline{\\{Y_{\mathbf{s}}(\omega):\mathbf{s}\in U\cap I\\}}$. Hence
$(X_{\mathbf{t}},Y_{\mathbf{t}})$ lies in

$\overline{\\{X_{\mathbf{s}}(\omega):\mathbf{s}\in U\cap
I\\}}\times\overline{\\{Y_{\mathbf{s}}(\omega):\mathbf{s}\in U\cap
I\\}}=\overline{\\{(X_{\mathbf{s}}(\omega),Y_{\mathbf{s}}(\omega)):\mathbf{s}\in
U\cap I\\}}.$

This allows us to conclude to the first point.

For the second, we use the definition rather than the characterization. Let
$I,\mathcal{N}$ be the sets that appear in the definition of the separability
of $(X,Y)$. Let $U\subseteq\Delta_{3}$ be an open set, and
$F\subset\mathbb{R}^{d}$ be a closed set. Let
$\pi:(\mathbb{R}^{d})^{2}\to\mathbb{R}^{d}$ be the map $(x,y)\mapsto x+y$.
Then, $\pi^{-1}(F)$ is closed, so that

$\displaystyle\\{\forall\mathbf{t}\in U,(X+Y)_{\mathbf{t}}(\omega)\in F\\}$
$\displaystyle=\\{\forall\mathbf{t}\in
U,(X_{\mathbf{t}}(\omega),Y_{\mathbf{t}}(\omega))\in\pi^{-1}(F)\\}$
$\displaystyle\subseteq\\{\forall\mathbf{t}\in U\cap
I,(X_{\mathbf{t}}(\omega),Y_{\mathbf{t}}(\omega))\in\pi^{-1}(F)\\}\cup\mathcal{N}$
$\displaystyle=\\{\forall\mathbf{t}\in U\cap I,(X+Y)_{\mathbf{t}}(\omega)\in
F\\}\cup\mathcal{N}.$

Hence $X+Y$ is separable. ∎

In particular, for
$\delta\mathbb{A}=(\mathbb{A}_{s,t}-\mathbb{A}_{s,u}-\mathbb{A}_{u,t}-\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t}))_{(s,u,t)\in\Delta_{3}}$
to be separable, it suffices that

  * $\scriptstyle\diamond$

$(\mathbb{A}_{s,t})_{(s,u,t)\in\Delta_{3}}$ be separable (or equivalently,
that $(\mathbb{A}_{s,t})_{(s,t)\in\Delta}$ be separable), and

  * $\scriptstyle\diamond$

$(\mathsf{M}(T_{s,u,t}))_{(s,u,t)\in\Delta_{3}}$ be separable.

For the first point, we know from Doob’s lemma that
$(\mathbb{A}_{s,t})_{(s,t)\in\Delta}$ admits a separable modification. For the
second point, the problem is posed in a slightly different way, because
$(\mathsf{M}(T_{s,u,t}))_{(s,u,t)\in\Delta_{3}}$ is defined not as a
collection of random variables indexed by $\Delta_{3}$, but really as a random
function on $\Delta_{3}$. Taking a modification of it to ensure its
separability would possibly destroy the structure given by the fact that
$\mathsf{M}$ is a measure: there is no reason why a modification
$(\tilde{m}_{s,u,t})_{(s,u,t)\in\Delta_{3}}$ of
$(\mathsf{M}(T_{s,u,t}))_{(s,u,t)\in\Delta_{3}}$ would be of the form
$\tilde{m}_{s,u,t}=\tilde{\mathsf{M}}(T_{s,u,t})$ for a random measure
$\tilde{\mathsf{M}}$.999Actually, random measures are entirely characterized
by their finite dimensional marginals $\mathsf{M}(A_{i})_{i\in I}$ (see [3]),
so that the only modifications of $\mathsf{M}$ which are random measures are
indistinguishable from $\mathsf{M}$.

