# Multifractal analysis of Gaussian multiplicative chaos and applications

Federico Bertacco111Imperial College London, United Kingdom. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Let $M_{\gamma}$ be a subcritical Gaussian multiplicative chaos measure
associated with a general log-correlated Gaussian field defined on a bounded
domain $D\subset\mathbb{R}^{d}$, $d\geq 1$. We find an explicit formula for
its singularity spectrum by showing that $M_{\gamma}$ satisfies almost surely
the multifractal formalism, i.e., we prove that its singularity spectrum is
almost surely equal to the Legendre–Fenchel transform of its $L^{q}$-spectrum.
Then, applying this result, we compute the lower singularity spectrum of the
multifractal random walk and of the Liouville Brownian motion.

##  1 Introduction

The main goal of this paper is to prove that subcritical _Gaussian
Multiplicative Chaos_ (GMC) measures associated with a large class of log-
correlated Gaussian fields in all dimensions satisfy the multifractal
formalism. This has been formally stated in [RV14, Section 4], but to the best
of our knowledge, it has never been proved rigorously. In order to achieve
this result, we perform a careful analysis of the local mass concentration of
GMC measures around thick points of the corresponding underlying field.
Moreover, using this result, we provide an explicit expression for the lower
singularity spectrum of the _Multifractal Random Walk_ (MRW) and of the
_Liouville Brownian Motion_ (LBM). Before entering into the details of the
main results, we briefly review the theory of GMC and of multifractal analysis
of measures.

##### Gaussian multiplicative chaos

Given a domain $D\subset\mathbb{R}^{d}$, $d\geq 1$, the theory of GMC,
originally developed by Kahane [Kah85], aims to define rigorously random
measures of the form

$M_{\gamma}({\mathrm{d}}x)=e^{\gamma
X(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X(x)^{2}]}{\mathrm{d}}x\,,$ (1.1)

where ${\mathrm{d}}x$ denotes the Lebesgue measure, $\gamma$ is a real
parameter, and $X$ is a log-correlated Gaussian field on $D$, i.e. a centred
Gaussian field whose covariance kernel can be formally written as

$\mathbb{E}[X(x)X(y)]=-\log|x-y|+g(x,y)\,,\quad x,y\in D\,,$

where $g:D\times D\to\mathbb{R}$ is say bounded and continuous. Since the
covariance kernel of $X$ has a logarithmic divergence along the diagonal, we
cannot define the field $X$ pointwise. However, we can make rigorous sense of
$X$ by viewing it as a random Schwartz distribution. Therefore, the definition
of (1.1) is non-trivial as, a priori, we cannot exponentiate a random
generalized function. In order to interpret (1.1) rigorously, we need to
approximate $X$ via a regularizing procedure which involves a suitable
collection of regularized random fields
$(X_{\varepsilon})_{\varepsilon\in(0,1]}$. The GMC measure associated with $X$
is then given by the limit of the sequence of approximating measures

$M_{\gamma}^{\varepsilon}({\mathrm{d}}x)=e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\,.$

As long as the parameter $\gamma^{2}$ is strictly less than the critical value
$2d$, which is usually called subcritical regime, it is well-known [Ber17,
Kah85, RV10b] that the sequence
$(M_{\gamma}^{\varepsilon})_{\varepsilon\in(0,1]}$ converges weakly in
probability towards a non-degenerate measure $M_{\gamma}$. Moreover, it is
known that $M_{\gamma}$ is almost surely non-atomic, but singular with respect
to the Lebesgue measure. Many further properties of such measures concerning,
among others, moments and multifractal behaviour are known. We refer to
Section 2 for more details.

The original interest in defining GMC measures stemmed from the need of making
rigorous Mandelbrot’s model for energy dissipation in fully developed
turbulence [Man72], but it has since been found applications in a wide range
of fields: from mathematical finance [DRV12] to mathematical physics [DS11],
but also random matrices [BWW18] as well as number theory [SW20]. For a review
on the theory of GMC and for further references and applications we refer to
[RV14].

##### Multifractal analysis

The purpose of multifractal analysis is to finely describe the heterogeneity
in distribution of measures whose mass is concentrated in a highly irregular
way. For $\mu$ a non-negative finite measure supported on a domain
$D\subset\mathbb{R}^{d}$, $d\geq 1$, we introduce the local Hölder exponent
(or local dimension) of $\mu$ at $x\in D$ by letting

$\dim_{\mu}(x):=\lim_{r\searrow 0}\frac{\log\mu(B(x,r))}{\log r}\,,$

provided the limit exists, where $B(x,r)$ denotes the closed ball centred at
$x$ with radius $r$. Then the irregularity on the mass concentration of $\mu$
can be described via the dimension of the sets

$\mathrm{E}_{\mu}(\alpha):=\left\\{x\in
D\,:\,\dim_{\mu}(x)=\alpha\right\\}\,,\quad\alpha\geq 0\,.$

In particular, we say that $\mu$ is a multifractal measure if the sets
$\mathrm{E}_{\mu}(\alpha)$ have positive Hausdorff dimensions for different
values of $\alpha\geq 0$ belonging to an interval with non-empty interior.
Hence, if $\mu$ is a multifractal measure, then the collection of sets
$(\mathrm{E}_{\mu}(\alpha))_{\alpha\geq 0}$ produces a decomposition of $D$
into a family of subfractals.

The main objective of multifractal analysis is to compute the size of
$\mathrm{E}_{\mu}(\alpha)$, i.e. to find an expression for the _singularity
spectrum_ of $\mu$, which is the function
${\mathrm{d}}_{\mu}:[0,\infty)\to[0,\infty)$ defined by

${\mathrm{d}}_{\mu}(\alpha):=\dim_{\mathcal{H}}(\mathrm{E}_{\mu}(\alpha))\,,\quad\alpha\in[0,\infty)\,,$

where $\dim_{\mathcal{H}}$ denotes the Hausdorff dimension. To this end,
Frisch and Parisi [PF85] introduced the notion of _multifractal formalism_ ,
which is a heuristic principle used to establish an explicit connection
between the singularity spectrum ${\mathrm{d}}_{\mu}$ and the $L^{q}$-spectrum
of the measure $\mu$. We define the $L^{q}$-spectrum of $\mu$ by

$\tau_{\mu}(q):=\limsup_{r\searrow 0}\frac{\log\sup\left\\{\sum_{i\in
I}\mu(B(x_{i},r))^{q}\right\\}}{-\log r}\,,\quad q\in\mathbb{R}\,,$

where $(B(x_{i},r))_{i\in I}$ is a countable family of disjoint closed balls
with radius $r$ centred at $x_{i}\in D$, and the supremum is taken over all
such families. We say that $\mu$ satisfies the multifractal formalism if the
following equality holds

${\mathrm{d}}_{\mu}(\alpha)=\tau_{\mu}^{*}(\alpha)\,,\quad\forall\alpha\geq
0\,.$ (1.2)

Here, $\tau_{\mu}^{*}$ refers to the Legendre–Fenchel transform of
$\tau_{\mu}$ which is defined by setting

$\tau_{\mu}^{*}(\alpha):=\inf_{q\in\mathbb{R}}\left\\{\alpha
q+\tau_{\mu}(q)\right\\}\wedge 0\,,\quad\alpha\geq 0\,.$

Let us mention that a rigorous mathematical version of multifractal formalism
was initially developed in [Ols95] and we refer to it for further details.

In order to investigate the local regularity of functions, or in our case of
paths of stochastic processes, we adopt a similar approach. More precisely, if
$I\subset\mathbb{R}$ is an interval and $f:I\to\mathbb{R}^{d}$, $d\geq 1$, is
a given function, we define the lower local Hölder exponent (or lower local
dimension) of $f$ at $x\in I$ by setting

$\underline{\dim}_{f}(x):=\liminf_{r\searrow 0}\frac{\log|f(x+r)-f(x-r)|}{\log
r}\,,$

and we define the sets in which $f$ has lower local Hölder exponent
$\alpha\geq 0$ as

$\underline{\mathrm{E}}_{f}(\alpha):=\left\\{x\in
I\,:\,\underline{\dim}_{f}(x)=\alpha\right\\}\,.$

As in the case of measures, we are interested in finding an explicit
expression for the lower singularity spectrum of $f$, which is the function
$\underline{{\mathrm{d}}}_{f}:[0,\infty)\to[0,\infty)$ defined by

$\underline{{\mathrm{d}}}_{f}(\alpha):=\dim_{\mathcal{H}}(\underline{\mathrm{E}}_{f}(\alpha))\,,\quad\alpha\in[0,\infty)\,.$

###  1.1 Main results

We state here the main results of this paper, and we refer to Section 3, 4 and
5 for the precise statements and for further details.

##### Multifractal analysis of GMC

For a bounded domain $D\subset\mathbb{R}^{d}$, $d\geq 1$, we consider a log-
correlated Gaussian field on $D$, and we show that the associated subcritical
GMC measure $M_{\gamma}$ satisfies the multifractal formalism, in the sense of
(1.2). This result has been heuristically discussed in [RV14, Section 4], but
to the best of our knowledge it has never been proved rigorously. To be
precise, in Theorem 3.1 we show that for $\gamma^{2}<2d$ and $\alpha\geq 0$,
it holds almost surely that

${\mathrm{d}}_{M_{\gamma}}(\alpha)=\tau_{M_{\gamma}}^{*}(\alpha)=\begin{cases}d-\frac{1}{2}\left(\frac{d-\alpha}{\gamma}+\frac{\gamma}{2}\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\sqrt{d}-\frac{|\gamma|}{\sqrt{2}}\right)^{2},\left(\sqrt{d}+\frac{|\gamma|}{\sqrt{2}}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

hence recovering precisely the prediction made in [RV14]. Furthermore, as a
by-product, we also obtain an explicit expression for the $L^{q}$-spectrum of
$M_{\gamma}$.

Let us briefly flesh out the intuition behind the proof of our main result.
The strategy of the proof is inspired by the multifractal analysis of
multiplicative cascades which has been performed in [Bar99]. The upper bound
for ${\mathrm{d}}_{M_{\gamma}}$, i.e. the inequality
${\mathrm{d}}_{M_{\gamma}}\leq\tau_{M_{\gamma}}^{*}$, follows from the general
theory of the multifractal analysis of measures. On the other hand, the lower
bound is more involved to prove and it is based on the following heuristic.
For $\gamma^{2}<2d$, it is known that the GMC measure $M_{\gamma}$ is carried
by the set of $\gamma$-thick points of the underlying field $X$ which is
defined by

$\mathcal{T}_{\gamma}:=\left\\{x\in D\,:\,\lim_{\varepsilon\searrow
0}\frac{X_{\varepsilon}(x)}{-\log\varepsilon}=\gamma\right\\}\,,$ (1.3)

where $(X_{\varepsilon})_{\varepsilon\in(0,1]}$ is a suitable regularization
of $X$. The key to our proof is to study the local mass concentration of
$M_{\gamma}$ around thick points of $X$. More precisely, following the idea of
[RV14, Theorem 4.1], we show that it exists a non-random exponent
$\alpha_{q}\geq 0$ such that $M_{\gamma}$ has local dimension $\alpha_{q}$ at
points in $\mathcal{T}_{q\gamma}$, for all $q^{2}<2d/\gamma^{2}$.
Consequently, this implies that $M_{\gamma}$ has local Hölder exponent
$\alpha_{q}$ on a set of full $M_{q\gamma}$-measure. This fact, together with
some known properties of GMC measures, is enough to prove the lower bound for
${\mathrm{d}}_{M_{\gamma}}$, i.e. the inequality
${\mathrm{d}}_{M_{\gamma}}\geq\tau_{M_{\gamma}}^{*}$.

##### Multifractal analysis of MRW and LBM

The MRW has been first introduced in [BDM01] as a stochastic volatility model,
and it can be simply defined as follows. Fix a time $T>0$ and let $d\geq 1$,
then the $d$-dimensional MRW $\mathcal{Z}^{\gamma}$ is defined for
$\gamma^{2}<2$ as

$\mathcal{Z}^{\gamma}_{t}:=B_{M_{\gamma}([0,t])}\,,\quad t\in[0,T]\,,$

where $M_{\gamma}$ is a GMC measure on $[0,T]$ and $B$ is an independent
$d$-dimensional Brownian motion. In Theorem 4.3, we find the relation between
the lower singularity spectrum of the paths of $\mathcal{Z}^{\gamma}$ and the
singularity spectrum of $M_{\gamma}$. More precisely, for $\gamma^{2}<2$ and
$\alpha\geq 0$, we show that

$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}(\alpha)={\mathrm{d}}_{M_{\gamma}}(2\alpha)=\begin{cases}d-\frac{1}{2}\left(\frac{1-2\alpha}{\gamma}+\frac{\gamma}{2}\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\frac{1}{\sqrt{2}}-\frac{|\gamma|}{2}\right)^{2},\left(\frac{1}{\sqrt{2}}+\frac{|\gamma|}{2}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

almost surely.

The LBM has been simultaneously defined in [Ber15, GRV16] as the canonical
planar diffusion associated with the Liouville quantum gravity metric tensor.
In this article, we will consider the $d$-dimensional LBM, for $d\geq 2$. More
precisely, let $D\subset\mathbb{R}^{d}$ be a bounded domain, let $B$ be a
$d$-dimensional Brownian motion started inside $D$, and let $X$ be an
independent log-correlated Gaussian field on $D$. Then the LBM
$\mathcal{B}^{\gamma}$ on $D$ can be formally defined for $\gamma^{2}<4$ as

$\mathcal{B}^{\gamma}_{t}:=B_{F_{\gamma}^{-1}(t)}\,,\qquad
F_{\gamma}(t):=\int_{0}^{t\wedge T}e^{\gamma
X(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X(B_{s})^{2}]}{\mathrm{d}}s\,,\quad
t\geq 0\,,$

where $T$ is the first exit time of $B$ from $D$. As we will see in Section 5,
adapting the original definition of planar LBM to higher dimensions is a
straightforward task. Then, we focus on computing the lower singularity
spectrum of the paths of $\mathcal{B}_{\gamma}$, and in Theorem 5.5, we show
that for $\gamma^{2}<4$ and $\alpha\geq 0$, it holds that

$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(\alpha)=\begin{cases}2\alpha-2\alpha\left(\frac{2\alpha-1}{2\alpha\gamma}+\frac{1}{4}\gamma\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\sqrt{2}+\frac{|\gamma|}{\sqrt{2}}\right)^{-2},\left(\sqrt{2}-\frac{|\gamma|}{\sqrt{2}}\right)^{-2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

almost surely. Along the proof of this result, we show that the measure
$\mu_{\gamma}$ defined by $\mu_{\gamma}([s,t]):=F_{\gamma}(t)-F_{\gamma}(s)$,
for $s\leq t\in[0,T]$, satisfies the multifractal formalism and we compute its
singularity spectrum.

###  1.2 Structure of the paper

The reminder of this article is structured as follows. In Section 2, we
collect some definitions and results that are used in the rest of the article.
More precisely, we start by recalling the definition of log-correlated
Gaussian field, and we prove a lemma concerning the fluctuations of its
convolution approximation. Then we recall some known properties of GMC
measures, we collect some facts about the Hausdorff dimension, and finally we
state some general results on multifractal analysis of measures. In Section 3,
we state precisely our main result and we perform its proof. Sections 4 and 5
are devoted to computing the lower singularity spectrum of the MRW and of the
LBM, respectively. In Appendix A, we prove Proposition 3.4, which is the main
step of the proof of our main result. Appendix B contains the proof of the
finiteness of positive moments of the measure involved in the definition of
the LBM. Finally, Appendix C collects some general results on Gaussian fields
that are used throughout the paper.

######  Acknowledgements.

The author would like to thank Prof. M. Hairer for his constant support and
guidance. We thank an anonymous referee for many helpful comments on an
earlier version of this article. The author is very grateful to the Royal
Society for financial support through Prof. M. Hairer’s research professorship
grant RP\R1\191065.

##  2 Preliminaries

In this section, after introducing the basic notation, we collect some
definitions and results on log-correlated Gaussian fields, on the theory of
GMC, on the Hausdorff dimension, and on multifractal analysis of measures.

### Basic notation

We let $\mathbb{N}:=\\{1,2,3,\dots\\}$. For $d\geq 1$, we use $\mathbb{R}^{d}$
to indicate the $d$-dimensional Euclidean space. If $a$ and $b$ are two
quantities, we use $a\lesssim b$ to denote the statement $a\leq Cb$ for some
constant $C>0$ independent from the parameters of interest. Given a subset
$D\subset\mathbb{R}^{d}$, we denote by $\bar{D}$ its closure and by $\partial
D$ its boundary.

###  2.1 Log-correlated Gaussian fields

Given a bounded domain $D\subset\mathbb{R}^{d}$, $d\geq 1$, a log-correlated
Gaussian field $X$ on $D$ is a Gaussian field whose covariance kernel takes
the form

$K(x,y)=-\log|x-y|+g(x,y)\,,\quad x,y\in D\,,$ (2.1)

where $g\in C(\bar{D}\times\bar{D})$. We adopt the convention to extend the
covariance kernel $K$ to $\mathbb{R}^{d}\times\mathbb{R}^{d}$ by setting
$K(x,y)=0$ whenever $(x,y)\not\in D\times D$. Moreover, for (2.1) to be a
covariance kernel, we need to require that it is symmetric and non-negative
semi-definite. Since the covariance kernel $K$ has a singularity on the
diagonal, the field $X$ does not make literal sense as a pointwise defined
Gaussian field, but it can be rigorously interpreted as a random Schwartz
distribution. Such a random generalized function can be characterized by the
property that, for any compactly supported test function $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$, the pairing $(X,\phi)$ produces a centred
Gaussian random variable with variance

$\mathbb{E}[(X,\phi)(X,\phi)]=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\phi(x)K(x,y)\phi(y){\mathrm{d}}x{\mathrm{d}}y\,.$

The existence of such a stochastic process follows from a direct construction.
Indeed, it can be easily verified that $K$, as defined in (2.1), is the kernel
of a self-adjoint Hilbert–Schmidt operator on $L^{2}(\mathbb{R}^{d})$. In
particular, such an operator is symmetric and compact, so by the spectral
theorem there exist a non-increasing sequence $(\lambda_{n})_{n\in\mathbb{N}}$
of strictly positive eigenvalues and corresponding eigenfunctions
$(\phi_{n})_{n\in\mathbb{N}}$ that form an orthonormal basis of
$\text{Ker}(K)^{\perp}$ in $L^{2}(\mathbb{R}^{d})$. Then the field $X$ can be
defined via its Karhunen–Loève expansion

$X(x)=\sum_{n\in\mathbb{N}}Z_{n}\sqrt{\lambda_{n}}\phi_{n}\,,$

where $(Z_{n})_{n\in\mathbb{N}}$ is a collection of i.i.d. standard normal
random variables defined on a common probability space
$(\Omega,\mathcal{F},\mathbb{P})$. Note that the functions $\phi_{n}$ are
supported on the domain $D$. It can be proved that the convergence of the
above series takes place in the space $H^{-\alpha}(\mathbb{R}^{d})$, for any
$\alpha>0$. We refer to [JSW20, Section 2] for more details.