In order to show that the map $(s,u,t)\mapsto\mathsf{M}(T_{s,u,t})$ is
separable (not up to modification), the only way that we found is to prove a
much stronger result, for which we need $\gamma$ to be smaller than
$2(\sqrt{2}-1)$. For $\mathbf{z}=(z_{1},z_{2},z_{3})\in(\mathbb{R}^{2})^{3}$,
we set $T^{\mathbf{z}}$ the convex hull of the three points
$z_{1},z_{2},z_{3}$. We then define $a:(\mathbb{R}^{2})^{3}\to\mathbb{R}^{+}$
the map given by $a(\mathbf{z})=\mathsf{M}(T^{\mathbf{z}})$.

######  Lemma 8.6.

For all $\gamma<2$, the random map $a$ is separable.

###### Proof.

Let $\mathcal{N}$ be the negligible event on which there exists a compact
subset of $\mathbb{R}^{2}$ with infinite $\mathsf{M}$-area. Any triangle $T$
is the intersection of a decreasing sequence of triangles with rational
vertices, and on the complement of $\mathcal{N}$, the measure of $T$ is the
decreasing limit of the measures of these rational triangles. Hence, $a$
satisfies the separability criterion of Lemma 8.4. ∎

We will now improve this result and show that the map $a$ is actually
continuous, and in fact Hölder continuous, under the additional assumption
that $\gamma<2(\sqrt{2}-1)$.

######  Lemma 8.7.

Let $\gamma<2(\sqrt{2}-1)$. Then, almost surely, the map $a$ is continuous,
and locally $\beta$-Hölder continous for any
$\beta<1-\sqrt{2}\gamma+\frac{\gamma^{2}}{4}$.

###### Proof.

We show the result for the restriction of $a$ on the set of triangles
contained in the box $[-1,1]^{2}$. The global continuity can be deduced by
scaling or by a covering argument.

For $x$ on the boundary of $[-1,1]^{2}$, $\varepsilon>0$ and
$\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$, we denote by
$R^{\varepsilon}_{x,\theta}$ the rectangle with length $4\sqrt{2}$ and width
$\varepsilon$, which is centered at $x$ and with angle $\theta$ with respect
to the $x$-axis (see Figure 5 below).

Figure 5.  Two triangles with close vertices, one of the six triangles that
cover there symmetric difference, and a rectangle $R^{\varepsilon}_{x,\theta}$
that contains this triangle.

Let $n$ be an integer such that $n\varepsilon>1$. Set

$I=\\{(1,\tfrac{k}{n}):|k|\leq n\\}\cup\\{(\tfrac{k}{n},1):|k|\leq n\\}\
\text{ and }\ J=\\{\tfrac{k\pi}{2n}:|k|\leq n\\}.$

Consider a triangle $T$ with vertices $z_{1},z_{2},z_{3}$ in $[0,1]^{2}$ such
that $|z_{2}-z_{3}|\leq\varepsilon$. The line through $z_{1}$ and the middle
of $z_{2}$ and $z_{3}$ crosses the boundary of $[-1,1]^{2}$ at a point close
of a point of $I$, with an angle close to an element of $J$. Thus, there
exists $(i,j)\in I\times J$ such that $T$ is included on
$R^{4\varepsilon}_{i,j}$.

Let us call $\varepsilon$-thin a triangle such as the one that we just
considered, that is, a triangle of which two vertices are $\varepsilon$-close.
The symmetric difference between two triangles, the vertices of which are
pairwise $\varepsilon$-close, is contained in the union of six
$\varepsilon$-thin triangles (see Figure 5 again). Thus, if
$\mathbf{z}=(z_{1},z_{2},z_{3})$ and
$\mathbf{z}^{\prime}=(z^{\prime}_{1},z^{\prime}_{2},z^{\prime}_{3})$ are such
that $|\mathbf{z}-\mathbf{z}^{\prime}|\leq\varepsilon$, then

$|a(\mathbf{z})-a(\mathbf{z}^{\prime})|\leq 6\max_{(i,j)\in I\times
J}R^{4\varepsilon}_{i,j}.$

We bound this supremum as in the proof of the Kolmogorov criterion, using the
fact that the cardinal of $I\times J$ is of order $\varepsilon^{-2}$. For any
$\beta<1-\sqrt{2}\gamma+\frac{\gamma^{2}}{4}$ and
$q\in[0,\frac{4}{\gamma^{2}})$,

$\displaystyle\mathbb{P}(\max_{(i,j)\in I\times
J}R^{4\varepsilon}_{i,j}\geq\varepsilon^{\beta})$
$\displaystyle\leq\varepsilon^{-\beta q}\mathbb{E}\sum_{(i,j)\in I\times
J}(R^{4\varepsilon}_{i,j})^{q}\leq C\varepsilon^{-2}\varepsilon^{-\beta
q}\mathbb{E}[(R^{4\varepsilon}_{i,j})^{q}]\leq C\varepsilon^{-2-\beta
q+\frac{\xi(q)}{2}}.$

The exponent is minimized by the choice of $q=\frac{2\sqrt{2}}{\gamma}$ (which
is less strictly than $\frac{4}{\gamma^{2}}$), and the bound on $\beta$ is
such that the exponent is then strictly positive. To conclude, we take
$\varepsilon=2^{-n}$ and we apply the Borel–Cantelli lemma. ∎

Let us summarize. For $\gamma<2$, for any map $\mathbb{A}:\Delta\to\mathbb{R}$
which satisfies the weak Chen relation relative to a continuous path $Z$, we
know that there exists a separable modification of $\tilde{\mathbb{A}}$ of
$\mathbb{A}$, which is easily seen to also satisfy the weak Chen relation. We
also know that the map $(s,u,t)\mapsto\mathsf{M}(T^{Z_{s},Z_{u},Z_{t}})$ is
separable.

Hence, we have obtained the following result.

######  Proposition 8.8.

For all $\gamma<2$, for all map $\mathbb{A}:\Delta\to\mathbb{R}$ which
satisfies the weak Chen relation relative to a continuous path $Z$, there
exists a modification of $\mathbb{A}$ which is separable and satisfies the
(strong) Chen relation relative to $Z$.

We will now adress the question of the regularity of such a map $\mathbb{A}$.
Before that, let us remark that the Chen relation, together with Lemma 8.7,
allows us to deduce Hölder continuity from regularity. Recall from (6) the
definition of $\beta$-regularity.

######  Lemma 8.9.

Let $\gamma<2(\sqrt{2}-1)$. Let $\mathbb{A}:\Delta\to\mathbb{R}$ be a map that
satisfies the Chen relation relative to a function $Z$ which is
$\alpha$-Hölder continuous, for some $\alpha>0$. Assume that $\mathbb{A}$ is
$\beta$-regular. Then, for all $\beta^{\prime}$ such that
$\beta^{\prime}\leq\beta$ and
$\beta^{\prime}<(1-\sqrt{2}\gamma+\frac{\gamma^{2}}{4})\alpha$, the map
$\mathbb{A}$ is $\beta^{\prime}$-Hölder continuous.

##  9\. A Kolmogorov type criterion

The goal of this section is to obtain a Kolmogorov type criterion that applies
to our situation, that is a result that allows us to deduce some pathwise
regularity (in particular, continuity) of a map
$\mathbb{A}(\omega):\Delta\to\mathbb{R}$ from regularity of the map
$\mathbb{A}:\Delta\to L^{q}(\Omega)$.

######  Proposition 9.1.

Assume that $\gamma<\sqrt{2}$,
$\alpha\in\big{(}\gamma^{2}\big{(}1+\frac{\gamma^{2}}{4}\big{)}^{-2},1\big{]}$,
and $\xi>\frac{1}{2}$. Set

$\beta_{1}=\left\\{\begin{array}[]{ll}\min(\alpha\nu-1,\xi-\frac{1}{2})&\text{if
}\alpha\geq\gamma^{-2},\\\
\min(2\alpha(1+\frac{\gamma^{2}}{4})-2\gamma\sqrt{\alpha},\alpha\nu-\frac{1}{2},\xi-\frac{1}{2})&\text{if
}\alpha\leq\gamma^{-2}.\end{array}\right.$

Let $Z\in\mathcal{C}^{\alpha}$ be a continuous function from $[0,1]$ to
$\mathbb{R}^{2}$, possibly random but independent from $\mathsf{M}$ (say,
defined in a probability space $\Omega^{X}$). Assume that $\mathbb{A}$ is
separable and satisfies the weak Chen relation relative to $Z$, and assume
that there exists a positive random variable $C_{0}$ on $\Omega^{X}$ such that
for all $(s,t)\in\Delta$, $\mathbb{P}^{X}$-almost surely,
$\|\mathbb{A}_{s,t}\|_{L^{2}(\Omega^{\mathsf{M}},\mathbb{P}^{\mathsf{M}})}\leq
C_{0}(t-s)^{\xi}$.