In order to give a meaning to GMC measures, we need to define the exponential
of a random Schwartz distribution. This is done through a regularization and
limiting procedure. The most general and natural way to obtain a
regularization of $X$ is through the so-called convolution approximation. If
we let $\psi\in C^{\infty}_{c}(\mathbb{R}^{d})$ be non-negative, radially
symmetric, with compact support and unit mass, then the
$\varepsilon$-convolution approximation of the field $X$ is defined by

$X_{\varepsilon}(x):=X*\psi_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}X(x^{\prime})\psi_{\varepsilon}(x-x^{\prime}){\mathrm{d}}x^{\prime}\,,\quad
x\in\mathbb{R}^{d}\,,\quad\varepsilon\in(0,1]\,,$ (2.2)

where $\psi_{\varepsilon}(x)=\varepsilon^{-d}\psi(\varepsilon^{-1}x)$ for
$x\in\mathbb{R}^{d}$. As we observed above, $X\in H^{-\alpha}(\mathbb{R}^{d})$
for any $\alpha>0$, and so the convolution $X*\psi_{\varepsilon}$ is actually
well-defined. Moreover, it is easy to check that the regularized field
$X_{\varepsilon}$ is a centred Gaussian field with covariance kernel given by

$K_{\varepsilon}(x,y)=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\psi_{\varepsilon}(x-x^{\prime})K(x^{\prime},y^{\prime})\psi_{\varepsilon}(y-y^{\prime}){\mathrm{d}}x^{\prime}{\mathrm{d}}y^{\prime}\,,\quad
x,y\in\mathbb{R}^{d}\,.$

Furthermore, the following properties of the convolution approximation
$(X_{\varepsilon})_{\varepsilon\in(0,1]}$ are well known (see e.g. [JSW20,
Lemma 2.8]):

  1. (P1)

for Lebesgue-almost every $(x,y)\in D\times D$,

$\lim_{\varepsilon\searrow 0}K_{\varepsilon}(x,y)=K(x,y)\,;$

  2. (P2)

there exists a finite constant $K>0$ such that,

$\sup_{0<\varepsilon^{\prime}<\varepsilon\leq 1}\sup_{x,y\in
D}\left|\mathbb{E}[X_{\varepsilon}(x)X_{\varepsilon^{\prime}}(y)]+\log\left(|x-y|+\varepsilon\right)\right|<K\,;$

  3. (P3)

for all $\varepsilon\in(0,1]$, the map $D\ni x\mapsto X_{\varepsilon}(x)$ is
almost surely continuous.

Let us mention that for particular types of log-correlated Gaussian fields
there are some other natural approximations having properties (P1), (P2), and
(P3). For example, if $X$ is a two-dimensional _Gaussian Free Field_ (GFF),
then one can use the regularization obtained through the circle average around
each point in the domain (cf. [DS11]).

Before proceeding, we state and prove a lemma that allows to control the
exponential moment of the fluctuations of the convolution approximation of
log-correlated Gaussian fields. Such lemma is crucial in the proof of our main
result, more specifically in the proof of Proposition 3.4.

######  Lemma 2.1.

For $d\geq 1$, let $D\subset\mathbb{R}^{d}$ be a bounded domain. Consider a
log-correlated Gaussian field $X$ on $D$ with covariance kernel (2.1) and let
$(X_{\varepsilon})_{\varepsilon\in(0,1]}$ be its convolution approximation, as
defined in (2.2). Then there exists a finite constant $C>0$ such that,

$\sup_{r\in(0,1]}\sup_{x\in D_{r}}\mathbb{E}\left[e^{\sup_{u\in
B(x,r)}X_{r}(u)-X_{r}(x)}\right]\leq C\,,$

where $D_{r}:=\\{x\in D\,:\,\operatorname{dist}(x,\partial D)<r\\}$.

###### Proof.

Fix $r\in(0,1]$ and $x\in D_{r}$. Consider the centred Gaussian field
$(X_{r}^{x}(u))_{u\in B(x,r)}$ defined by

$X_{r}^{x}(u):=X_{r}(u)-X_{r}(x)\,,\quad u\in B(x,r)\,,$

and the random variable $\Omega_{r}^{x}$ given by

$\Omega_{r}^{x}:=\sup_{u\in B(x,r)}X_{r}^{x}(u)\,.$

The variance of the field $(X_{r}^{x}(u))_{u\in B(x,r)}$ can be uniformly
bounded in $r\in(0,1]$ and $x\in D_{r}$. Indeed, for $u$, $v\in B(x,r)$,
thanks to property (P2), it exists a finite constant $K>0$ such that

$\displaystyle\mathbb{E}[X_{r}^{x}(u)X_{r}^{x}(v))]$
$\displaystyle\leq-\log(|u-v|+r)+\log(|u-x|+r)+\log(|v-x|+r)-\log r+4K$
$\displaystyle\leq-2\log r+2\log 2r+4K$ $\displaystyle=2\log 2+4K\,,$ (2.3)

where the last inequality is due to the fact that $|u-x|\leq r$ and $|v-x|\leq
r$. Thanks to property (P3), we know that the the field $X_{r}^{x}$ is almost
surely continuous and so we can apply Borell-TIS inequality (cf. Lemma C.3) to
obtain that

$\mathbb{P}\left(\left|\Omega_{r}^{x}-\mathbb{E}[\Omega_{r}^{x}]\right|>t\right)\leq
2e^{-\frac{t^{2}}{2\sigma_{x,r}^{2}}}\,,$ (2.4)

for all $t\geq 0$, where $\sigma_{x,r}^{2}:=\sup_{u\in
B(x,r)}\mathbb{E}\left[X_{r}^{x}(u)^{2}\right]$. We claim that both
$\sigma_{x,r}^{2}$ and $\mathbb{E}[\Omega_{r}^{x}]$ can be uniformly bounded
in $x$ and $r$. For $\sigma_{x,r}^{2}$, this follows easily from (2.1). For
$\mathbb{E}[\Omega_{r}^{x}]$, we can apply Dudley’s entropy bound (cf. Lemma
C.4). Indeed, doing computation similar to the one in (2.1), and using the
properties of the convolution approximation (cf. also [JSW20, Lemma 2.8]), one
can easily verify that there exists a finite constant $A>0$, that does not
depend on $x$ and $r$, such that

$\mathbb{E}[(X_{r}^{x}(u)-X_{r}^{x}(v))^{2}]=\mathbb{E}[(X_{r}(u)-X_{r}(v))^{2}]\leq
A\frac{|u-v|}{r}\,,\quad\forall u,v\in B(x,r)\,.$

Therefore, applying Lemma C.4 to the field $(X_{r}^{x}(u))_{u\in B(x,r)}$, one
can readily obtain the desired uniform upper bound. Hence, from (2.4), it
follows that there exist finite constants $a_{0}$, $a_{1}>0$, independent of
$x$ and $r$, such that

$\mathbb{P}\left(\Omega_{r}^{x}>t\right)\leq
2e^{-\frac{(t-a_{0})^{2}}{2a_{1}}}\,.$

Finally, remembering that for every non-negative random variable $X$ it holds
that $\mathbb{E}[X]=\int_{0}^{\infty}\mathbb{P}(X>t){\mathrm{d}}t$, we have
that

$\mathbb{E}\left[e^{\Omega_{r}^{x}}\right]=\int_{0}^{\infty}\mathbb{P}(e^{\Omega_{r}^{x}}>t){\mathrm{d}}t=\int_{0}^{\infty}\mathbb{P}\left(\Omega_{r}^{x}>\log
t\right){\mathrm{d}}t\leq\int_{0}^{\infty}2e^{-\frac{(\log(t)-a_{0})^{2}}{2a_{1}}}{\mathrm{d}}t<\infty\,,$

uniformly in $x$ and $r$, which concludes the proof. ∎

###  2.2 Properties of Gaussian multiplicative chaos

We recall here the main properties of GMC measures that we need in the
following sections. For $d\geq 1$, let $D\subset\mathbb{R}^{d}$ be a bounded
domain. Consider a log-correlated Gaussian field $X$ defined on $D$ with
covariance kernel (2.1) and let $(X_{\varepsilon})_{\varepsilon\in(0,1]}$ be
its convolution approximation as defined in (2.2).

We start by recalling the following standard result concerning the existence
and the non-degeneracy of GMC measures.

######  Proposition 2.2 ([Ber17, Theorem 1.1]).

If $\gamma^{2}<2d$, then the sequence of approximating measures

$M^{\varepsilon}_{\gamma}({\mathrm{d}}x):=e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\,,\quad\varepsilon\in(0,1]\,,$

converges in probability and in $L^{1}(\mathbb{P})$ to some non-degenerate
limit $M_{\gamma}$, called GMC measure, in the space of Radon measures with
respect to the topology of weak convergence. Furthermore, $M_{\gamma}$ does
not depend on the choice of the mollifier $\psi$ used in (2.2).

For $\gamma^{2}<2d$, it can be easily verified that the random measure
$M_{\gamma}$ is almost surely supported on the whole domain $D$ (cf. [RV14]).
On the other hand, $M_{\gamma}$ gives full measure to the set of
$\gamma$-thick points $\mathcal{T}_{\gamma}$, defined in (1.3). It is known
that the Hausdorff dimension of $\mathcal{T}_{\gamma}$ is equal to
$d-\gamma^{2}/2$ (cf. [RV14, Theorem 4.2]). Immediate consequences of this
fact are that $M_{\gamma}$ is almost surely atomless, and that $M_{\gamma}$ is
singular with respect to $M_{\gamma^{\prime}}$ for any
$\gamma\neq\gamma^{\prime}$. In particular, for $\gamma\neq 0$, $M_{\gamma}$
is singular with respect to the Lebesgue measure. Let us also observe that
when $\gamma^{2}$ gets closer to $2d$, the measure $M_{\gamma}$ is carried by
a set whose Hausdorff dimension gets closer to $0$. In particular, this means
that $M_{\gamma}$ tends to cluster as $\gamma^{2}$ increases, until it
degenerates at $\gamma^{2}=2d$. Let us emphasize that there is a rich
literature on critical GMC measures, i.e. when $\gamma^{2}=2d$, and we refer
to [Pow20] for a review.

We have the following standard result concerning the existence of uniform
bounds on positive and negative moments of GMC measures.

######  Proposition 2.3 ([RV14, Theorems 2.11 and 2.12]).

Let $\gamma^{2}<2d$ and $q<2d/\gamma^{2}$. Then for any non-empty compact set
$A\subset D$ it holds that

$\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{A}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{q}\right]<\infty\,.$

An important feature of GMC measures is their multifractal behaviour. More
precisely, it can be shown that the moments of $M_{\gamma}$ have a power law
behaviour in which the exponents can be expressed through a non-linear
function $\xi_{M_{\gamma}}$, called the power law spectrum, which is defined
by

$\xi_{M_{\gamma}}(q):=\left(d+\frac{1}{2}\gamma^{2}\right)q-\frac{1}{2}\gamma^{2}q^{2}\,,\quad
q\in\mathbb{R}\,.$ (2.5)

More formally, we have the following proposition.

######  Proposition 2.4 ([RV14, Theorem 2.14]).

Let $\gamma^{2}<2d$ and $q<2d/\gamma^{2}$. Then there exists a finite constant
$C>0$ such that for all $x\in D$ and $r\in(0,1]$ it holds that

$\mathbb{E}[M_{\gamma}(B(x,r))^{q}]\leq Cr^{\xi_{M_{\gamma}}(q)}\,.$

Finally, we state below a result on the local modulus of continuity of GMC
measures which is an easy consequence of Proposition 2.4.

######  Proposition 2.5.

Let $\gamma^{2}<2d$ and $\varepsilon\in(0,1]$. Then, almost surely, there
exist finite constants $C_{1}$, $C_{2}>0$ such that for all $x\in D$ and
$r\in(0,1]$ it holds that

$C_{1}r^{\left(\sqrt{d}+|\gamma|/\sqrt{2}\right)^{2}+\varepsilon}\leq
M_{\gamma}(B(x,r))\leq
C_{2}r^{\left(\sqrt{d}-|\gamma|/\sqrt{2}\right)^{2}-\varepsilon}$

###### Proof.

A proof of the upper bound is given in [GRV16, Theorem 2.2] in the case $d=2$.
However, the proof can be easily adapted to all dimensions. The lower bound
can be proved similarly. Indeed, let us assume for simplicity that
$D=(-1,1)^{d}$ and let $\alpha:=(\sqrt{d}+|\gamma|/\sqrt{2})^{2}$. For
$n\in\mathbb{N}$, we write $\Sigma_{n}:=2^{-n}\mathbb{Z}^{d}\cap(-1,1)^{d}$
for the lattice $2^{-n}\mathbb{Z}^{d}$ restricted to the open box
$(-1,1)^{d}$. Thanks to an union bound, Markov’s inequality, and Proposition
2.4, it holds for each $n\in\mathbb{N}$, $\varepsilon\in(0,1]$, and $q<0$ that

$\displaystyle\mathbb{P}\left(\min_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n}))\leq
2^{-n(\alpha+\varepsilon)}\right)$
$\displaystyle=\mathbb{P}\left(\max_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n}))^{q}\geq
2^{-nq(\alpha+\varepsilon)}\right)$
$\displaystyle\leq\sum_{z\in\Sigma_{n}}2^{nq(\alpha+\epsilon)}\mathbb{E}[M_{\gamma}(B(z,2^{-n})^{q})]$
$\displaystyle\lesssim 2^{nq\varepsilon-n(\xi_{M_{\gamma}}(q)-d-\alpha q)}\,,$

where the implicit constant does not depend on $n$. If we chose
$q=-\sqrt{2d}/|\gamma|$, then one can easily check that
$\xi_{M_{\gamma}}(q)-d-\alpha q=0$. Therefore, thanks to the Borel–Cantelli
lemma, we have that it almost surely exists a finite constant $C>0$ such that

$\min_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n})\geq
C2^{-n(\alpha+\varepsilon)}\,,\quad\forall n\in\mathbb{N}\,.$

To conclude, it is sufficient to notice that for all $x\in(-1,1)^{d}$ and
$r\in(0,1]$, there exist $n\in\mathbb{N}$ and $z\in\Sigma_{n+1}$ such that
$2^{-n}<r\leq 2^{-n+1}$ and $B(z,2^{-n-1})\subset B(x,r)$. ∎

###  2.3 Hausdorff dimension

We collect here the definition and some properties of the Hausdorff dimension
that we need in the sequel. We refer to [Fal97] for further details.

######  Definition 2.6.

Let $D\subset\mathbb{R}^{d}$, $d\geq 1$, and $s\geq 0$. We define the
$s$-dimensional Hausdorff measure of $D$ by

$\mathcal{H}^{s}(D):=\lim_{\delta\searrow 0}\mathcal{H}^{s}_{\delta}(D)\,,$

where

$\mathcal{H}_{\delta}^{s}(D):=\inf\left\\{\sum_{n\in\mathbb{N}}\operatorname{diam}(U_{n})^{s}\,:\,(U_{n})_{n\in\mathbb{N}}\text{
is a }\delta\text{-cover of }D\right\\}\,,\quad\delta\in(0,1]\,.$

The Hausdorff dimension of $D$ is defined by the following equivalent formulas

$\dim_{\mathcal{H}}(D):=\inf\\{s\geq
0\,:\,\mathcal{H}^{s}(D)=0\\}=\sup\\{s\geq
0\,:\,\mathcal{H}^{s}(D)=\infty\\}\,,$

with the convention that $\dim_{\mathcal{H}}(\emptyset)=0$.

Note that $0\leq\dim_{\mathcal{H}}(D)\leq d$ for any $D\subset\mathbb{R}^{d}$,
$d\geq 1$. Moreover, from the definition of Hausdorff dimension, it
immediately follows that if $\mathcal{H}^{s}(D)<\infty$, then
$\dim_{\mathcal{H}}(D)\leq s$. Furthermore, the Hausdorff dimension enjoys
monotonicity and countable stability, i.e.
$\dim_{\mathcal{H}}(D)\leq\dim_{\mathcal{H}}(D^{\prime})$ for any $D\subset
D^{\prime}$, and
$\dim_{\mathcal{H}}(\cup_{n\in\mathbb{N}}D_{n})=\sup_{n\in\mathbb{N}}\dim_{\mathcal{H}}(D_{n})$
for any collection of subsets $(D_{n})_{n\in\mathbb{N}}$.

Other useful properties of the Hausdorff dimension are collected in the
following proposition.

######  Proposition 2.7 ([Jac18, Proposition 2.15]).

Let $f:[0,\infty)\to\mathbb{R}$ be a continuous and increasing function.
Consider a set $D\subset[0,\infty)$ and assume there exist finite constants
$C$, $R$, $\alpha>0$ such that $|f(x+r)-f(x-r)|\leq Cr^{\alpha}$ for all
$r\in[0,R)$ and $x\in D$. Then it holds that

$\dim_{\mathcal{H}}(f(D))\leq\frac{1}{\alpha}\dim_{\mathcal{H}}(D)\,.$

We conclude this subsection with the following result.