Then, $\mathbb{A}$ is almost surely $\beta_{1}$-regular: $\mathbb{P}$-almost
surely, for all $\beta<\beta_{1}$, there exists $C$ such that for all
$(s,t)\in\Delta$,

$|\mathbb{A}_{s,t}|\leq C(t-s)^{\beta}.$

###### Proof.

We write

$\mathbb{D}=\\{s\in[0,1):\exists i,j\in\mathbb{N},s=i2^{-n}\\}$

the set of dyadic numbers. We set $\beta_{0}=\alpha\nu-1$ if
$\alpha\geq\gamma^{-2}$, and
$\beta_{0}=2\alpha(1+\frac{\gamma^{2}}{4})-2\gamma\sqrt{\alpha}$ if
$\alpha\leq\gamma^{-2}$. In particular,
$\beta_{1}=\min(\beta_{0},\alpha\nu-\frac{1}{2},\xi-\frac{1}{2})$. We rely on
the classical proof of Kolmogorov criterion for rough paths. We follow in
particular the proof of Theorem 3.1 in [6] (with $q=2$), but with some
modification. By localization, we can assume that both $C_{0}$ and
$\|Z\|_{\mathcal{C}^{\alpha}}$ admit poynomial moments of all order.

For all $n\geq 0$, we denote by $\mathbb{D}_{n}$ the set of integer multiples
of $2^{-n}$ in $[0,1)$.

We now introduce a few notations for some rectangles and triangles. Figure 6
should help understand the notations. For $s\leq t$ and $Z$ equal to either
$X$ (a Brownian motion) or $Y$ (an $\alpha$-Hölder continuous curve), we
denote by $T_{s,t}$ the triangle with vertices $Z_{s}$ ,$Z_{t}$, and
$(Z_{t}^{1},Z_{s}^{2})$. For $s\leq u\leq t$, we denote (as before)
$T_{s,u,t}$ the triangle with vertices $Z_{s}$, $Z_{u}$ and $Z_{t}$, and we
denote $R_{s,u,t}$ the rectangle
$[Z^{1}_{u},Z^{1}_{t}]\times[Z^{2}_{s},Z^{2}_{u}]$. Finally, we define
$R_{s,t,u,v}$ as the rectangle
$[Z^{1}_{s},Z^{1}_{t}]\times[Z^{2}_{u},Z^{2}_{v}]$.

Figure 6.  The triangles $T_{s,u,t}$ and$T_{s,t}$, and the rectangles
$R_{s,u,t}$ and $R_{s,t,u,v}$.

Let us recall that $\varepsilon_{s,u,t}\in\\{\pm 1\\}$ is equal to $1$ if the
points $Z_{s}$, $Z_{t}$ and $Z_{u}$ appears in trigonometric order along the
boundary of $T_{s,u,t}$. We similarily define $\varepsilon_{s,t}$ as equal to
$1$ if the points $Z_{s}$, $(Z^{1}_{t},Z^{2}_{s})$ and $Z_{t}$ appears in
trigonometric order along the boundary of $T_{s,t}$, and equal to $-1$
otherwise. In Figure 6, $\varepsilon_{s,t}=\varepsilon_{s,u,t}=1$. These
values are such that for all $s\leq u\leq t$,

$\varepsilon_{s,u,t}\mathsf{M}(T_{s,u,t})+\varepsilon_{s,t}\mathsf{M}(T_{s,t})=\varepsilon_{s,u,t}\mathsf{M}(R_{s,u,t})+\varepsilon_{s,u}\mathsf{M}(T_{s,u})+\varepsilon_{u,t}\mathsf{M}(T_{u,t}).$

Besides, $R_{s,u,t}=R_{s,u,u,t}$. We denote

$\mathcal{A}^{Y}_{s,t}=\mathbb{A}^{Y}_{s,t}+\varepsilon_{s,t}\mathsf{M}(T_{s,t}).$
(37)

We also set

$J_{n,n^{\prime}}=\max_{\begin{subarray}{c}i\in\mathbb{D}_{n}\\\
j\in\mathbb{D}_{n^{\prime}}\end{subarray}}\mathsf{M}(R_{i,i+2^{-n},j,j+2^{-n^{\prime}}}).$
(38)

The second moment of this variable can be estimated as follows:

$\mathbb{E}[J_{n,n^{\prime}}^{2}]\leq\sum_{\begin{subarray}{c}i\in\mathbb{D}_{n}\\\
j\in\mathbb{D}_{n^{\prime}}\end{subarray}}\mathbb{E}[\mathsf{M}(R_{i,i+2^{-n},j,j+2^{-{n^{\prime}}}})^{2}]\leq
C\sum_{\begin{subarray}{c}i\in\mathbb{D}_{n}\\\
j\in\mathbb{D}_{n^{\prime}}\end{subarray}}|R_{i,i+2^{-n},j,j+2^{-{n^{\prime}}}}|^{\nu}\leq
C2^{(1-\alpha\nu)(n+n^{\prime})}$ (39)

with $C=\mathbb{E}[\|Z\|_{\mathcal{C}^{\alpha}}^{2}]$.

Actually, we obtain a better estimation by looking at the moment of order $q$
of $J_{n,n^{\prime}}$. We obtain the following bound, the proof of which is
postponed to Section 10 (Lemma 10.5):

$\mathbb{E}[J_{n,n^{\prime}}^{2}]\leq C2^{-\beta_{0}(n+n^{\prime})}.$ (40)

For $s<t\in\mathbb{D}=\bigcup_{n\geq 0}\mathbb{D}_{n}$, we define
$s=\tau_{0}<\dots<\tau_{N}=t$ as follows101010This sequence
$\tau_{0},\dots,\tau_{N}$ plays the same role as the one defined in [6],
though the construction is not exactly the same.. Let $m$ be the integer part
of $-\log_{2}(t-s)$ and $r$ be the unique element of
$\mathbb{D}_{m}\cap[s,t)$. Write

$s=r-\sum_{i=1}^{M}2^{-\phi(i)},\qquad t=r+\sum_{i=1}^{P}2^{-\psi(i)},$

with $\phi,\psi$ two stricly increasing functions from $\mathbb{N}$ to
$\mathbb{N}$ with $\phi(1)\geq m+1,\psi(1)\geq m+1$. Such a decomposition
always exists. We set $N=M+P$, $\tau_{M}=r$. For $j\in\\{0,\dots,M-1\\}$, we
set $\tau_{j}=r-\sum_{i=1}^{M-j}2^{-\phi(i)}$. For $j\in\\{M+1,\dots,M+P\\}$,
we set $\tau_{j}=r+\sum_{i=1}^{j-M}2^{-\psi(i)}$. For each
$j\in\\{0,\dots,M+P\\}\setminus\\{M\\}$, set $n_{j}$ such that
$\tau_{j}\in\mathbb{D}_{n_{j}}\setminus\mathbb{D}_{n_{j}-1}$. Set also
$n_{M}=m$. Then, it is easily seen that $(n_{j})_{j\in\\{0,M\\}}$ is stricly
decreasing, whilst $(n_{j})_{j\in\\{M,M+P\\}}$ is stricly increasing. In
particular, the sequence $(n_{j})_{j\in\\{0,M+P\\}}$ takes each value at most
twice, and takes no value smaller than $m$. In particular, for any sequence
$(a_{n})_{n\in\mathbb{N}}$ of positive terms,

$\sum_{i=0}^{N}a_{n_{i}}\leq 2\sum_{n=m}^{+\infty}a_{n}.$

For $u<v\in\mathbb{D}$, we define $u=\sigma_{0}<\dots<\sigma_{N^{\prime}}=t$
and $(n^{\prime}_{j})_{j\in\\{0,\dots,N^{\prime}\\}}$ in an identical way, and
we set $m^{\prime}$ the integer part of $-\log_{2}(v-u)$.