######  Proposition 2.8 ([Fal97, Proposition 2.3]).

Let $\mu$ be a non-negative finite Radon measure on $\mathbb{R}^{d}$ supported
on a bounded domain $D\subset\mathbb{R}^{d}$, $d\geq 1$. Let $E\subset D$ such
that $\mu(E)>0$. For each $x\in E$, we let $\dim_{\mu}(x)$ to be the local
dimension of $\mu$ at $x$ as defined in (2.10) below. If $\dim_{\mu}(x)=s$ for
all $x\in E$, then $\dim_{\mathcal{H}}(E)=s$.

###  2.4 Multifractal analysis of measures

We collect here some general facts about multifractal analysis of measures.
Let $\mu$ be a non-negative finite Radon measure on $\mathbb{R}^{d}$, $d\geq
1$, which may be random or non-random, supported on a bounded domain
$D\subset\mathbb{R}^{d}$.

######  Definition 2.9.

The $L^{q}$-spectrum of $\mu$ is the function
$\tau_{\mu}:\mathbb{R}\to\mathbb{R}$ defined by

$\tau_{\mu}(q):=\limsup_{r\searrow 0}\frac{\log\sup\sum_{i\in
I}\mu(B(x_{i},r))^{q}}{-\log r}\,,\quad q\in\mathbb{R}\,,$ (2.6)

where $(B(x_{i},r))_{i\in I}$ is a countable family of disjoint closed balls
with radius $r$ centred at $x_{i}\in D$, and the supremum is taken over all
such families. Moreover, the Legendre–Fenchel transform of the
$L^{q}$-spectrum $\tau_{\mu}$ is the function
$\tau_{\mu}^{*}:[0,\infty)\to[0,\infty)$ defined by

$\tau_{\mu}^{*}(\alpha):=\inf_{q\in\mathbb{R}}\left\\{\alpha
q+\tau_{\mu}(q)\right\\}\wedge 0\,,\quad\alpha\in[0,\infty)\,.$ (2.7)

We have the following classical result concerning the $L^{q}$-spectrum.

######  Proposition 2.10 ([LN99, Section 3]).

The $L^{q}$-spectrum $\tau_{\mu}$ is a decreasing convex function with
$\tau_{\mu}(1)=0$. Moreover,
$\operatorname{dom}(\tau_{\mu}):=\\{q\in\mathbb{R}\,:\,\tau_{\mu}(q)<\infty\\}=\mathbb{R}$
if and only if

$\limsup_{r\searrow 0}\frac{\log\inf_{x\in D}\\{\mu(B(x,r))\\}}{\log
r}<\infty\,.$

Let us collect here some notation. For $\alpha\geq 0$, we introduce the sets
$\mathrm{E}_{\mu}(\alpha)$, $\underline{\mathrm{U}}_{\mu}(\alpha)$ and
$\overline{\mathrm{B}}_{\mu}(\alpha)$ defined by

$\displaystyle\mathrm{E}_{\mu}(\alpha):=\left\\{x\in
D\,:\,\dim_{\mu}(x)=\alpha\right\\}\,,$ (2.8)
$\displaystyle\underline{\mathrm{U}}_{\mu}(\alpha):=\\{x\in
D\,:\,\underline{\dim}_{\mu}(x)\leq\alpha\\}\,,\quad\overline{\mathrm{B}}_{\mu}(\alpha):=\\{x\in
D\,:\,\overline{\dim}_{\mu}(x)\geq\alpha\\}\,,$ (2.9)

where

$\displaystyle\dim_{\mu}(x):=\lim_{r\searrow 0}\frac{\log\mu(B(x,r))}{\log
r}\,,$ (2.10) $\displaystyle\underline{\dim}_{\mu}(x):=\liminf_{r\searrow
0}\frac{\log\mu(B(x,r))}{\log
r}\,,\quad\overline{\dim}_{\mu}(x):=\limsup_{r\searrow
0}\frac{\log\mu(B(x,r))}{\log r}\,,$

which are called the local dimension, the lower local dimension and the upper
local dimension of $\mu$ at $x$, respectively.

######  Definition 2.11.

The singularity spectrum of $\mu$ is the function
${\mathrm{d}}_{\mu}:[0,\infty)\to[0,\infty)$ defined by

${\mathrm{d}}_{\mu}(\alpha):=\dim_{\mathcal{H}}(\mathrm{E}_{\mu}(\alpha))\,,\quad\alpha\in[0,\infty)\,.$
(2.11)

We have the following fundamental result.

######  Proposition 2.12.

For $\alpha\geq 0$, it holds that
$\dim_{\mathcal{H}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))\leq\tau_{\mu}^{*}(\alpha)$.

A proof of Proposition 2.12, in a slightly different setting, can be found in
[BMP92, Theorem 1]. However, the proof we provide here is a simple
generalization of [BJ10, Proposition A.1]. The proof is based on Besicovitch’s
covering theorem (cf. [Fal97]).

######  Lemma 2.13 (Besicovitch’s covering theorem).

Consider a set $E\subset\mathbb{R}^{d}$ and for each $x\in E$ fix a number
$r_{x}>0$ such that $\sup_{x\in E}r_{x}<\infty$. Then there exists an integer
$\sigma_{d}$, depending only on the dimension $d$, for which there exist
countable subfamilies $\mathcal{B}_{1},\dots,\mathcal{B}_{\sigma_{d}}$ of
$\\{B(x,r_{x})\,:\,x\in E\\}$ such that
$E\subset\cup_{i\in\\{1,\dots,\sigma_{d}\\}}\cup_{B\in\mathcal{B}_{i}}B$ and
$\mathcal{B}_{i}$ is a collection of disjoint sets for each
$i\in\\{1,\dots,\sigma_{d}\\}$.

###### Proof of Proposition 2.12.

Fix $\alpha\geq 0$. By definition (2.7) of $\tau_{\mu}^{*}$, it is sufficient
to prove that

$\dim_{\mathcal{H}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))\leq\alpha
q+\tau_{\mu}(q)\,,\quad\forall q\in R\,.$ (2.12)

Without loss of generality, we can assume that $\tau_{\mu}(q)<\infty$, for all
$q\in\mathbb{R}$. We split the proof into two parts. In the first part we
prove inequality (2.12) for $q\geq 0$, while in the second part we prove
inequality (2.12) for $q<0$.

Case $q\geq 0$. Define
$\overline{s}_{q,\varepsilon}:=(\alpha+\varepsilon)q+\tau_{\mu}(q)+\varepsilon$,
for $\varepsilon>0$ fixed. Using the notation introduced in Subsection 2.3, it
is sufficient to show that
$\sup_{\delta\in(0,1]}\mathcal{H}_{\delta}^{\overline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))<\infty$.
By definition, for every
$x\in\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)$,
it exists a decreasing sequence $(r_{x,n})_{n\in\mathbb{N}}$ converging to $0$
such that

$\mu(B(x,r_{x,n}))\geq r_{x,n}^{\alpha+\varepsilon}\,,\quad\forall
n\in\mathbb{N}\,.$

Fix $\delta\in(0,1]$, then for each
$x\in\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)$,
we choose $n_{x}$ such that $r_{x,n_{x}}\in(0,\delta)$. For each
$m\in\mathbb{N}$, we let

$F_{m}:=\left\\{x\in\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)\,:\,2^{-m}<r_{x,n_{x}}\leq
2^{-m+1}\right\\}\,.$

Thanks to the Besicovitch’s covering theorem (cf. Lemma 2.13), it exists a
positive integer $\sigma_{d}$ such that for every $m\in\mathbb{N}$, we can
find $\sigma_{d}$ disjoint subsets $F_{m,1},\dots,F_{m,\sigma_{d}}$ of $F_{m}$
such that each $F_{m,j}$ is at most countable, the balls $B(x,r_{x,n_{x}})$
with centres at $x\in F_{m,j}$ are pairwise disjoint and

$\left(\left(\left(B(x,r_{x,n_{x}})\right)_{x\in
F_{m,j}}\right)_{j\in\\{1,\dots,\sigma_{d}\\}}\right)_{m\in\mathbb{N}}$

is a $\delta$-cover of
$\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)$.
Then we have that

$\displaystyle\mathcal{H}_{\delta}^{\overline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))$
$\displaystyle\leq\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{d}}\sum_{x\in
F_{m,j}}r_{x,n_{x}}^{(\alpha+\varepsilon)q+\tau_{\mu}(q)+\varepsilon}$
$\displaystyle\leq\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{d}}\sum_{x\in
F_{m,j}}\mu(B(x,r_{x,n_{x}}))^{q}r_{x,n_{x}}^{\tau_{\mu}(q)+\varepsilon}$
$\displaystyle\lesssim\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{d}}\sum_{x\in
F_{m,j}}\mu(B(x,2^{-m+1}))^{q}2^{-m(\tau_{\mu}(q)+\varepsilon)}\,,$

where the implicit constant does not depend on $m$. We observe that, for all
$j\in{1,\dots,\sigma_{d}}$, the family $(B(x,2^{-m+1}))_{x\in F_{m,j}}$ can be
divided into two countable families of disjoint closed balls with centres in
$D$. Hence, by definition of $\tau_{\mu}$, for $m$ large enough, we get

$\sum_{x\in F_{m,j}}\mu(B(x,2^{-m+1}))^{q}\lesssim
2^{m(\tau_{\mu}(q)+\varepsilon/2)}\,,$

which implies that
$\sup_{\delta\in(0,1]}\mathcal{H}_{\delta}^{\overline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))<\infty$.

Case $q<0$. Define
$\underline{s}_{q,\varepsilon}:=(\alpha-\varepsilon)q+\tau_{\mu}(q)+\varepsilon$,
for $\varepsilon>0$ fixed. As before, it is sufficient to show that
$\sup_{\delta\in(0,1]}\mathcal{H}_{\delta}^{\underline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))<\infty$.
By definition, for every
$x\in\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)$,
it exists a decreasing sequence $(u_{x,n})_{n\in\mathbb{N}}$ converging to $0$
such that

$\mu(B(x,u_{x,n}))\leq u_{x,n}^{\alpha-\varepsilon}\,,\quad\forall
n\in\mathbb{N}\,.$

Proceeding similarly to the previous case, we get

$\mathcal{H}_{\delta}^{\underline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))\lesssim\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{d}}\sum_{x\in
F_{m,j}}\mu(B(x,2^{-m}))^{q}2^{-m(\tau_{\mu}(q)+\varepsilon)}\,,$

where the implicit constant does not depend on $m$, and by definition of
$\tau_{\mu}$, for $m$ large enough, we get

$\sum_{x\in F_{m,j}}\mu(B(x,2^{-m}))^{q}\lesssim
2^{m(\tau_{\mu}(q)+\varepsilon/2)}\,,$

which implies that
$\sup_{\delta\in(0,1]}\mathcal{H}_{\delta}^{\underline{s}_{q,\varepsilon}}(\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha))<\infty$.
∎

Since
$\mathrm{E}_{\mu}(\alpha)\subset\underline{\mathrm{U}}_{\mu}(\alpha)\cap\overline{\mathrm{B}}_{\mu}(\alpha)$,
an immediate consequence of Proposition 2.12 is the following corollary.

######  Corollary 2.14.

For $\alpha\geq 0$, it holds that
${\mathrm{d}}_{\mu}(\alpha)\leq\tau_{\mu}^{*}(\alpha)$.

In particular, Corollary 2.14 implies that to prove the validity of the
multifractal formalism for a non-negative Radon measure $\mu$, we only need to
prove the bound ${\mathrm{d}}_{\mu}(\alpha)\geq\tau_{\mu}^{*}(\alpha)$ for all
$\alpha\geq 0$.

##  3 Multifractal analysis of Gaussian multiplicative chaos

Let us now turn to the multifractal analysis of GMC measures. For $d\geq 1$,
let $D\subset\mathbb{R}^{d}$ be a bounded domain. Consider a log-correlated
Gaussian field $X$ on $D$ with covariance kernel (2.1) and let
$(X_{\varepsilon})_{\varepsilon\in(0,1]}$ be its convolution approximation, as
defined in (2.2). For every $\gamma^{2}<2d$, consider the GMC measure
$M_{\gamma}$ on $D$ associated with $X$, as defined in Proposition 2.2.

The main result of this article is contained in the following theorem.

######  Theorem 3.1.

For $\gamma^{2}<2d$, the GMC measure $M_{\gamma}$ satisfies the multifractal
formalism. More precisely, for $\alpha\geq 0$ it holds that

${\mathrm{d}}_{M_{\gamma}}(\alpha)=\tau^{*}_{M_{\gamma}}(\alpha)=\begin{cases}d-\frac{1}{2}\left(\frac{d-\alpha}{\gamma}+\frac{\gamma}{2}\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\sqrt{d}-\frac{|\gamma|}{\sqrt{2}}\right)^{2},\left(\sqrt{d}+\frac{|\gamma|}{\sqrt{2}}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$ (3.1)

almost surely. Moreover, the $L^{q}$-spectrum of $M_{\gamma}$ is given by

$\tau_{M_{\gamma}}(q)=\begin{cases}-\xi^{\prime}_{M_{\gamma}}(q_{-})q\,,&\text{
if }q\in(-\infty,q_{-}]\,,\\\ d-\xi_{M_{\gamma}}(q)\,,&\text{ if
}q\in[q_{-},q_{+}]\,,\\\ -\xi^{\prime}_{M_{\gamma}}(q_{+})q\,,&\text{ if
}q\in[q_{+},\infty)\,,\end{cases}$ (3.2)

almost surely, where $q_{\pm}:=\pm\sqrt{2d}/|\gamma|$ and $\xi_{M_{\gamma}}$
is the power law spectrum of $M_{\gamma}$ as defined in (2.5).

######  Remark 3.2.

We emphasize that the range of values of $\alpha\geq 0$ for which the set
$\mathrm{E}_{M_{\gamma}}(\alpha)$ is non-empty increases with $\gamma^{2}$.
This is a consequence of the well known fact that when $\gamma^{2}$ gets
closer to the critical value $2d$, the concentration of mass of $M_{\gamma}$
becomes more and more clustered and so it gives rise to a larger spectrum of
singularities.

######  Remark 3.3.

Since the GMC measure $M_{\gamma}$ is almost surely a non-negative finite
Radon measure supported on a bounded domain, Proposition 2.10 implies that the
$L^{q}$-spectrum $\tau_{M_{\gamma}}$ is a decreasing convex function and
$\tau_{M_{\gamma}}(1)=0$. Moreover, thanks to the lower bound in Proposition
2.5, one can easily verify that
$\operatorname{dom}(\tau_{M_{\gamma}})=\mathbb{R}$.

The rest of this section is devoted to the proof of Theorem 3.1 which is based
on the following result.

######  Proposition 3.4.

Let $\gamma^{2}<2d$ and $q^{2}<2d/\gamma^{2}$. For $M_{q\gamma}$-almost every
$x\in D$, the GMC measure $M_{\gamma}$ satisfies

$\lim_{r\searrow 0}\frac{\log M_{\gamma}(B(x,r))}{\log
r}=d+\left(\frac{1}{2}-q\right)\gamma^{2}\,,$

almost surely.

###### Proof.

The proof can be found in Appendix A. ∎

######  Remark 3.5.

Roughly speaking, since for $q^{2}<2d/\gamma^{2}$ the measure $M_{q\gamma}$ is
carried by the set of $q\gamma$-thick points of $X$, Proposition 3.4 implies
that around such points the measure $M_{\gamma}$ has local dimension equal to

$\alpha_{q}:=d+\left(\frac{1}{2}-q\right)\gamma^{2}\,.$ (3.3)

Thanks to Corollary 2.14, we already know that, for $\alpha\geq 0$, it holds
almost surely that
${\mathrm{d}}_{M_{\gamma}}(\alpha)\leq\tau_{M_{\gamma}}^{*}(\alpha)$.
Therefore, to prove the validity of the multifractal formalism for
$M_{\gamma}$, it suffices to show that for $\alpha\geq 0$ the lower bound
${\mathrm{d}}_{M_{\gamma}}(\alpha)\geq\tau_{M_{\gamma}}^{*}(\alpha)$ holds
almost surely. Let us start with the following lemma which is an immediate
consequence of Proposition 3.4.

######  Lemma 3.6.

For $\gamma^{2}<2d$, it holds almost surely that
${\mathrm{d}}_{M_{\gamma}}(\alpha_{1})=\alpha_{1}$, where $\alpha_{1}$ is as
in (3.3) with $q=1$. Moreover, if $E\subset D$ is such that
$\dim_{\mathcal{H}}(E)<\alpha_{1}$, then $M_{\gamma}(E)=0$ almost surely.

###### Proof.

For $x\in\mathrm{E}_{M_{\gamma}}(\alpha_{1})$, it obviously holds that
$\dim_{M_{\gamma}}(x)=\alpha_{1}$. Furthermore, thanks to Proposition 3.4, we
know that $M_{\gamma}(\mathrm{E}_{M_{\gamma}}(\alpha_{1}))>0$ almost surely.
Therefore, the first part of the lemma follows from Proposition 2.8.
Concerning the second part of the lemma, let us proceed by contradiction.
Consider a subset $E\subset D$ with $\dim_{\mathcal{H}}(E)<\alpha_{1}$ and
such that $M_{\gamma}(E)>0$. Thanks to Proposition 3.4, we know that the
subset $\mathrm{E}_{M_{\gamma}}(\alpha_{1})$ is of full $M_{\gamma}$-measure,
and so we have that
$M_{\gamma}(E\cap\mathrm{E}_{M_{\gamma}}(\alpha_{1}))=M_{\gamma}(E)>0$.
Moreover, since for all $x\in E\cap\mathrm{E}_{M_{\gamma}}(\alpha_{1})$ it
holds that $\dim_{M_{\gamma}}(x)=\alpha_{1}$, then Proposition 2.8 and the
monotonicity of Hausdorff dimension imply that

$\dim_{\mathcal{H}}(E)\geq\dim_{\mathcal{H}}(E\cap\mathrm{E}_{M_{\gamma}}(\alpha_{1}))=\alpha_{1}\,,$

which is clearly a contradiction. ∎

Now, we introduce the structure function $\phi_{M_{\gamma}}$ associated with
$M_{\gamma}$ which is defined by

$\phi_{M_{\gamma}}(q):=d-\xi_{M_{\gamma}}(q)=\frac{1}{2}\gamma^{2}q^{2}-\left(d+\frac{1}{2}\gamma^{2}\right)q+d\,,\quad
q\in\mathbb{R}\,,$ (3.4)

where we recall that $\xi_{M_{\gamma}}$ is the power law spectrum of
$M_{\gamma}$ defined in (2.5). A direct computation shows that the
Legendre–Fenchel transform of $\phi_{M_{\gamma}}$ can be written as follows

$\phi_{M_{\gamma}}^{*}(\alpha)=\left(d-\frac{1}{2}\left(\frac{d-\alpha}{\gamma}+\frac{1}{2}\gamma\right)^{2}\right)\wedge
0\,,\quad\alpha\geq 0\,.$

The following two lemmas provide a lower bound for
${\mathrm{d}}_{M_{\gamma}}(\alpha_{q})$ in terms of
$\phi^{*}_{M_{\gamma}}(q)$, for all $q^{2}<2d/\gamma^{2}$, and an upper bound
for $\tau_{M_{\gamma}}(q)$ in terms of $\phi_{M_{\gamma}}(q)$, for all
$q\in\mathbb{R}$, respectively.