Then, for any $\delta<\frac{\beta_{0}}{2}$, we claim that
$\mathbb{E}\big{[}\max_{s<t,u<v}\big{(}(t-s)^{-\delta}(v-u)^{-\delta}\mathsf{M}(R_{s,t,u,v})\big{)}^{2}\big{]}$
is finite. Indeed, it is bounded above by

$\mathbb{E}\bigg{[}\max_{s<t,u<v}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{N^{\prime}}(t-s)^{-\delta}(v-u)^{-\delta}\mathsf{M}(R_{\tau_{i-1},\tau_{i},\sigma_{i-1},\sigma_{i}})\Big{)}^{2}\bigg{]},$

which in turn is not greater than

$\displaystyle\mathbb{E}\bigg{[}\max_{s<t,u<v}\Big{(}\sum_{i=1}^{N}\sum_{j=1}^{N^{\prime}}2^{-\delta
m}2^{-\delta m^{\prime}}J_{n_{i},n^{\prime}_{j}}\Big{)}^{2}\bigg{]}$
$\displaystyle\leq\mathbb{E}\big{[}\max_{s<t,u<v}4\big{(}\sum_{\begin{subarray}{c}n\geq
m\\\ n^{\prime}\geq
m^{\prime}\end{subarray}}2^{\delta(m+m^{\prime})}J_{n,n^{\prime}}\big{)}^{2}\big{]}$
$\displaystyle\leq 4\Big{(}\sum_{\begin{subarray}{c}n\geq m\\\ n^{\prime}\geq
m^{\prime}\end{subarray}}2^{\delta(n+n^{\prime})}\mathbb{E}[J_{n,n^{\prime}}^{2}]^{\frac{1}{2}}\Big{)}^{2}\leq
4\Big{(}\sum_{\begin{subarray}{c}n\geq m\\\ n^{\prime}\geq
m^{\prime}\end{subarray}}2^{(\delta-\frac{\beta_{0}}{2})(n+n^{\prime})}\Big{)}^{2}<+\infty.$

We denote by $C_{\mathsf{M}}$ the expectation of which we just proved that it
is finite.

Let us now look at $\mathbb{A}_{s,t}^{Z}$. Remark that

$\mathbb{E}[|\mathsf{M}(T_{s,t})|^{2}]\leq
C\mathbb{E}^{Z}[|T_{s,t}|^{\nu}]=C\frac{\mathbb{E}[|Z_{t}^{1}-Z_{s}^{1}|^{\nu}|Z_{t}^{2}-Z_{s}^{2}|^{\nu}]}{2^{\nu}}\leq
C^{\prime}(t-s)^{2\alpha\nu},$

with
$C^{\prime}=\frac{C}{2^{\nu}}\mathbb{E}[\|Z\|_{\mathcal{C}^{\alpha}}^{2\nu}]$,
and $C$ the constant of Lemma 2.2. Since we assumed
$\mathbb{E}[(\mathbb{A}^{Z}_{s,t})^{2}]\leq C(t-s)^{2\xi}$, this implies

$\mathbb{E}^{\mathsf{M}}[\mathcal{A}^{Z}_{s,t}]\leq
C(t-s)^{2\min(\xi,\alpha\nu)}.$

Let us define
$\mathbb{K}_{n}=\max_{t\in\mathbb{D}_{n}}|\mathcal{A}^{Z}_{t,t+2^{-n}}|$. Then

$\mathbb{E}^{\mathsf{M}}[\mathbb{K}_{n}^{2}]\leq\sum_{t\in\mathbb{D}_{n}}\mathbb{E}^{\mathsf{M}}[|\mathcal{A}^{Z}_{t,t+2^{-n}}|^{2}]\leq
C2^{(1-2\min(\xi,\alpha\nu))n}.$

We set $s=\tau_{0}<\dots<\tau_{n}=t$ as before. Then, for any
$\beta^{\prime}\in(0,\beta)$ and
$\delta\in(\frac{\beta^{\prime}}{2},\frac{\beta}{2})$,