######  Lemma 3.7.

Let $\gamma^{2}<2d$ and $q^{2}<2d/\gamma^{2}$. Then it holds almost surely
that
${\mathrm{d}}_{M_{\gamma}}(\alpha_{q})\geq\phi_{M_{\gamma}}^{*}(\alpha_{q})$,
where $\alpha_{q}$ is as defined in (3.3).

###### Proof.

On one hand, thanks to Proposition 3.4, we know that the set
$\mathrm{E}_{M_{\gamma}}(\alpha_{q})$ is almost surely of full
$M_{q\gamma}$-measure. On the other hand, Lemma 3.6 implies that $M_{q\gamma}$
cannot give positive measure to a set of Hausdorff dimension strictly less
than $d-q^{2}\gamma^{2}/2$. Therefore, it should hold almost surely that

${\mathrm{d}}_{M_{\gamma}}(\alpha_{q})\geq
d-\frac{1}{2}q^{2}\gamma^{2}=\phi_{M_{\gamma}}^{*}(\alpha_{q})\,,$

where the last equality can be checked directly. ∎

######  Lemma 3.8.

For $\gamma^{2}<2d$, it holds almost surely that
$\tau_{M_{\gamma}}(q)\leq\phi_{M_{\gamma}}(q)$ for all $q\in\mathbb{R}$.

###### Proof.

Thanks to Remark 3.3, the function $\tau_{M_{\gamma}}:\mathbb{R}\to\mathbb{R}$
is convex and hence continuous. Therefore, it is sufficient to prove that for
all $q\in\mathbb{R}$, it holds almost surely that
$\tau_{M_{\gamma}}(q)\leq\phi_{M_{\gamma}}(q)$. For simplicity, we assume that
$D=(-1,1)^{d}$. We split the proof into two parts. In the first part we focus
on $q\geq 2d/\gamma^{2}$, while in the second part we focus on
$q<2d/\gamma^{2}$.

Case $q\geq 2d/\gamma^{2}$. Since $2d/\gamma^{2}>1$, the conclusion in this
case is trivial. Indeed, since $\tau_{M_{\gamma}}$ is a decreasing function
and $\tau_{M_{\gamma}}(1)=0$ (cf. Remark 3.3), it holds almost surely that
$\tau_{M_{\gamma}}(q)\leq 0\leq\phi_{M_{\gamma}}(q)$, where the last
inequality can be checked directly.

Case $q<2d/\gamma^{2}$. For $n\in\mathbb{N}$, we write
$\Sigma_{n}:=2^{-n-1}\mathbb{Z}^{d}\cap(-1,1)^{d}$ for the lattice
$2^{-n-1}\mathbb{Z}^{d}$ restricted to the open box $(-1,1)^{d}$. Then for
each $n\in\mathbb{N}$ and $\varepsilon>0$, by Markov’s inequality and
Proposition 2.4, we have that

$\displaystyle\mathbb{P}\left(\sum_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n}))^{q}\geq
2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\right)$ $\displaystyle\leq
2^{-n(\phi_{M_{\gamma}}(q)+\varepsilon)}\sum_{z\in\Sigma_{n}}\mathbb{E}\left[M_{\gamma}(B(z,2^{-n}))^{q}\right]$
$\displaystyle\lesssim
2^{-n(\phi_{M_{\gamma}}(q)+\varepsilon)}2^{-n\xi_{M_{\gamma}}(q)}2^{nd}=2^{-n\varepsilon}\,,$

where the implicit constant does not depend on $n$. Hence, it holds that

$\sum_{n\in\mathbb{N}}\mathbb{P}\left(\sum_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n}))^{q}\geq
2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\right)<\infty\,,$

and so, thanks to the Borel–Cantelli lemma, we have that it almost surely
exists a finite constant $C>0$ independent of $n$ such that

$\sum_{z\in\Sigma_{n}}M_{\gamma}(B(z,2^{-n}))^{q}\leq
C2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\,,\quad\forall n\in\mathbb{N}\,.$
(3.5)

Let $(B(x_{i},r))_{i\in I}$ be a countable family of disjoint closed balls
with radius $r\in(0,1]$ centred at $x_{i}\in(-1,1)^{d}$, and fix
$n\in\mathbb{N}$ such that $2^{-n}<r\leq 2^{-n+1}$. We observe that there
exists a finite constant $\sigma_{d}>0$ depending only on the dimension $d$,
such that every ball $B(x_{i},r)$ is contained in the union of at most
$\sigma_{d}$ balls of the form $B(z_{ij},2^{-n+1})$, for some
$z_{ij}\in\Sigma_{n-1}$, $j=1,\dots,\sigma_{d}$. Here, we adopt the convention
that if for $i\in I$, the ball $B(x_{i},r)$ is contained in
$\cup_{j=1,\dots,J}B(z_{ij},2^{-n+1})$ for some $J<\sigma_{d}$, then
$z_{ij}=z_{iJ}$ for $j=J+1,\dots,\sigma_{d}$. Furthermore, since the balls
$(B(x_{i},r))_{i\in I}$ are disjoint, we can choose the balls
$(B(z_{ij},2^{-n+1}))_{i\in I,j\in{1,\dots,\sigma_{d}}}$ in such a way that
for fixed $i_{1}\neq i_{2}\in I$, it holds that $z_{i_{1}j}\neq{z_{i_{2}j}}$
for $j=1,\dots,\sigma_{d}$. Now, if $q\in(0,1]$, thanks to (3.5) and the sub-
additivity of the function $x\mapsto x^{q}$, it holds almost surely that

$\sum_{i\in
I}M_{\gamma}(B(x_{i},r))^{q}\leq\sum_{j=1,..,\sigma_{d}}\sum_{z\in\Sigma_{n-1}}M_{\gamma}(B(z,2^{-n+1}))^{q}\lesssim
2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\lesssim
r^{-(\phi_{M_{\gamma}}(q)+\varepsilon)}\,,$ (3.6)

where the implicit constant does not depend on $n$. Similarly, if
$q\in(1,2d/\gamma^{2})$, thanks to (3.5) and the convexity of the function
$x\mapsto x^{q}$ for $q>1$, using Jensen’s inequality, it holds almost surely
that

$\sum_{i\in
I}M_{\gamma}(B(x_{i},r))^{q}\leq\sigma_{d}^{q-1}\sum_{j=1,..,\sigma_{d}}\sum_{z\in\Sigma_{n-1}}M_{\gamma}(B(z,2^{-n+1}))^{q}\lesssim
2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\lesssim
r^{-(\phi_{M_{\gamma}}(q)+\varepsilon)}\,,$ (3.7)

where the implicit constant does not depend on $n$. Moreover, for every ball
$B(x_{i},r)$, we have that $B(z,2^{-n-1})\subset B(x_{i},r)$ for some
$z\in\Sigma_{n+1}$. Therefore, if $q\leq 0$, then thanks to (3.5) it holds
almost surely that

$\sum_{i\in
I}M_{\gamma}(B(x_{i},r))^{q}\leq\sum_{z\in\Sigma_{n+1}}M_{\gamma}(B(z,2^{-n-1}))^{q}\lesssim
2^{n(\phi_{M_{\gamma}}(q)+\varepsilon)}\lesssim
r^{-(\phi_{M_{\gamma}}(q)+\varepsilon)}\,,$ (3.8)

where the implicit constant does not depend on $n$. Finally, the conclusion
follows by taking logs in (3.6), (3.7), (3.8), and thanks to the arbitrariness
of $\varepsilon>0$. ∎

Thanks to Corollary 2.14, Lemmas 3.7 and 3.8, we obtain that for
$q^{2}<2d/\gamma^{2}$ it holds almost surely that

$0<\phi_{M_{\gamma}}^{*}(\alpha_{q})\leq{\mathrm{d}}_{M_{\gamma}}(\alpha_{q})\leq\tau_{M_{\gamma}}^{*}(\alpha_{q})\leq\phi_{M_{\gamma}}^{*}(\alpha_{q})\,,$
(3.9)

which implies that
${\mathrm{d}}_{M_{\gamma}}(\alpha_{q})=\tau_{M_{\gamma}}^{*}(\alpha_{q})$ and
$\tau_{M_{\gamma}}(q)=\phi_{M_{\gamma}}(q)$. For $q^{2}<2d/\gamma^{2}$, we
observe that the range of values of $\alpha\geq 0$ of the form $\alpha_{q}$ is
equal to the interval $(\alpha_{-},\alpha_{+})$, where

$\alpha_{-}=\left(\sqrt{d}-\frac{|\gamma|}{\sqrt{2}}\right)^{2}\,,\quad\alpha_{+}=\left(\sqrt{d}+\frac{|\gamma|}{\sqrt{2}}\right)^{2}\,.$

Thanks to Corollary 2.14 and Lemma 3.8, we can easily observe that for
$\alpha\not\in(\alpha_{-},\alpha_{+})$ we have that
${\mathrm{d}}_{M_{\gamma}}(\alpha)\leq\tau^{*}_{M_{\gamma}}(\alpha)\leq\phi^{*}_{M_{\gamma}}(\alpha)=0$.
Therefore, these results imply that for all $\alpha\geq 0$, it holds almost
surely that

${\mathrm{d}}_{M_{\gamma}}(\alpha)=\tau_{M_{\gamma}}^{*}(\alpha)=\begin{cases}d-\frac{1}{2}\left(\frac{d-\alpha}{\gamma}+\frac{\gamma}{2}\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\sqrt{d}-\frac{|\gamma|}{\sqrt{2}}\right)^{2},\left(\sqrt{d}+\frac{|\gamma|}{\sqrt{2}}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

which coincides with (3.1).

To conclude the proof of Theorem 3.1 we need to show the validity of (3.2).
This is exactly the objective of the subsequent lemma whose proof follows
similar steps of those in [BJ10, Section 2.3].

######  Lemma 3.9.

If $\gamma^{2}<2d$, then for $q\in\mathbb{R}$ it holds almost surely that

$\tau_{M_{\gamma}}(q)=\begin{cases}-\xi^{\prime}_{M_{\gamma}}(q_{-})q\,,&\quad\
\text{ if }q\in(-\infty,q_{-}]\,,\\\ d-\xi_{M_{\gamma}}(q)\,,&\quad\ \text{ if
}q\in[q_{-},q_{+}]\,,\\\ -\xi^{\prime}_{M_{\gamma}}(q_{+})q\,,&\quad\ \text{
if }q\in[q_{+},\infty)\,,\end{cases}$

where $q_{\pm}=\pm\sqrt{2d}/|\gamma|$.

###### Proof.

From (3.9), we know that $\tau_{M_{\gamma}}(q)=\phi_{M_{\gamma}}(q)$ for
$q\in(q_{-},q_{+})$. Thanks to the continuity of the convex function
$\tau_{M_{\gamma}}$, the above equality can be extended to the close interval
$[q_{-},q_{+}]$. Therefore, it remains to prove that $\tau_{M_{\gamma}}$ is
derivable at $q_{+}$ and linear on $[q_{+},\infty)$, and the same also for
$q_{-}$ and $(-\infty,q_{-}]$. Thanks to the equality
$\tau_{M_{\gamma}}(q)=\phi_{M_{\gamma}}(q)$ on the interval $[q_{-},q_{+}]$,
we know that

$\tau^{\prime}_{M_{\gamma}}(q_{+}^{-})=\phi^{\prime}_{M_{\gamma}}(q_{+})=\frac{\phi_{M_{\gamma}}(q_{+})}{q_{+}}\,,\quad\tau^{\prime}_{M_{\gamma}}(q_{-}^{+})=\phi^{\prime}_{M_{\gamma}}(q_{-})=\frac{\phi_{M_{\gamma}}(q_{-})}{q_{-}}\,,$

where $\tau^{\prime}_{M_{\gamma}}(q_{+}^{-})$ (resp.
$\tau^{\prime}_{M_{\gamma}}(q_{-}^{+})$) denotes the left-hand (resp. right-
hand) derivative of $\tau_{M_{\gamma}}$ at $q_{+}$ (resp. at $q_{-}$). Since
the function $\tau_{M_{\gamma}}$ is convex, we have that

$\tau_{M_{\gamma}}(q)\geq\tau_{M_{\gamma}}(q_{+})+\tau^{\prime}_{M_{\gamma}}(q_{+}^{-})(q-q_{+})=\phi^{\prime}_{M_{\gamma}}(q_{+})q\,,$

and similarly $\tau_{M_{\gamma}}(q)\geq\phi^{\prime}_{M_{\gamma}}(q_{-})q$.
For $q\geq q_{+}$, the reverse inequality can be obtained using the fact that
$q_{+}/q\in(0,1]$ and the sub-additivity of the function $x\mapsto
x^{q_{+}/q}$. Indeed, if we consider a countable family $(B(x_{i},r))_{i\in
I}$ of disjoint closed balls with radius $r$ centred at $x_{i}\in D$, then

$\left(\sum_{i\in
I}M_{\gamma}(B(x_{i},r))^{q}\right)^{\frac{q_{+}}{q}}\leq\sum_{i\in
I}M_{\gamma}(B(x_{i},r))^{q_{+}}\,.$

Hence, taking logs and by definition of $\tau_{M_{\gamma}}$, we have that

$\tau_{M_{\gamma}}(q)\leq\frac{\tau_{M_{\gamma}}(q_{+})}{q_{+}}q=\frac{\phi_{M_{\gamma}}(q_{+})}{q_{+}}q=\phi^{\prime}_{M_{\gamma}}(q_{+})q\,.$

Therefore, for all $q\geq q_{+}$ it holds that
$\tau_{M_{\gamma}}(q)=\phi^{\prime}_{M_{\gamma}}(q_{+})q$, and the same exact
computation yields also that
$\tau_{M_{\gamma}}(q)=\phi^{\prime}_{M_{\gamma}}(q_{-})q$ for all $q\leq
q_{-}$. Since $\phi_{M_{\gamma}}(q)=d-\xi_{M_{\gamma}}(q)$, for all
$q\in\mathbb{R}$, the result is proved. ∎

This concludes the proof of Theorem 3.1.

##  4 Multifractal analysis of multifractal random walk

Theorem 3.1 gives a full characterization of the singularity spectrum of GMC
measures. A natural question that may arise is whether we can use this result
to perform the multifractal analysis of objects that are built by means of
these random measures. In particular, in this section, we investigate the
multifractal behaviour of the _Multifractal Random Walk_ (MRW).

###  4.1 Definition of the MRW

Let $T>0$ and consider a log-correlated Gaussian field $X$ on $[0,T]$ with
covariance kernel of the from (2.1). For $\gamma^{2}<2$, consider the GMC
measure $M_{\gamma}$ on $[0,T]$ associated with $X$, as defined in Proposition
2.2. In addition, for $d\geq 1$, let $B$ be a Brownian motion on
$\mathbb{R}^{d}$ independent from the field $X$.

######  Remark 4.1.

As the Brownian motion $B$ is independent from the log-correlated Gaussian
field $X$, it will be useful to decompose the probability measure $\mathbb{P}$
as follows

$\mathbb{P}=\mathbb{P}_{B}\otimes\mathbb{P}_{X}\,.$

When we write “almost surely”, we mean $\mathbb{P}$-almost surely. If we need
to specify that a property holds almost surely with respect to the Brownian
motion (resp. the field $X$), we write $\mathbb{P}_{B}$-almost surely (resp.
$\mathbb{P}_{X}$-almost surely).

######  Definition 4.2.

Fix $\gamma^{2}<2$ and $d\geq 1$. The $d$-dimensional MRW on $[0,T]$
associated with the field $X$ is the stochastic process
$(\mathcal{Z}^{\gamma}_{t})_{t\in[0,T]}$ defined by

$\mathcal{Z}^{\gamma}_{t}:=B_{M_{\gamma}([0,t])}\,,\quad t\in[0,T]\,.$ (4.1)

The MRW has been originally introduced in [BDM01] as a stochastic volatility
model. Indeed, assuming that the Brownian motion $B$ is one-dimensional, the
MRW is used for modelling the price fluctuations of financial assets since it
accounts for most of the stylized facts of financial time series. For example,
it reproduces the sudden jumps and the periods of intense activity usually
observed in stock prices. We refer to [BKM08] for a short overview on this
topic.

###  4.2 Study of the lower singularity spectrum

The main goal of this section is to analyse the local fluctuations of the
paths of the MRW $\mathcal{Z}^{\gamma}$. More precisely, we introduce for
$\alpha\geq 0$ the sets

$\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha):=\left\\{t\in[0,T]\,:\,\underline{\dim}_{\mathcal{Z}^{\gamma}}(t)=\alpha\right\\}\,,$

where $\underline{\dim}_{\mathcal{Z}^{\gamma}}(t)$ is the lower local
dimension of $\mathcal{Z}^{\gamma}$ at $t\in[0,T]$ defined by

$\underline{\dim}_{\mathcal{Z}^{\gamma}}(t):=\liminf_{r\searrow
0}\frac{\log|\mathcal{Z}^{\gamma}_{t+r}-\mathcal{Z}^{\gamma}_{t-r}|}{\log
r}\,.$

We want to compute the Hausdorff dimension of
$\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha)$ for each $\alpha\geq
0$, i.e. we are interested in finding an explicit expression for the lower
singularity spectrum
$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}:[0,\infty)\to[0,\infty)$
defined by

$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}(\alpha):=\dim_{\mathcal{H}}(\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha))\,,\quad\alpha\in[0,\infty)\,.$

In particular, we want to find the relation between
$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}(\alpha)$ and
${\mathrm{d}}_{M_{\gamma}}(\alpha)$. This is exactly the content of the
subsequent theorem.