$\displaystyle\mathbb{E}^{\mathsf{M}}[$
$\displaystyle\max_{s<t\in[0,1]}\big{(}(t-s)^{-\beta^{\prime}}\mathbb{A}_{s,t}^{Z}\big{)}^{2}]\leq\mathbb{E}^{\mathsf{M}}[\max_{s<t\in[0,1]}\big{(}(t-s)^{-\beta^{\prime}}\sum_{i=0}^{N}(\mathbb{A}^{Z}_{\tau_{i},\tau_{i+1}}+\mathsf{M}(R_{s,\tau_{i},\tau_{i+1}}))\big{)}^{2}]$
$\displaystyle\leq
2\mathbb{E}^{\mathsf{M}}[\max_{s<t\in[0,1]}\big{(}(t-s)^{-\beta^{\prime}}\sum_{i=0}^{N}\mathbb{A}^{Z}_{\tau_{i},\tau_{i+1}}\big{)}^{2}]+2\mathbb{E}^{\mathsf{M}}[\max_{s<t\in[0,1]}\big{(}\sum_{i=0}^{N}(t-s)^{-\beta^{\prime}}\mathsf{M}(R_{s,\tau_{i},\tau_{i+1}})\big{)}^{2}]$
$\displaystyle\leq
8\mathbb{E}^{\mathsf{M}}\big{[}\big{(}\sum_{n=0}^{+\infty}2^{\beta^{\prime}n}\mathbb{K}_{n}\big{)}^{2}\big{]}$
$\displaystyle\hskip
42.67912pt+8\mathbb{E}^{\mathsf{M}}\Big{[}\max_{s<t\in[0,1]}\big{(}(t-s)^{-\beta^{\prime}}\sum_{n=0}^{+\infty}\max_{u\in[s,t]}(s-u)^{\delta}2^{-\delta
n}(s-u)^{-\delta}2^{\delta n}\mathsf{M}(R_{s,u,u+2^{-n}})\big{)}^{2}\Big{]}$
$\displaystyle\leq
8\Big{(}\sum_{n=0}^{+\infty}2^{\beta^{\prime}n}\mathbb{E}^{\mathsf{M}}\big{[}\big{(}\mathbb{K}_{n}\big{)}^{2}\big{]}^{\frac{1}{2}}\Big{)}^{2}$
$\displaystyle\hskip
85.35826pt+8\mathbb{E}^{\mathsf{M}}\Big{[}\max_{s<t\in[0,1]}\big{(}\sum_{n=0}^{+\infty}2^{(\beta^{\prime}-\delta)n}2^{-\delta
n}\max_{u\in[s,t]}(s-u)^{-\delta}2^{\delta
n}\mathsf{M}(R_{s,u,u+2^{-n}})\big{)}^{2}\Big{]}$ $\displaystyle\leq
C\Big{(}\sum_{n=0}^{+\infty}2^{(\beta^{\prime}+\frac{1}{2}-\max(\xi,\alpha\nu))n}\Big{)}^{2}+8\Big{(}\sum_{n=0}^{+\infty}2^{(\beta^{\prime}-2\delta)n}C_{\mathsf{M}}\big{)}^{2}]\Big{)}^{2}<+\infty.$

This proves that, for all $\beta<\beta_{1}$, $\mathbb{P}$-almost surely, there
exists a constant $C$ such that for all $(s,t)\in\Delta\cap\mathbb{D}^{2}$,
$|\mathbb{A}_{s,t}|\leq C(t-s)^{\beta}$.

Our assumption is that $\mathbb{A}$ is separable, but with respect to a
countable subset $I$ of $\Delta$ which we do not know. Let us explain how this
general situation can be reduced to the dyadic situation that we treated
above. Firstly, $I$ can be replaced by a countable set with a product
structure, namely the set of all points of $\Delta$ which share each of their
coordinates with a point of $I$. Thus, $I$ is the intersection with $\Delta$
of a set of the form $\tilde{\mathbb{D}}^{2}$, for some countable dense subset
$\tilde{\mathbb{D}}$ of $[0,1]$. Then, we can write $\tilde{\mathbb{D}}$ as an
infinite union $\bigcup_{N\in\mathbb{N}}\tilde{\mathbb{D}}_{N}$, in a way that
mimics the decomposition of $\mathbb{D}$ that we used above, namely in such a
way that the points of $\tilde{\mathbb{D}}_{N}$ are close enough to being
evenly spaced for our arguments to work.

This being taken into account, the separability of $\mathbb{A}$ allows us to
conclude that for all $(s,t)\in\Delta$, $|\mathbb{A}_{s,t}|\leq
C(t-s)^{\beta}$ and the proposition is proved. ∎

######  Corollary 9.2.

Assume that $\gamma<\sqrt{2}$ and $Y$ is $\alpha$-Hölder continuous for
$\alpha>\frac{1}{2}(1-\frac{\gamma^{2}}{4})^{-1}$. Then $\mathbb{A}^{Y}$
admits a separable modification which satisfies the Chen relation and which is
$\beta$-regular for all

$\beta<\beta_{0}=\left\\{\begin{array}[]{ll}\alpha\nu-1&\mbox{if
}\alpha\geq\gamma^{-2},\\\
2\alpha(1+\frac{\gamma^{2}}{4})-2\gamma\sqrt{\alpha}&\mbox{if
}\alpha\in[\frac{3-2\sqrt{2}}{2}\gamma^{-2},\gamma^{-2}]\\\
\alpha\nu-\frac{1}{2}&\mbox{if
}\alpha\leq\frac{3-2\sqrt{2}}{2}\gamma^{-2}.\end{array}\right.$

If $\gamma<2(\sqrt{2}-1)$, this modification is $\beta^{\prime}$-Hölder
continuous for
$\beta^{\prime}=\min((1-\sqrt{2}\gamma+\frac{\gamma^{2}}{4})\alpha,\beta)$.

###### Proof.

The condition $\alpha>\frac{1}{2}(1-\frac{\gamma^{2}}{4})^{-1}$ implies
$\alpha>\gamma^{2}(1+\frac{\gamma^{2}}{4})^{-2}$. Corollary 6.4 and Lemma 7.2
ensure that we can take $\xi=\alpha\nu-\varepsilon$ for any $\varepsilon>0$ in
the conditions of Proposition 9.1. This suffices to conclude to the first
family of properties. For the Hölder regularity when $\gamma<2(\sqrt{2}-1)$,
we also use Proposition 8.8 and then Lemma 8.9. ∎

For the equivalent result in the Brownian situation, we need to obtain a
scaling relation for $\mathbb{A}^{X}$. It is obtained from the scaling
properties of the Brownian motion and the measure $\mathsf{M}$.

######  Lemma 9.3.

Assume that $\gamma<\sqrt{4/3}$. There exists an increasing family of events
$(E_{n})_{n\geq 1}$ on $\Omega^{X}$, with $\mathbb{P}^{X}(E_{n})\to 1$, and a
family of constants $(C_{n})_{n\geq 1}$, such that for all $n\geq 1$, for all
$(s,t)\in\Delta$,

$\mathbb{E}[\mathbbm{1}_{E_{n}}\mathbb{A}_{s,t}^{2}]\leq C_{n}(t-s)^{\nu}.$

###### Proof.

For $N$ a positive integer and $(s,t)\in\Delta$, we set

$\mathcal{D}_{N,s,t}=\\{z\in\mathbb{R}^{2}:\theta_{X_{|[s,t]}}(z)\geq N\\},\ \
\mathcal{D}_{-N,s,t}=\\{z\in\mathbb{R}^{2}:\theta_{X_{|[s,t]}}(z)\leq-N\\}.$

Assume first that $K$ is given by $K(z,w)=\log_{+}(|z-w|^{-1})$ and write
$\mathsf{M}_{0}$ the associated measure. Let $E_{n}$ be the event
$E_{n}=\\{\forall(s,t)\in\Delta,\
t-s<\frac{1}{n}\implies|X_{t}-X_{s}|\leq\frac{1}{2}\\}$. Clearly,
$\mathbb{P}(E_{n})\to 1$. For $n\geq 1$ and $(s,t)\in\Delta$ with
$t-s<\frac{1}{n}$, using the exact scale invariance of $\mathsf{M}_{0}$ (see