######  Theorem 4.3.

Let $\gamma^{2}<2$ and $d\geq 1$. Then, for $\alpha\geq 0$, the lower
singularity spectrum of the $d$-dimensional MRW $\mathcal{Z}^{\gamma}$ is
given by

$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}(\alpha)={\mathrm{d}}_{M_{\gamma}}(2\alpha)=\begin{cases}1-\frac{1}{2}\left(\frac{1-2\alpha}{\gamma}+\frac{\gamma}{2}\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\frac{1}{\sqrt{2}}-\frac{|\gamma|}{2}\right)^{2},\left(\frac{1}{\sqrt{2}}+\frac{|\gamma|}{2}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

almost surely.

For $\alpha\geq 0$, we define the set $\mathrm{E}_{M_{\gamma}}(\alpha)$ as in
(2.8), and the sets $\underline{\mathrm{U}}_{M_{\gamma}}(\alpha)$,
$\overline{\mathrm{B}}_{M_{\gamma}}(\alpha)$ as in (2.9). Then Theorem 4.3 is
an easy consequence of the following lemma.

######  Lemma 4.4.

If $\gamma^{2}<2$, then for $\alpha\geq 0$, it holds $\mathbb{P}_{B}$-almost
surely that

  1. (i)

$\mathrm{E}_{M_{\gamma}}(2\alpha)\subset\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha)$;

  2. (ii)

$\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha)\subset\underline{\mathrm{U}}_{M_{\gamma}}(2\alpha)\cap\overline{\mathrm{B}}_{M_{\gamma}}(2\alpha)$.

###### Proof.

Let $(\mathcal{Z}^{\gamma}_{t})_{t\in[0,T]}$ be the $d$-dimensional MRW as
defined in (4.1). Thanks to well-known properties of Brownian motion, the
following facts holds $\mathbb{P}_{B}$-almost surely. Fix $\varepsilon>0$,
then it exists a finite constant $C>0$ such that, for every $t\in[0,T]$ and
$r>0$ small enough, it holds that

$|\mathcal{Z}^{\gamma}_{t+r}-\mathcal{Z}^{\gamma}_{t-r}|=|B_{M_{\gamma}([0,t+r])}-B_{M_{\gamma}([0,t-r])}|\leq
CM_{\gamma}(B(t,r))^{\frac{1}{2}-\varepsilon}\,.$ (4.2)

Moreover, we also know that for each $\varepsilon>0$ and $t\in[0,T]$, for
$h>0$ small enough, it holds that

$\sup_{0<r\leq
h}|\mathcal{Z}^{\gamma}_{t+r}-\mathcal{Z}^{\gamma}_{t-r}|=\sup_{0<r\leq
h}|B_{M_{\gamma}([0,t+r])}-B_{M_{\gamma}([0,t-r])}|\geq
M_{\gamma}(B(t,r))^{\frac{1}{2}+\varepsilon}\,.$ (4.3)

Proof of item (i). For $\alpha\geq 0$, we fix
$t\in\mathrm{E}_{M_{\gamma}}(2\alpha)$. For every $\varepsilon>0$, we know
that for $r>0$ small enough, it holds that

$r^{2\alpha+\varepsilon}\leq M_{\gamma}(B(t,r))\leq
r^{2\alpha-\varepsilon}\,.$ (4.4)

Consequently, thanks to (4.2) and the right-hand side of (4.4), we get that
for $r>0$ small enough, it holds that

$|\mathcal{Z}^{\gamma}_{t+r}-\mathcal{Z}^{\gamma}_{t-r}|\leq
CM_{\gamma}(B(t,r))^{\frac{1}{2}-\varepsilon}\leq
Cr^{(2\alpha-\varepsilon)\left(\frac{1}{2}-\varepsilon\right)}\,.$

On the other hand, thanks to (4.3) and the left-hand side of (4.4), there
exists a decreasing sequence $(r_{n})_{n\in\mathbb{N}}$ converging to $0$ such
that

$|\mathcal{Z}^{\gamma}_{t+r_{n}}-\mathcal{Z}^{\gamma}_{t-r_{n}}|\geq
M_{\gamma}(B(t,r_{n}))^{\frac{1}{2}+\varepsilon}\geq
r_{n}^{(2\alpha+\varepsilon)\left(\frac{1}{2}+\varepsilon\right)}\,,\quad\forall
n\in\mathbb{N}\,.$

Therefore, taking logs, by arbitrariness of $\varepsilon>0$, this implies that

$\liminf_{r\searrow
0}\frac{\log|\mathcal{Z}^{\gamma}_{t+r}-\mathcal{Z}^{\gamma}_{t-r}|}{\log
r}=\alpha\,.$

which means that $t\in\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha)$,
$\mathbb{P}_{B}$-almost surely.

Proof of item (ii). For $\alpha\geq 0$, we fix
$t\in\underline{\mathrm{E}}_{\mathcal{Z}^{\gamma}}(\alpha)$. Similarly as
above, for each $\varepsilon>0$, there exist decreasing sequences
$(r_{n})_{n\in\mathbb{N}}$, $(r^{\prime}_{n})_{n\in\mathbb{N}}$ both
converging to $0$ such that

$r_{n}^{\alpha+\varepsilon}\leq|\mathcal{Z}^{\gamma}_{t+r_{n}}-\mathcal{Z}^{\gamma}_{t-r_{n}}|\leq
CM_{\gamma}(B(t,r_{n}))^{\frac{1}{2}-\varepsilon}\,,\quad\forall
n\in\mathbb{N}\,,$

where the last inequality is given by (4.2), and

${r^{\prime}_{n}}^{\alpha-\varepsilon}\geq|\mathcal{Z}^{\gamma}_{t+r^{\prime}_{n}}-\mathcal{Z}^{\gamma}_{t-r^{\prime}_{n}}|\geq
M_{\gamma}(B(t,r^{\prime}_{n}))^{\frac{1}{2}+\varepsilon}\,,\quad\forall
n\in\mathbb{N}\,,$

where the last inequality is given by (4.3). By arbitrariness of
$\varepsilon>0$, these facts imply that
$t\in\underline{\mathrm{U}}_{M_{\gamma}}(2\alpha)\cap\overline{\mathrm{B}}_{M_{\gamma}}(2\alpha)$,
$\mathbb{P}_{B}$-almost surely. ∎

Finally, we show how the proof of Theorem 4.3 follows immediately from Lemma
4.4.

###### Proof of Theorem 4.3.

From item (i) of Lemma 4.4, for all $\alpha\geq 0$, we have the inequality

$\underline{{\mathrm{d}}}_{\mathcal{Z}^{\gamma}}(\alpha)\geq{\mathrm{d}}_{M_{\gamma}}(2\alpha)\,,$

almost surely. On the other hand, item (ii) of Lemma 4.4, Proposition 2.12 and
Theorem 3.1 imply that, for all $\alpha\geq 0$, it holds

${\mathrm{d}}_{\mathcal{Z}^{\gamma}}(\alpha)\leq\dim_{\mathcal{H}}\left(\underline{\mathrm{U}}_{M_{\gamma}}(2\alpha)\cap\overline{\mathrm{B}}_{M_{\gamma}}(2\alpha)\right)\leq\tau^{*}_{M_{\gamma}}(2\alpha)={\mathrm{d}}_{M_{\gamma}}(2\alpha)\,,$

almost surely. Since Theorem 3.1 gives a full characterization of the
singularity spectrum ${\mathrm{d}}_{M_{\gamma}}$, this concludes the proof. ∎

##  5 Multifractal analysis of Liouville Brownian motion

The main goal of this section is to study the multifractal behaviour of the
_Liouville Brownian motion_ (LBM). Roughly speaking, the LBM can be defined as
the canonical diffusion associated with the _Liouville quantum gravity_ metric
tensor given by

$e^{\gamma X}({\mathrm{d}}x^{2}+{\mathrm{d}}y^{2})\,,$

where $X$ is a GFF on a domain $D\subset\mathbb{R}^{2}$ and
${\mathrm{d}}x^{2}+{\mathrm{d}}y^{2}$ is the Euclidean metric tensor. The LBM
has been introduced simultaneously in [Ber15, GRV16], and it can be built as a
time changed planar Brownian motion with respect to a GMC measure associated
with the field $X$. We refer to the above mentioned references for more
details.

In what follows, we take a slightly more general approach, and we generalize
the construction of the LBM in dimension $d\geq 2$. Once this is done, we
consider the problem of studying the lower singularity spectrum of the
$d$-dimensional LBM.

###  5.1 Definition of the LBM

For $d\geq 2$, consider a bounded domain $D\subset\mathbb{R}^{d}$ containing
the ball $B(0,1)$. Let $B$ be a $d$-dimensional Brownian motion started from
the origin and define $T$ to be its first exit time from $D$, i.e.

$T:=\inf\left\\{t\geq 0\,:\,B_{t}\not\in D\right\\}\,.$ (5.1)

Moreover, consider a log-correlated Gaussian field $X$ on $D$ with covariance
kernel of the form (2.1) and independent from the Brownian motion $B$. Then
the $d$-dimensional LBM on $D$ can be formally defined by

$\mathcal{B}^{\gamma}_{t}:=B_{F_{\gamma}^{-1}(t)}\,,\quad t\geq 0\,,$

where the clock process $F_{\gamma}$ is given by

$F_{\gamma}(t):=\int_{0}^{t\wedge T}e^{\gamma
X(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X(B_{s})^{2}]}{\mathrm{d}}s\,,\quad
t\geq 0\,.$

In the section, we adopt the same notation specified in Remark 4.1, i.e., we
decompose $\mathbb{P}$ as $\mathbb{P}_{B}\otimes\mathbb{P}_{X}$.

To give rigorous sense to the definition of the LBM, it is sufficient to show
the existence of the clock process $F_{\gamma}$. Following [Ber15, GRV16], we
proceed with a regularization and limiting procedure. Let
$(X_{\varepsilon})_{\varepsilon\in(0,1]}$ be the convolution approximation of
the field $X$ as defined in (2.2), then we have the following lemma.

######  Lemma 5.1.

If $\gamma^{2}<4$, then $\mathbb{P}_{B}$-almost surely, for all $t\geq 0$, the
sequence

$F_{\gamma}^{\varepsilon}(t):=\int_{0}^{t\wedge T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s\,,\quad\varepsilon\in(0,1]\,,$

converges in $\mathbb{P}_{X}$-probability and in $L^{1}(\mathbb{P}_{X})$
towards a non-degenerate limit $F_{\gamma}(t)$ that does not depend on the
choice of the mollifier $\psi$ used in (2.2).

###### Proof.

This fact is a standard result and it is an easy consequence of [Ber17,
Theorem 1.1]. Indeed, for $t\geq 0$, let $\nu^{T}_{t}$ be the occupation
measure between time $0$ and time $t\wedge T$ of the Brownian motion $B$, so
that

$F_{\gamma}^{\varepsilon}(t)=\int_{\mathbb{R}^{d}}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}\nu^{T}_{t}({\mathrm{d}}x)\,,\quad\varepsilon\in(0,1]\,.$

Then it is sufficient to prove that $\mathbb{P}_{B}$-almost surely, for all
$t\geq 0$, the measure $\nu^{T}_{t}$ has dimension two, i.e., we need to show
that for $\alpha\in(0,2)$, it holds that

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu^{T}_{t}({\mathrm{d}}x)\nu^{T}_{t}({\mathrm{d}}y)<\infty\,.$

Fix $t\geq 0$ and let $\nu_{t}$ be the occupation measure of the (unstopped)
Brownian motion $B$ between time $0$ and time $t$, then it obviously holds
that

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu^{T}_{t}({\mathrm{d}}x)\nu^{T}_{t}({\mathrm{d}}y)\leq\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu_{t}({\mathrm{d}}x)\nu_{t}({\mathrm{d}}y)\,.$

Therefore, taking expectation, it is enough to prove that for
$\alpha\in(0,2)$, it holds that

$\mathbb{E}_{B}\left[\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu_{t}({\mathrm{d}}x)\nu_{t}({\mathrm{d}}y)\right]=\mathbb{E}_{B}\left[\int_{0}^{t}\int_{0}^{t}|B_{r}-B_{s}|^{-\alpha}{\mathrm{d}}r{\mathrm{d}}s\right]<\infty\,.$
(5.2)

Using Fubini’s theorem, we have that

$\displaystyle\mathbb{E}_{B}\left[\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu_{t}({\mathrm{d}}x)\nu_{t}({\mathrm{d}}y)\right]$
$\displaystyle=\int_{0}^{t}\int_{0}^{t}\mathbb{E}_{B}\left[|B_{r}-B_{s}|^{-\alpha}\right]{\mathrm{d}}r{\mathrm{d}}s$
$\displaystyle=\mathbb{E}_{B}\left[{|B_{1}|^{-\alpha}}\right]\int_{0}^{t}\int_{0}^{t}|r-s|^{-\frac{\alpha}{2}}{\mathrm{d}}r{\mathrm{d}}s$
$\displaystyle\leq
2t\mathbb{E}_{B}\left[{|B_{1}|^{-\alpha}}\right]\int_{0}^{t}r^{-\frac{\alpha}{2}}{\mathrm{d}}r\,$

which is clearly finite if $\alpha\in(0,2)$. Therefore, this implies that for
fixed $t\geq 0$, the measure $\nu_{t}^{T}$ has dimension two,
$\mathbb{P}_{B}$-almost surely. To prove that the measure $\nu_{t}^{T}$ has
dimension two for all $t\geq 0$, $\mathbb{P}_{B}$-almost surely, it is
sufficient to take a sequence $(t_{n})_{n\in\mathbb{N}}$ converging to
infinity and noting that for $0\leq s\leq t$, it holds that

$\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu^{T}_{s}({\mathrm{d}}x)\nu^{T}_{s}({\mathrm{d}}y)\leq\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{-\alpha}\nu^{T}_{t}({\mathrm{d}}x)\nu^{T}_{t}({\mathrm{d}}y)\,.$

The conclusion follows thanks to [Ber17, Theorem 1.1]. ∎

######  Definition 5.2.

Fix $\gamma^{2}<4$ and $d\geq 2$. The $d$-dimensional LBM on $D$ associated
with the field $X$ is the stochastic process
$(\mathcal{B}^{\gamma}_{t})_{t\geq 0}$ defined by

$\mathcal{B}^{\gamma}_{t}:=B_{F_{\gamma}^{-1}(t)}\,,\quad t\geq 0\,,$ (5.3)

where $F_{\gamma}$ is defined as in Lemma 5.1.

######  Remark 5.3.

Let us emphasize that when $d\geq 3$, Lemma 5.1 yields a construction of the
LBM only for $\gamma^{2}<4$, and not for $\gamma^{2}<2d$ as for the case of
GMC measures. Therefore, there is a gap, for $4\leq\gamma^{2}<2d$, where a
construction of the LBM does not follow from the general theory of GMC. This
is due to the well-known fact that the dimension of the occupation measure of
the $d$-dimensional Brownian motion is at most two, for all $d\geq 2$ (see
e.g. [MP10, Chapter 4]). To the best of our knowledge, if $d\geq 3$, a
construction of the $d$-dimensional LBM for $4\leq\gamma^{2}<2d$ has not be
done yet.

Before proceeding, let us introduce some notation. For $\gamma^{2}<4$, we can
define on the interval $[0,T]$ the measure $\mu_{\gamma}$ by setting

$\mu_{\gamma}([s,t]):=F_{\gamma}(t)-F_{\gamma}(s)\,,\quad s\leq t\in[0,T]\,.$
(5.4)

It may be worth emphasizing that an immediate consequence of Lemma 5.1 is that
the measure $\mu_{\gamma}$ can be interpreted as the weak limit of the
sequence of approximating measures
$(\mu_{\gamma}^{\varepsilon})_{\varepsilon\in(0,1]}$, where

$\mu_{\gamma}^{\varepsilon}({\mathrm{d}}t):=e^{\gamma
X_{\varepsilon}(B_{t})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{t})^{2}]}\mathbbm{1}_{\\{T>t\\}}{\mathrm{d}}t\,,\quad\varepsilon\in(0,1]\,.$
(5.5)

Similarly, we can define on the interval $[0,F_{\gamma}(T)]$ the measure
$\mu_{\gamma}^{-}$ by letting

$\mu^{-}_{\gamma}([s,t]):=F^{-1}_{\gamma}(t)-F^{-1}_{\gamma}(s)\,,\quad s\leq
t\in[0,F_{\gamma}(T)]\,,$ (5.6)

so that for all $t\in[0,F_{\gamma}(T)]$, we can write
$\mathcal{B}_{t}=B_{\mu_{\gamma}^{-}([0,t])}$.

######  Proposition 5.4.

For $\gamma^{2}<4$, the random function $F_{\gamma}:[0,T]\to\mathbb{R}$ is
almost surely continuous and strictly increasing on the interval $[0,T]$.

###### Proof.

The function $F_{\gamma}$ is continuous and strictly increasing as long as the
measure $\mu_{\gamma}$ has no atoms and it has full support on the interval
$[0,T]$. To verify that $\mu_{\gamma}$ is almost surely atomless on $[0,T]$,
one needs to use the power law behaviour of $\mu_{\gamma}$ (cf. Proposition
5.6 below). We refer to [Ber15, Remark 3.5] for details. The proof of the fact
that $\mu_{\gamma}$ has almost surely full support on $[0,T]$ can be found in
[GRV16, Theorem 2.7]. To be precise, in the above mentioned references the
proofs are given in the planar case. However, they immediately generalize to
all dimensions and hence the details are omitted. ∎

###  5.2 Study of the lower singularity spectrum

In the same spirit as in Section 4, our goal is to study the local regularity
of the paths of the LBM $\mathcal{B}^{\gamma}$ defined in (5.3). More
precisely, we are interested in the lower singularity spectrum
$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}:[0,\infty)\to[0,\infty)$
defined by

$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(\alpha):=\dim_{\mathcal{H}}(\underline{\mathrm{E}}_{\mathcal{B}^{\gamma}}(\alpha))\,,\quad\alpha\in[0,\infty)\,,$

where $\underline{\mathrm{E}}_{\mathcal{B}^{\gamma}}(\alpha)$ denotes the set
of times at which $\mathcal{B}^{\gamma}$ has lower local dimension equal to
$\alpha\geq 0$, i.e.

$\underline{\mathrm{E}}_{\mathcal{B}^{\gamma}}(\alpha):=\left\\{t\in[0,F_{\gamma}(T)]\,:\,\underline{\dim}_{\mathcal{B}^{\gamma}}(t)=\alpha\right\\}\,,$

where

$\underline{\dim}_{\mathcal{B}^{\gamma}}(t):=\liminf_{r\searrow
0}\frac{\log|\mathcal{B}^{\gamma}_{t+r}-\mathcal{B}^{\gamma}_{t-r}|}{\log
r}\,.$

######  Theorem 5.5.

Let $\gamma^{2}<4$ and $d\geq 2$. Then, for $\alpha\geq 0$, the lower
singularity spectrum of the LBM $\mathcal{B}^{\gamma}$ is given by

$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(\alpha)=\begin{cases}2\alpha-2\alpha\left(\frac{2\alpha-1}{2\alpha\gamma}+\frac{1}{4}\gamma\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(\sqrt{2}+\frac{|\gamma|}{\sqrt{2}}\right)^{-2},\left(\sqrt{2}-\frac{|\gamma|}{\sqrt{2}}\right)^{-2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

almost surely.

Since the LBM $\mathcal{B}^{\gamma}$ is a time changed Brownian motion, the
lower singularity spectrum $\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}$
can be obtained through the singularity spectrum of the measure
$\mu_{\gamma}^{-}$ defined in (5.6), as in the case of the MRW. However, in
this setting the situation is less trivial since we do not know the
singularity spectrum of $\mu_{\gamma}^{-}$ at this point. Instead of working
directly with $\mu_{\gamma}^{-}$, we start by proving that the measure
$\mu_{\gamma}$, defined in (5.4), satisfies the multifractal formalism, and we
find an explicit formula for its singularity spectrum. Once this is done, we
can move to the proof of Theorem 5.5 which consists in finding the relations
between $\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}$,
${\mathrm{d}}_{\mu_{\gamma}^{-}}$ and ${\mathrm{d}}_{\mu_{\gamma}}$.

####  5.2.1 Multifractal analysis of $\mu_{\gamma}$

The main goal of this section is to prove that the measure $\mu_{\gamma}$
defined in (5.4) satisfies the multifractal formalism. We will follow a
strategy similar to that developed in Section 3. As we will see, the presence
of the Brownian motion creates some more technical difficulties, but the
underlying ideas are the same.

We start by computing the power law spectrum of the measure $\mu_{\gamma}$.

######  Proposition 5.6.

Let $\gamma^{2}<4$ and $q<4/\gamma^{2}$. For $r\in(0,1]$, define the stopping
time $\tau_{\sqrt{r}}:=\inf\\{t\geq 0\,:\,B_{t}\not\in B(0,\sqrt{r})\\}$. Then
there exists a finite constant $C>0$ such that for all $r\in(0,1]$, it holds
that

$\mathbb{E}[\mu_{\gamma}([0,\tau_{\sqrt{r}}])^{q}]\leq
Cr^{\xi_{\mu_{\gamma}}(q)}\,,$

where

$\xi_{\mu_{\gamma}}(q):=\left(1+\frac{1}{4}\gamma^{2}\right)q-\frac{1}{4}\gamma^{2}q^{2}\,,\quad
q\in\mathbb{R}\,,$

is the power law spectrum of $\mu_{\gamma}$.

###### Proof.

Thanks to Kahane’s convexity inequality (cf. Lemma C.2), it is sufficient to
prove the result only for the $d$-dimensional exactly scale invariant field
$(X(x))_{x\in\mathbb{R}^{d}}$ defined in [RV10a]. More precisely, for
$\varepsilon\in(0,1]$, we let $(X_{\varepsilon}(x))_{x\in\mathbb{R}^{d}}$ be
the approximation of $X$ as defined in [BP21, Lemma 3.19]. Then, for
$\lambda\in(0,1)$, we can assume that

$\left(X_{\lambda\varepsilon}(\lambda x)\right)_{x\in
B(0,1)}\overset{d}{=}\left(X_{\varepsilon}(x)+\Omega_{\lambda}\right)_{x\in
B(0,1)}\,,$ (5.7)

where $\Omega_{\lambda}$ is an independent centred Gaussian random variable
with variance $-\log\lambda$ (cf. [BP21, Corollay 3.20]). Let
$0\leq\varepsilon<\sqrt{r}$, then scaling time by a factor $r$ and using
(5.7), we get that

$\displaystyle\mu^{\varepsilon}_{\gamma}([0,\tau_{\sqrt{r}}])$
$\displaystyle=\int_{0}^{\tau_{\sqrt{r}}}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s$
$\displaystyle=r\int_{0}^{\tau_{\sqrt{r}}/r}e^{\gamma
X_{\varepsilon}(B_{rs})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{rs})^{2}]}{\mathrm{d}}s$
$\displaystyle\overset{d}{=}r\int_{0}^{\tilde{\tau}}e^{\gamma
X_{\varepsilon}(\sqrt{r}\tilde{B}_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(\sqrt{r}\tilde{B}_{s})^{2}]}{\mathrm{d}}s$
$\displaystyle\overset{d}{=}re^{\gamma\Omega_{\sqrt{r}}-\frac{1}{2}\gamma^{2}\mathbb{E}[\Omega_{\sqrt{r}}^{2}]}\int_{0}^{\tilde{\tau}}e^{\gamma
X_{\varepsilon/\sqrt{r}}(\tilde{B}_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon/\sqrt{r}}(\tilde{B}_{s})^{2}]}{\mathrm{d}}s\,,$

where $\tilde{B}$ is an independent Brownian motion and $\tilde{\tau}$ is the
first exit time of $\tilde{B}$ from $B(0,1)$. Now, raising to the power
$q<4/\gamma^{2}$ and taking expectations, one can easily verify that

$\mathbb{E}[\mu^{\varepsilon}_{\gamma}([0,\tau_{\sqrt{r}}])^{q}]=r^{\xi(q)}\mathbb{E}\left[\left(\int_{0}^{\tilde{\tau}}e^{\gamma
X_{\varepsilon/\sqrt{r}}(\tilde{B}_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon/\sqrt{r}}(\tilde{B}_{s})^{2}]}{\mathrm{d}}s\right)^{q}\right]\
$

Removing the approximation, i.e. taking the limit when $\varepsilon\searrow
0$, gives us the desired result provided that the quantities
$\mu^{\varepsilon}_{\gamma}([0,\tau_{\sqrt{r}}])$ and
$\mu_{\gamma}^{\varepsilon/\sqrt{r}}([0,\tilde{\tau}])^{q}$ are uniformly
integrable in $\varepsilon\in(0,1]$, for $q<4/\gamma^{2}$. For $q\leq 0$ this
fact is proved in the planar case in [GRV16, Proposition 2.12]), but the proof
generalizes almost verbatim in higher dimensions. For $q\in(0,4/\gamma^{2})$ a
proof of this fact can be found in Appendix B. ∎

We now state the main result that allows to prove that $\mu_{\gamma}$
satisfies the multifractal formalism. Let us emphasize that such a result is
the analogue of Proposition 3.4, but for the measure $\mu_{\gamma}$.

######  Proposition 5.7.

Let $\gamma^{2}<4$ and $q^{2}<4/\gamma^{2}$. For $\mu_{q\gamma}$-almost every
$t\in[0,T]$, the measure $\mu_{\gamma}$ satisfies

$\lim_{r\searrow 0}\frac{\log\mu_{\gamma}(B(t,r))}{\log
r}=1+\left(\frac{1}{2}-q\right)\frac{\gamma^{2}}{2}\,,$

almost surely.

###### Proof.

The case in which the underlying field is a zero-boundary GFF on a bounded
domain $D\subset\mathbb{R}^{2}$ has already been treated in [Jac18]. The proof
follows similar lines as the proof of Proposition 3.4 in Appendix A.
Therefore, we only highlight here the main changes that one needs to do in
order for the proof to work in the general setting of a log-correlated
Gaussian field $X$ on a bounded domain $D\subset\mathbb{R}^{d}$, $d\geq 2$.

The main steps of the proof in [Jac18] are [Jac18, Lemmas 3.9 and 3.11]. Let
us focus on [Jac18, Lemma 3.9]. The only main change that one needs to do is
in [Jac18, Equation 3.19]. Indeed, in that step the author uses the domain
Markov property of the GFF to bound the exponential moment of the fluctuations
of the circle average approximation. However, in the case of a general log-
correlated Gaussian field, we can bypass the use of such a property by using
instead Lemma 2.1 in order to control the fluctuations of the convolution
approximation. To be more precise, let us briefly introduce some notation. Let
$B$ and $\tilde{B}$ be two independent $d$-dimensional Brownian motions
started from the origin, let $\mathcal{F}_{t}:=\sigma(B_{s}\,:\,s\leq t)$ be
the natural filtration associated with $B$ up to a certain fixed time $t\geq
0$, and define the stopping time

$\tilde{\tau}:=\inf\\{s\geq 0\,:\,\sqrt{r}\tilde{B}_{s}+B_{t}\not\in
B(0,1/2)\\}\wedge\inf\\{s\geq 0\,:\,\tilde{B}_{s}\not\in B(0,1/2)\\}\,,$

where $r\in(0,1]$. Then, letting $\Omega_{\sqrt{r}}^{B_{t}}:=\inf_{x\in
B(B_{t},\sqrt{r})}X_{\sqrt{r}}(x)-X_{\sqrt{r}}(B_{t})$ the following lower
bound trivially holds on the interval $[0,\tilde{\tau}]$,

$X_{\varepsilon\sqrt{r}}(\sqrt{r}\tilde{B}_{s}+B_{t})\geq(X_{\varepsilon\sqrt{r}}(\sqrt{r}\tilde{B}_{s}+B_{t})-X_{\sqrt{r}}(\sqrt{r}\tilde{B}_{s}+B_{t}))+\Omega_{\sqrt{r}}^{B_{t}}+X_{\sqrt{r}}(B_{t})\,.$

Therefore, plugging this estimate into the second line of [Jac18, Equation
3.19], using the fact that in [Jac18, Equation 3.20] the probability is
conditioned on $\mathcal{F}_{t}$, and proceeding in the same exact way as in
the proof of Proposition A.1 below, we can reach the same conclusion of
[Jac18, Lemma 3.9] also in our more general setting.

Finally, the changes that one needs to do in the proof of [Jac18, Lemma 3.11]
are similar and therefore not discussed. ∎

We can now state the following proposition concerning the singularity spectrum
of $\mu_{\gamma}$.

######  Proposition 5.8.

If $\gamma^{2}<4$, then the measure $\mu_{\gamma}$ satisfies the multifractal
formalism. More precisely, for $\alpha\geq 0$ it holds that

${\mathrm{d}}_{\mu_{\gamma}}(\alpha)=\tau^{*}_{\mu_{\gamma}}(\alpha)=\begin{cases}1-\left(\frac{1-\alpha}{\gamma}+\frac{1}{4}\gamma\right)^{2}\,,&\quad\text{
if
}\alpha\in\left[\left(1-\frac{|\gamma|}{2}\right)^{2},\left(1+\frac{|\gamma|}{2}\right)^{2}\right]\,,\\\
0\,,&\quad\text{ otherwise}\,,\end{cases}$

almost surely.

###### Proof.

Given Propositions 5.6 and 5.7, and defining the structure function
$\phi_{\mu_{\gamma}}(q):=1-\xi_{\mu_{\gamma}}(q)$, for all $q\in\mathbb{R}$,
one can proceed similarly to Section 3 to prove this result. Therefore, the
details are omitted. ∎

####  5.2.2 Conclusion of the proof

Now that we have found an explicit formula for the singularity spectrum of
$\mu_{\gamma}$, we can study the relations between the various (lower)
singularity spectra. For $\alpha\geq 0$, we define the set
$\mathrm{E}_{\mu_{\gamma}}(\alpha)$ as in (2.8), the sets
$\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha)$,
$\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha)$ as in (2.9), and similarly also
for $\mu^{-}_{\gamma}$. Let us start with the following lemma.

######  Lemma 5.9.

If $\gamma^{2}<4$, then for $\alpha\geq 0$, it holds $\mathbb{P}_{B}$-almost
surely that

  1. (i)

$\mathrm{E}_{\mu^{-}_{\gamma}}(2\alpha)\subset\underline{\mathrm{E}}_{\mathcal{B}^{\gamma}}(\alpha)$;

  2. (ii)

$\underline{\mathrm{E}}_{\mathcal{B}^{\gamma}}(\alpha)\subset\underline{\mathrm{U}}_{\mu^{-}_{\gamma}}(2\alpha)\cap\overline{\mathrm{B}}_{\mu^{-}_{\gamma}}(2\alpha)$.

###### Proof.

Since for all $t\in[0,T]$, it holds that
$\mathcal{B}^{\gamma}_{t}=B_{\mu_{\gamma}^{-}((0,t])}$, one can use the same
argument used in the proof of Lemma 4.4 to prove this result. ∎

Thanks to Lemma 5.9, the problem has been reduced to computing the Hausdorff
dimensions of $\mathrm{E}_{\mu^{-}_{\gamma}}(\alpha)$ and of
$\underline{\mathrm{U}}_{\mu^{-}_{\gamma}}(\alpha)\cap\overline{\mathrm{B}}_{\mu^{-}_{\gamma}}(2\alpha)$,
for $\alpha\geq 0$. We start with the following trivial lemma.

######  Lemma 5.10.

If $\gamma^{2}<4$, then for $\alpha>0$, it holds almost surely that

  1. (i)

$\mathrm{E}_{\mu^{-}_{\gamma}}(\alpha)=F_{\gamma}(\mathrm{E}_{\mu_{\gamma}}(\alpha^{-1}))$;

  2. (ii)

$\underline{\mathrm{U}}_{\mu^{-}_{\gamma}}(\alpha)\cap\overline{\mathrm{B}}_{\mu^{-}_{\gamma}}(\alpha)=F_{\gamma}(\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1}))$.

###### Proof.

The proof follows thanks to the fact that random function $F_{\gamma}$ is
almost surely continuos and strictly increasing on the interval $[0,T]$ (cf.
Proposition 5.4). ∎

The following lemma is concerned with the relation between
${\mathrm{d}}_{\mu_{\gamma}}$ and ${\mathrm{d}}_{\mu^{-}_{\gamma}}$.

######  Lemma 5.11.

If $\gamma^{2}<4$, then for $\alpha>0$, it holds almost surely that
${\mathrm{d}}_{\mu^{-}_{\gamma}}(\alpha)=\alpha{\mathrm{d}}_{\mu_{\gamma}}(\alpha^{-1})$.

###### Proof.

Thanks to item (i) of Lemma 5.10, we know that for $\alpha>0$, it holds almost
surely that
$\mathrm{E}_{\mu^{-}_{\gamma}}(\alpha)=F_{\gamma}(\mathrm{E}_{\mu_{\gamma}}(\alpha^{-1}))$.
Fix $\varepsilon>0$, and let $t\in\mathrm{E}_{\mu_{\gamma}}(\alpha^{-1})$,
then we define $\delta_{t}>0$ as follows

$\delta_{t}:=\sup\left\\{\delta>0\,:\,|F_{\gamma}(t+r)-F_{\gamma}(t-r)|\leq
r^{\alpha^{-1}-\varepsilon},\,\forall r\in(0,\delta)\right\\}\wedge 1\,.$

We define the collection of sets
$(E^{n}_{\mu_{\gamma}}(\alpha^{-1}))_{n\in\mathbb{N}}$ by letting

$\mathrm{E}^{n}_{\mu_{\gamma}}(\alpha^{-1}):=\left\\{t\in\mathrm{E}_{\mu_{\gamma}}(\alpha^{-1})\,:\,2^{-n}<\delta_{t}\leq
2^{-n+1}\right\\}\,,\quad n\in\mathbb{N}\,.$

Then
$\mathrm{E}_{\mu_{\gamma}}(\alpha^{-1})=\cup_{n\in\mathbb{N}}\mathrm{E}^{n}_{\mu_{\gamma}}(\alpha^{-1})$,
and for all $t\in\mathrm{E}^{n}_{\mu_{\gamma}}(\alpha^{-1})$, it holds that
$|F_{\gamma}(t+r)-F_{\gamma}(t-r)|\leq r^{\alpha^{-1}-\varepsilon}$ for all
$r\in(0,2^{-n})$. Thanks to Proposition 5.4, we know that the random function
$F_{\gamma}$ is almost surely continuous and strictly increasing. Hence, using
the countable stability of the Hausdorff dimension and Proposition 2.7, it
holds almost surely that

$\displaystyle{\mathrm{d}}_{\mu^{-}_{\gamma}}(\alpha)=\sup_{n\in\mathbb{N}}\dim_{\mathcal{H}}(F_{\gamma}(\mathrm{E}^{n}_{\mu_{\gamma}}(\alpha^{-1})))\leq\frac{\alpha}{1-\alpha\varepsilon}\sup_{n\in\mathbb{N}}\dim_{\mathcal{H}}(\mathrm{E}^{n}_{\mu_{\gamma}}(\alpha^{-1}))=\frac{\alpha}{1-\alpha\varepsilon}{\mathrm{d}}_{\mu_{\gamma}}(\alpha^{-1})\,.$

The reverse inequality can be obtained similarly. Indeed, if we fix
$\varepsilon>0$ and we let $t\in\mathrm{E}_{\mu^{-}_{\gamma}}(\alpha)$, then
we can define $\delta_{t}>0$ as follows

$\delta_{t}:=\sup\left\\{\delta>0\,:\,|F^{-1}_{\gamma}(t+r)-F^{-1}_{\gamma}(t-r)|\leq
r^{\alpha-\varepsilon},\,\forall r\in(0,\delta)\right\\}\wedge 1\,.$

Defining the collection of sets
$(\mathrm{E}^{n}_{\mu^{-}_{\gamma}}(\alpha))_{n\in\mathbb{N}}$ similarly as
above, and proceeding in the same exact way, we obtain that it holds almost
surely that

${\mathrm{d}}_{\mu_{\gamma}}(\alpha^{-1})\leq\frac{1}{\alpha-\varepsilon}{\mathrm{d}}_{\mu^{-}_{\gamma}}(\alpha)\,.$

The conclusion follows by arbitrariness of $\varepsilon>0$. ∎

We also need the following lemma which allows to conclude the proof.

######  Lemma 5.12.

If $\gamma^{2}<4$, then for $\alpha>0$ it holds almost surely that
$\dim_{\mathcal{H}}(\underline{\mathrm{U}}_{\mu^{-}_{\gamma}}(\alpha)\cap\overline{\mathrm{B}}_{\mu^{-}_{\gamma}}(\alpha))\leq\alpha\tau^{*}_{\mu_{\gamma}}(\alpha^{-1})$.

###### Proof.

Thanks to item (ii) of Lemma 5.10 and to the definition of
$\tau_{\mu_{\gamma}}^{*}$, it is sufficient to prove that for $\alpha>0$ it
holds that

$\dim_{\mathcal{H}}(F_{\gamma}(\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1})))\leq
q+\alpha\tau_{\mu_{\gamma}}(q)\,,\quad\forall q\in R\,,$ (5.8)

almost surely. We will be brief here since this result can be proved using the
same procedure used in the proof of Proposition 2.12. We will only focus in
the case $q\geq 0$. Fix $\varepsilon>0$ and let
$\overline{s}_{q,\varepsilon}:=q+(\tau_{\mu_{\gamma}}(q)+\varepsilon)/(\alpha^{-1}-\varepsilon)$.
By definition, for every
$t\in\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1})$,
it exists a decreasing sequence $(r_{t,n})_{n\in\mathbb{N}}$ converging to $0$
such that

$|F_{\gamma}(t+r_{t,n})-F_{\gamma}(t-r_{t,n})|=\mu_{\gamma}(B(t,r_{t,n}))\leq
r_{t,n}^{\alpha^{-1}-\varepsilon}\,,\quad\forall n\in\mathbb{N}\,.$

Fix $\delta\in(0,1]$, then for each
$t\in\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1})$,
we choose $n_{t}$ such that $r_{t,n_{t}}\in(0,\delta)$. For each
$m\in\mathbb{N}$, we let

$F_{m}:=\left\\{t\in\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1})\,:\,2^{-m}<r_{t,n_{t}}\leq
2^{-m+1}\right\\}\,.$

Proceeding as in the proof of Proposition 2.12, it exists a positive integer
$\sigma_{1}$ such that for every $m\in\mathbb{N}$, we can find $\sigma_{1}$
disjoint subsets $F_{m,1},\dots,F_{m,\sigma_{1}}$ of $F_{m}$ such that each
$F_{m,j}$ is at most countable, the balls $B(t,r_{t,n_{t}})$ with centres at
$t\in F_{m,j}$ are pairwise disjoint, and the collection of sets

$\left(\left(\left(F_{\gamma}(B(t,r_{t,n_{t}}))\right)_{t\in
F_{m,j}}\right)_{j\in\\{1,\dots,\sigma_{1}\\}}\right)_{m\in\mathbb{N}}$

is a $\delta$-cover of
$F_{\gamma}(\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1}))$.
Then, thanks to the definition of $\mu_{\gamma}$ and to the fact that
$F_{\gamma}$ is almost surely strictly increasing, we have that

$\displaystyle\mathcal{H}_{\delta}^{\overline{s}_{q,\varepsilon}}(F_{\gamma}(\underline{\mathrm{U}}_{\mu_{\gamma}}(\alpha^{-1})\cap\overline{\mathrm{B}}_{\mu_{\gamma}}(\alpha^{-1})))$
$\displaystyle\leq\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{1}}\sum_{t\in
F_{m,j}}|F_{\gamma}(t+r_{t,n_{t}})-F_{\gamma}(t-r_{t,n_{t}})|^{\overline{s}_{q,\varepsilon}}$
$\displaystyle\leq\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{1}}\sum_{t\in
F_{m,j}}\mu_{\gamma}(B(t,r_{t,n_{t}}))^{q}r_{t,n_{t}}^{(\tau_{\mu}(q)+\varepsilon)}$
$\displaystyle\lesssim\sum_{m\in\mathbb{N}}\sum_{j=1}^{\sigma_{1}}\sum_{t\in
F_{m,j}}\mu_{\gamma}(B(t,2^{-m+1}))^{q}2^{-m(\tau_{\mu}(q)+\varepsilon)}\,,$

where the implicit constant does not depend on $m$. Hence, the conclusion
follows as in the proof of Proposition 2.12. Similarly, we can prove
inequality (5.8) for $q<0$. ∎

Finally, let us see how we can put everything together in order to prove
Theorem 5.5.

###### Proof of Theorem 5.5.

Let us start by observing that almost surely it holds that
$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(0)=0$ since
$\mathcal{B}^{\gamma}$ is almost surely continuous. Thanks to item (i) of
Lemma 5.9 and Lemma 5.11, for $\alpha>0$, we have that

$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(\alpha)\geq{\mathrm{d}}_{\mu^{-}_{\gamma}}(2\alpha)=2\alpha{\mathrm{d}}_{\mu_{\gamma}}((2\alpha)^{-1})\,,$

almost surely. On the other hand, item (ii) of Lemma 5.9, Lemma 5.12 and
Proposition 5.8 imply that, for all $\alpha>0$, it holds that

$\underline{{\mathrm{d}}}_{\mathcal{B}^{\gamma}}(\alpha)\leq\dim_{\mathcal{H}}\left(\underline{\mathrm{U}}_{\mu^{-}_{\gamma}}(2\alpha)\cap\overline{\mathrm{B}}_{\mu^{-}_{\gamma}}(2\alpha)\right)\leq
2\alpha\tau^{*}_{\mu_{\gamma}}((2\alpha)^{-1})=2\alpha{\mathrm{d}}_{\mu_{\gamma}}((2\alpha)^{-1})\,,$

almost surely. Since the singularity spectrum ${\mathrm{d}}_{\mu_{\gamma}}$ is
known, this concludes the proof. ∎

##  Appendix A Proof of Proposition 3.4

The proof of Proposition 3.4 follows similar steps of the proof in [RV14,
Theorem 4.1]. However, our proof accommodates general log-correlated Gaussian
fields. Indeed, it is not enough to prove the result for an exactly
stochastically scale invariant field, and then deduce the result for all log-
correlated fields, by means of Kahane’s convexity inequality (cf. Lemma C.2).
This is due to the fact that in the proof we need to consider moments of
certain integrals of the field that involve indicator functions depending on
the field itself.

Since the conclusion of Proposition 3.4 trivially holds for $\gamma=0$, we may
assume $\gamma\neq 0$. The proof is essentially based on the following
proposition and on a standard Borel–Cantelli argument.

######  Proposition A.1.

Let $\gamma^{2}<2d$, $q^{2}<2d/\gamma^{2}$ and define
$\alpha:=d+(1/2-q)\gamma^{2}$. For $\beta>0$ and $E>0$, there exist constants
$C$, $b$, $C^{\prime}$, $b^{\prime}>0$ such that for $r\in(0,1]$ small enough
it holds that

$\mathbb{E}\left[M_{q\gamma}\left(\left\\{x\in
D_{r}\,:\,M_{\gamma}(B(x,r))>Er^{\alpha-\beta}\right\\}\right)\right]\leq
Cr^{b}\,,$ (A.1)

and

$\mathbb{E}\left[M_{q\gamma}\left(\left\\{x\in
D_{r}\,:\,M_{\gamma}(B(x,r))<Er^{\alpha+\beta}\right\\}\right)\right]\leq
C^{\prime}r^{b^{\prime}}\,,$ (A.2)

where $D_{r}:=\\{x\in D\,:\,\operatorname{dist}(x,\partial D)<r\\}$.

###### Proof.

The proof can be found in Subsection A.1. ∎

We are now ready to prove Proposition 3.4.

###### Proof of Proposition 3.4.

Fix $\beta>0$ and let $\alpha=d+(1/2-q)\gamma^{2}$. For $n\in\mathbb{N}$, we
define the following sets

$U_{\gamma}^{n}:=\left\\{x\in D_{2^{-n}}\,:\,M_{\gamma}(B(x,r))\leq
r^{\alpha-\beta},\quad\forall r\in[0,2^{-n})\right\\}\,,$

and

$L_{\gamma}^{n}:=\left\\{x\in D_{2^{-n}}\,:\,M_{\gamma}(B(x,r))\geq
r^{\alpha+\beta}\quad\forall r\in[0,2^{-n})\right\\}\,,$

where we recall that $D_{r}:=\\{x\in D\,:\,\operatorname{dist}(x,\partial
D)<r\\}$. We claim that for every $\varepsilon\in(0,1]$, it exists almost
surely a finite $N\in\mathbb{N}$ such that

$M_{q\gamma}\left(\left(U_{\gamma}^{N}\right)^{c}\right)<\varepsilon\,,\quad
M_{q\gamma}\left(\left(L_{\gamma}^{N}\right)^{c}\right)<\varepsilon\,.$

Thanks to Proposition A.1 with $E=2^{-(\alpha-\beta)}$ and to Markov’s
inequality, we know that there exist constants $C>0$ and $b>0$ such that for
$r\in(0,1]$ small enough, it holds that

$\displaystyle\mathbb{P}\left(M_{q\gamma}\left(\left\\{x\in
D_{r}\,:\,M_{\gamma}(B(x,r))>2^{-(\alpha-\beta)}r^{\alpha-\beta}\right\\}\right)\geq
r^{\frac{b}{2}}\right)$ $\displaystyle\qquad\qquad\leq
r^{-\frac{b}{2}}\mathbb{E}\left[M_{q\gamma}\left(\left\\{x\in
D_{r}\,:\,M_{\gamma}(B(x,r))>2^{-(\alpha-\beta)}r^{\alpha-\beta}\right\\}\right)\right]$
$\displaystyle\qquad\qquad\leq Cr^{\frac{b}{2}}\,.$

For $n\in\mathbb{N}$, taking $r=2^{-n}$, thanks to the Borel–Cantelli lemma,
we get that the events

$\left(\left\\{M_{q\gamma}\left(\left\\{x\in
D_{2^{-n}}\,:\,M_{\gamma}(B(x,2^{-n}))>(2^{-n-1})^{\alpha-\beta}\right\\}\right)\geq(2^{-n})^{\frac{b}{2}}\right\\}\right)_{n\in\mathbb{N}}$

occur only finitely often almost surely. Therefore, for all
$\varepsilon\in(0,1]$, it exists almost surely a finite $N\in\mathbb{N}$ such
that

$M_{q\gamma}\left(\bigcup_{n\geq N}\left\\{x\in
D_{2^{-n}}\,:\,M_{\gamma}(B(x,2^{-n}))>(2^{-n-1})^{\alpha-\beta}\right\\}\right)\leq\sum_{n\geq
N}(2^{-n})^{\frac{b}{2}}<\varepsilon\,.$

Let $x\in(U_{\gamma}^{N})^{c}$, then it exists $r\in[0,2^{-N})$ such that
$M_{\gamma}(B(x,r))>r^{\alpha-\beta}$. Since it exists $n\geq N$ such that
$2^{-n-1}\leq r<2^{-n}$, we get

$M_{\gamma}(B(x,2^{-n}))\geq
M_{\gamma}(B(x,r))>r^{\alpha-\beta}\geq(2^{-n-1})^{\alpha-\beta}\,,$

which implies that

$M_{q\gamma}\left(\left(U_{\gamma}^{N}\right)^{c}\right)\leq
M_{q\gamma}\left(\bigcup_{n\geq N}\left\\{x\in
D_{2^{-n}}\,:\,M_{\gamma}(B(x,2^{-n}))>(2^{-n-1})^{\alpha-\beta}\right\\}\right)<\varepsilon\,.$

A similar argument can be used to prove that for every $\varepsilon\in(0,1]$,
it exists almost surely a finite $N\in\mathbb{N}$ such that
$M_{q\gamma}((L_{\gamma}^{N})^{c})<\varepsilon$. Therefore, introducing the
sets

$\displaystyle U_{\gamma}:=\bigcup_{n\in\mathbb{N}}U_{\gamma}^{n}=\left\\{x\in
D\,:\,\liminf_{r\searrow 0}\frac{\log M_{\gamma}(B(x,r))}{\log
r}\geq\alpha-\beta\right\\}\,,$ $\displaystyle
L_{\gamma}:=\bigcup_{n\in\mathbb{N}}L_{\gamma}^{n}=\left\\{x\in
D\,:\,\limsup_{r\searrow 0}\frac{\log M_{\gamma}(B(x,r))}{\log
r}\leq\alpha+\beta\right\\}\,,$

since $\varepsilon>0$ is arbitrary and $U_{\gamma}^{N}$, $L_{\gamma}^{N}$ are
increasing sets, we obtain that

$M_{q\gamma}\left(\left(L_{\gamma}\cap U_{\gamma}\right)^{c}\right)=0\,.$

The conclusion then follows from the arbitrariness of $\beta>0$. ∎

###  A.1 Proof of Proposition A.1

The objective of this subsection is to prove Proposition A.1. For simplicity,
but without loss of generality, we assume that $D=(-1,1)^{d}$. The proof is
based on Lemma 2.1 and on the following two lemmas.

######  Lemma A.2.

For $\gamma^{2}<2d$ and $q^{2}<2d/\gamma^{2}$, there exist $\bar{a}\in(0,1]$
and $C>0$ such that

$\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{\bar{a}}\right]\leq
C$

###### Proof.

The proof can be found in Subsection A.2. ∎

######  Lemma A.3.

For $\gamma^{2}<2d$ and $q^{2}<2d/\gamma^{2}$, there exists $C>0$ such that

$\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{[-1/2,1/2]^{d}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{-1}\right]\leq
C\,.$

###### Proof.

The proof can be found in Subsection A.2. ∎

###### Proof of Proposition A.1.

We split the proof into two parts. We start by proving (A.1) and then we move
to the proof of (A.2). For simplicity, but without loss of generality, we
consider the case $E=1$.

Proof of (A.1). Let $r\in(0,1]$ and $x\in D_{r}$, where we recall that we are
assuming that $D=(-1,1)^{d}$. Fix $\varepsilon$, $\varepsilon^{\prime}>0$ so
that $\varepsilon^{\prime}<r\varepsilon$. Then by Girsanov’s theorem (cf.
Lemma C.1), we have that

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}e^{q\gamma
X_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[X_{\varepsilon^{\prime}}(x)^{2}]}\right]$
$\displaystyle\qquad\qquad=\mathbb{P}\left(\int_{B(x,r)}e^{\gamma
X_{r\varepsilon}(u)+q\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)X_{\varepsilon^{\prime}}(x)]-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right)$
$\displaystyle\qquad\qquad\leq\mathbb{P}\left(e^{\gamma^{2}K|q|}\int_{B(x,r)}\frac{e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}}{(|u-x|+r\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u>r^{\alpha-\beta}\right)\,,$
(A.3)

where the last inequality is justified by the fact that, thanks to property
(P2), for $x$, $y\in(-1,1)^{d}$, it exists $K>0$ such that

$-\log\left(|x-y|+r\varepsilon\right)-K\leq\mathbb{E}[X_{r\varepsilon}(x)X_{\varepsilon^{\prime}}(y)]\leq-\log\left(|x-y|+r\varepsilon\right)+K\,.$

Now, let $\Omega_{r}^{x}:=\sup_{u\in B(x,r)}X_{r}(u)-X_{r}(x)$, then by
changing variables, we see that the integral in (A.1) can be upper bounded as
follows

$\displaystyle\int_{B(x,r)}\frac{e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}}{(|u-x|+r\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u$
$\displaystyle\qquad\qquad\leq
e^{|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(x,r)}\frac{e^{\gamma(X_{r\varepsilon}(u)-X_{r}(u))-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}}{(|u-x|+r\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u$
$\displaystyle\qquad\qquad=r^{d-q\gamma^{2}}e^{|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(0,1)}\frac{e^{\gamma(X_{r\varepsilon}(ru+x)-X_{r}(ru+x))-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(ru+x)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u$
$\displaystyle\qquad\qquad\leq
r^{d+(1/2-q)\gamma^{2}}e^{\frac{1}{2}\gamma^{2}K+|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(0,1)}\frac{e^{\gamma(X_{r\varepsilon}(ru+x)-X_{r}(ru+x))+\frac{1}{2}\gamma^{2}\log\varepsilon}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u$
$\displaystyle\qquad\qquad\leq
r^{\alpha}e^{\gamma^{2}Kc+|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(0,1)}\frac{e^{\gamma
X^{\prime}_{\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X^{\prime}_{\varepsilon}(u)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u\,,$
(A.4)

for some log-correlated field $X^{\prime}$ in $B(0,1)$, and some finite
constant $c>0$ depending only on $K$. Substituting the last expression of
(A.1) back into (A.1), we obtain that

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}e^{q\gamma
X_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[X_{\varepsilon^{\prime}}(x)^{2}]}\right]$
(A.5)
$\displaystyle\quad\leq\mathbb{P}\left(e^{\gamma^{2}K(c+|q|)+|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(0,1)}\frac{e^{\gamma
X^{\prime}_{\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X^{\prime}_{\varepsilon}(u)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u>r^{-\beta}\right)$
$\displaystyle\quad\leq\mathbb{P}\left(e^{\gamma^{2}K(c+|q|)+|\gamma|\Omega_{r}^{x}}>r^{-\frac{\beta}{3}}\right)+\mathbb{P}\left(e^{|\gamma|X_{r}(x)}>r^{-\frac{\beta}{3}}\right)+\mathbb{P}\left(\int_{B(0,1)}\frac{e^{\gamma
X^{\prime}_{\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X^{\prime}_{\varepsilon}(u)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u>r^{-\frac{\beta}{3}}\right)\,.$

Let us notice that all the three terms decay polynomially in $r$, for
$r\in(0,1]$ small enough, uniformly in $x\in D_{r}$ and $\varepsilon$,
$\varepsilon^{\prime}\in(0,1]$. Indeed, for the first term this follows from
Lemma 2.1 and Markov’s inequality. For the second term, this follows from the
fact that $X_{r}(x)$ is a centred Gaussian random variable with variance
$\mathbb{E}[X_{r}(x)^{2}]=-\log r+\mathcal{O}(1)$ (cf. [MP10, Lemma 12.9]).
Finally, for the third term this follows from Lemma A.2 and Markov’s
inequality again. Therefore, for $r\in(0,1]$ small enough, integrating over
$x\in D_{r}$ the expression in (A.5) and putting everything together, we can
see that there exist finite constants $C$, $b>0$ such that for all
$\varepsilon$, $\varepsilon^{\prime}\in(0,1]$, it holds that

$\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
Y_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[Y_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}e^{q\gamma
Y_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[Y_{\varepsilon^{\prime}}(x)^{2}]}{\mathrm{d}}x\right]\leq
Cr^{b}\,.$

Now, thanks to Proposition 2.2, we know that the sequence of measures
$(M_{q\gamma}^{\varepsilon^{\prime}})_{\varepsilon^{\prime}\in(0,1]}$
converges to $M_{q\gamma}$ in $L^{1}(\mathbb{P})$ in the topology of weak
convergence of measures. Therefore, thanks to Portmanteau lemma and Fatou’s
lemma, we get that

$\displaystyle\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
Y_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[Y_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}M_{q\gamma}({\mathrm{d}}x)\right]$
$\displaystyle\qquad\qquad\leq\liminf_{\varepsilon^{\prime}\searrow
0}\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
Y_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[Y_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}e^{q\gamma
Y_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[Y_{\varepsilon^{\prime}}(x)^{2}]}{\mathrm{d}}x\right]\,.$

Therefore, since $C$ and $b$ are independent from
$\varepsilon^{\prime}\in(0,1]$, we have that

$\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
Y_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[Y_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}M_{q\gamma}({\mathrm{d}}x)\right]\leq
Cr^{b}\,.$ (A.6)

Finally, since $C$ and $b$ are independent also from $\varepsilon\in(0,1]$,
using (A.6) and Fatou’s lemma, we see that

$\displaystyle\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{M_{\gamma}(B(x,r))>r^{\alpha-\beta}\right\\}}M_{q\gamma}({\mathrm{d}}x)\right]$
$\displaystyle\qquad\qquad\leq\liminf_{\varepsilon\searrow
0}\mathbb{E}\left[\int_{D_{r}}\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
Y_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[Y_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u>r^{\alpha-\beta}\right\\}}M_{q\gamma}({\mathrm{d}}x)\right]$
$\displaystyle\qquad\qquad\leq Cr^{b}\,.$

Proof of (A.2). We shall be brief here since the argument is similar to that
used in the proof of (A.1). Let $r\in(0,1]$ and $x\in D_{r}$, where we recall
that we are assuming that $D=(-1,1)^{d}$. Fix $\varepsilon$,
$\varepsilon^{\prime}\in(0,1]$ so that $\varepsilon^{\prime}<r\varepsilon$.
Then by Girsanov’s theorem (cf. Lemma C.1) and property (P2), we have

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u<r^{\alpha+\beta}\right\\}}e^{q\gamma
X_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[X_{\varepsilon^{\prime}}(x)^{2}]}\right]$
$\displaystyle\qquad\qquad\leq\mathbb{P}\left(e^{-\gamma^{2}K|q|}\int_{B(x,r)}\frac{e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}}{(|u-x|+r\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u<r^{\alpha+\beta}\right)\,.$
(A.7)

Now, let $\Omega_{r}^{x}:=\inf_{u\in B(x,r)}X_{r}(u)-X_{r}(x)$, then by
proceeding in the same way as in the proof of (A.1), we can see that the
right-hand side of (A.1) can be lower bounded as follows

$\displaystyle\int_{B(x,r)}\frac{e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}}{(|u-x|+r\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u$
$\displaystyle\qquad\qquad\geq
r^{\alpha}e^{-\gamma^{2}Kc+|\gamma|(\Omega_{r}^{x}+X_{r}(x))}\int_{B(0,1)}\frac{e^{\gamma
X^{\prime}_{\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X^{\prime}_{\varepsilon}(u)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u\,,$
(A.8)

for some log-correlated field $X^{\prime}$ in $B(0,1)$, and some finite
constant $c>0$ depending only on $K$. Substituting the last expression of
(A.1) back into (A.1), we obtain that

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{\left\\{\int_{B(x,r)}e^{\gamma
X_{r\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{r\varepsilon}(u)^{2}]}{\mathrm{d}}u<r^{\alpha+\beta}\right\\}}e^{q\gamma
X_{\varepsilon^{\prime}}(x)-\frac{1}{2}q^{2}\gamma^{2}\mathbb{E}[X_{\varepsilon^{\prime}}(x)^{2}]}\right]$
$\displaystyle\quad\leq\mathbb{P}\left(e^{-\gamma^{2}K(c+|q|)+|\gamma|\Omega_{r}^{x}}<r^{\frac{\beta}{3}}\right)+\mathbb{P}\left(e^{|\gamma|X_{r}(x)}<r^{\frac{\beta}{3}}\right)+\mathbb{P}\left(\int_{B(0,1)}\frac{e^{\gamma
X^{\prime}_{\varepsilon}(u)-\frac{1}{2}\gamma^{2}\mathbb{E}[X^{\prime}_{\varepsilon}(u)^{2}]}}{(|u|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}u<r^{\frac{\beta}{3}}\right)\,.$

Thanks to Lemmas 2.1 and A.3, the conclusion then follows as in the proof of
(A.1). ∎

###  A.2 Proofs of Lemmas A.2 and A.3

Let us finish by proving Lemmas A.2 and A.3. We recall once again that, for
simplicity, we are assuming that $D=(-1,1)^{d}$.

###### Proof of Lemma A.2.

For $\varepsilon\in(0,1]$ and $a\in(0,1]$, we define the quantity

$U^{a}_{\varepsilon}:=\mathbb{E}\left[\left(\int_{(-1,1)^{d}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{a}\right]\,,$

and we need to prove that it exists $\bar{a}\in(0,1]$ such that
$U^{\bar{a}}_{\varepsilon}$ can be uniformly bounded in $\varepsilon\in(0,1]$.
We split the proof into two parts. In the first part we focus on
$q\in(-\sqrt{2d}/|\gamma|,0]$, while in the second part we focus on
$q\in(0,\sqrt{2d}/|\gamma|)$.

Case $q\in(-\sqrt{2d}/|\gamma|,0]$. Since $|x|+\varepsilon\leq\sqrt{d}+1$ for
all $x\in(-1,1)^{d}$ and $\varepsilon\in(0,1]$, we can write

$U^{a}_{\varepsilon}\leq(\sqrt{d}+1)^{-aq\gamma^{2}}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{a}\right]\leq
C(\sqrt{d}+1)^{-aq\gamma^{2}}\,,$

where the existence of the constant $C>0$, independent of
$\varepsilon\in(0,1]$, follows from Jensen’s inequality.

Case $q\in(0,\sqrt{2d}/|\gamma|)$. If $\varepsilon\in(1/2,1]$, then we have
that

$U_{\varepsilon}^{a}\leq
2^{aq\gamma^{2}}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{a}\right]\leq
C2^{aq\gamma^{2}}\,.$

where the existence of the constant $C>0$, independent of
$\varepsilon\in(1/2,1]$, follows from Jensen’s inequality. Let us now assume
that $\varepsilon\in(0,1/2]$. Then it exists a unique $n\in\mathbb{N}$ such
that $2^{-n-1}<\varepsilon\leq 2^{-n}$. For all $k\in\\{0,1,\dots,n\\}$, we
let $Q_{k}:=(-2^{-k},2^{-k})^{d}$. Then by sub-additivity of the function
$x\mapsto x^{a}$ for $a\in(0,1]$, we have that

$\displaystyle U^{a}_{\varepsilon}$
$\displaystyle\leq\mathbb{E}\left[\left(\int_{Q_{n}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{a}\right]+\sum_{k=0}^{n-1}\mathbb{E}\left[\left(\int_{Q_{k}\setminus
Q_{k+1}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\leq
2^{na(q\gamma^{2}-d)}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}\frac{e^{\gamma
X_{\varepsilon}(2^{-n}x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-n}x)^{2}]}}{(|x|+2^{n}\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad+\sum_{k=0}^{n-1}2^{(k+1)aq\gamma^{2}}\mathbb{E}\left[\left(\int_{Q_{k}\setminus
Q_{k+1}}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\leq
2^{na(q\gamma^{2}-d)}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(2^{-n}x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-n}x)^{2}]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad+2^{ad}\sum_{k=0}^{n-1}2^{(k+1)a(q\gamma^{2}-d)}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(2^{-k}x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-k}x)^{2}]}{\mathrm{d}}x\right)^{a}\right]\,.$
(A.9)

Now let $k\in\\{0,1,\dots,n\\}$, then we want to estimate the following
quantity

$\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(2^{-k}x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-k}x)^{2}]}{\mathrm{d}}x\right)^{a}\right]\,.$

Thanks to property (P2), for $x$, $y\in(-1,1)^{d}$, it exists $K>0$ such that

$\mathbb{E}[X_{\varepsilon}(2^{-k}x)X_{\varepsilon}(2^{-k}y)]\geq-\log\left(|x-y|+2^{k}\varepsilon\right)+\log\left(2^{k}\right)-K\,,$

and therefore, if we let $Z$ and $\Omega_{2^{-k}}$ be two real normal random
variables independent from everything else with variance $2K$ and $\log
2^{k}$, respectively, then

$\mathbb{E}[(X_{\varepsilon}(2^{-k}x)+Z)(X_{\varepsilon}(2^{-k}y)+Z)]\geq\mathbb{E}[\left(X_{2^{k}\varepsilon}(x)+\Omega_{2^{-k}}\right)\left(X_{2^{k}\varepsilon}(y)+\Omega_{2^{-k}}\right)]\,.$

Since the function $x\mapsto x^{a}$ for $a\in(0,1]$ is concave, thanks to
Kahane’s convexity inequality (cf. Lemma C.2), we have that

$\displaystyle\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{\varepsilon}(2^{-k}x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-k}x)^{2}]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad=e^{\gamma^{2}aK(1-a)}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma\left(X_{\varepsilon}(2^{-k}x)+Z\right)-\frac{1}{2}\gamma^{2}\mathbb{E}\left[\left(X_{\varepsilon}(2^{-k}x)+Z\right)^{2}\right]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad\leq
e^{\gamma^{2}aK(1-a)}\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma\left(X_{2^{k}\varepsilon}(x)+\Omega_{2^{-k}}\right)-\frac{1}{2}\gamma^{2}\mathbb{E}\left[\left(X_{2^{k}\varepsilon}(x)+\Omega_{2^{-k}}\right)^{2}\right]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad=e^{\gamma^{2}aK(1-a)}\mathbb{E}\left[e^{\gamma
a\Omega_{2^{-k}}-\frac{1}{2}a\gamma^{2}\mathbb{E}\left[\Omega_{2^{-k}}^{2}\right]}\right]\mathbb{E}\left[\left(\int_{(-1,1)^{d}}e^{\gamma
X_{2^{k}\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}\left[X_{2^{k}\varepsilon}(x)^{2}\right]}{\mathrm{d}}x\right)^{a}\right]$
$\displaystyle\qquad\qquad\leq
Ce^{\gamma^{2}aK(1-a)}2^{\frac{1}{2}\gamma^{2}ka(a-1)}\,,$

where the existence of the constant $C>0$, independent of
$\varepsilon\in(0,1/2]$, follows from Jensen’s inequality. Hence, going back
to (A.2), letting

$s_{a}:=\left(d+\frac{1}{2}\gamma^{2}-q\gamma^{2}\right)a-\frac{1}{2}\gamma^{2}a^{2}\,,$
(A.10)

we obtain that

$U^{a}_{\varepsilon}\lesssim\sum_{k=0}^{n}2^{-ks_{a}}\,,$

where the implicit constant does not depend on $n$. It can be easily verified
that for each $\gamma^{2}<2d$ and $q\in(0,\sqrt{2d}/|\gamma|)$ it exists
$\bar{a}\in(0,1]$ such that $s_{\bar{a}}>0$. In particular, this implies that
$U^{\bar{a}}_{\varepsilon}$ is uniformly bounded in $\varepsilon\in[0,1/2)$,
which proves the result. ∎

###### Proof of Lemma A.3.

For every $\varepsilon\in(0,1]$, we define

$U^{-1}_{\varepsilon}:=\mathbb{E}\left[\left(\int_{[-1/2,1/2]^{d}}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{-1}\right]\,.$

We need to prove that $U^{-1}_{\varepsilon}$ can be uniformly bounded in
$\varepsilon\in(0,1]$. We split the proof into two cases.

Case $q\in[0,\sqrt{2d}/|\gamma|)$. In this case, since
$|x|+\varepsilon\leq\sqrt{d/2}+1$ for all $x\in[-1/2,1/2]^{d}$ and
$\varepsilon\in(0,1]$, we can write

$\sup_{\varepsilon\in(0,1]}U_{\varepsilon}^{-1}\leq(\sqrt{d/2}+1)^{q\gamma^{2}}\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{[-1/2,1/2]^{d}}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{-1}\right]\leq
C(\sqrt{d/2}+1)^{q\gamma^{2}}\,,$

where the existence of the constant $C>0$, independent of
$\varepsilon\in(0,1]$, follows from Proposition 2.3.

Case $q\in(-\sqrt{2d}/|\gamma|,0)$. In this case, we can simply proceed as
follows

$U_{\varepsilon}^{-1}\leq\mathbb{E}\left[\left(\int_{B(1/4,1/8)}\frac{e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}}{(|x|+\varepsilon)^{q\gamma^{2}}}{\mathrm{d}}x\right)^{-1}\right]\,,$

and since $|x|+\varepsilon\geq 1/8$ for all $x\in B(1/4,1/8)$ and
$\varepsilon\in(0,1]$, we can write

$\displaystyle\sup_{\varepsilon\in(0,1]}U_{\varepsilon}^{-1}\leq(1/8)^{-q\gamma^{2}}\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{B(1/4,1/8)}e^{\gamma
X_{\varepsilon}(x)-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(x)^{2}]}{\mathrm{d}}x\right)^{-1}\right]\leq
C(1/8)^{-q\gamma^{2}}\,,$

and also in this case the existence of the constant $C>0$, independent of
$\varepsilon\in(0,1]$, follows from Proposition 2.3. ∎

##  Appendix B Finiteness of positive moments of $\mu_{\gamma}$

In this section, we adopt the same notation used in Section 5. The main goal
is to prove that the moments of order $q\in(0,4/\gamma^{2})$ of the measure
$\mu_{\gamma}$, defined in (5.4), are uniformly bounded. The techniques used
in the proof of this result are inspired from [BP21, Section 3]. Moreover, let
us mention that this result has been proved in the planar case in [Ber15,
Subsection 3.2], but the proof does not trivially generalize to all
dimensions. Before stating and proving the result, it may be worth recalling
that the measure $\mu_{\gamma}$ can be interpreted as the weak limit of the
sequence of approximated measures
$(\mu_{\gamma}^{\varepsilon})_{\varepsilon\in(0,1]}$ defined in (5.5).

######  Proposition B.1.

Let $\gamma^{2}<4$ and $q\in(0,4/\gamma^{2})$. Then it holds that

$\sup_{\varepsilon\in(0,1]}\mathbb{E}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}]{\mathrm{d}}s\right)^{q}\right]<\infty\,,$

where we recall that $T$ denotes the first exit time of $B$ from the bounded
domain $D$.

###### Proof.

As in the proof of Proposition 5.6, thanks to Kahane’s convexity inequality
(cf. Lemma C.2), we can assume $X$ to be the $d$-dimensional exactly scale
invariant field. Moreover, without loss of generality, we can consider $q>1$.
In order to keep the notation not too cumbersome, we will give the proof only
for $q\in(1,2\wedge 4/\gamma^{2})$. Notice that this is not restrictive if
$\gamma^{2}\in[2,4)$. The case in which $\gamma^{2}<2$ and
$q\in[2,4/\gamma^{2})$ is discussed at the end.

For simplicity, we consider the case $D=(0,1)^{d}$. Let us fix
$\varepsilon\in(0,1]$, then letting $a=q-1\in(0,1)$, using Fubini’s theorem,
Girsanov’s theorem (cf. Lemma C.1), and the analogue of property (P2) for the
exactly scale invariant field $X_{\varepsilon}$, we obtain that

$\displaystyle\mathbb{E}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s\right)^{q}\right]$
$\displaystyle\qquad\qquad=\mathbb{E}_{B}\left[\int_{0}^{T}\mathbb{E}_{X}\left[e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}{\mathrm{d}}r\right)^{a}\right]{\mathrm{d}}s\right]$
$\displaystyle\qquad\qquad=\mathbb{E}_{B}\left[\int_{0}^{T}\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})+\gamma^{2}\mathbb{E}_{X}[X_{\varepsilon}(B_{s})X_{\varepsilon}(B_{r})]-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}{\mathrm{d}}r\right)^{a}\right]{\mathrm{d}}s\right]$
$\displaystyle\qquad\qquad\lesssim\mathbb{E}_{B}\left[\int_{0}^{T}\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}{\mathrm{d}}r\right)^{a}\right]{\mathrm{d}}s\right]\,.$
(B.1)

If $\varepsilon\in(1/2,1]$, then thanks to the concavity of the function
$x\mapsto x^{a}$ for $a\in(0,1)$, using Jensen’s inequality, we have that

$\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}{\mathrm{d}}r\right)^{a}\right]\leq
2^{a\gamma^{2}}\mathbb{E}_{X}\left[\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}{\mathrm{d}}r\right]^{a}=2^{a\gamma^{2}}T^{a}\,.$

Therefore, substituting this estimate into (B), we get that

$\mathbb{E}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s\right)^{q}\right]\leq
2^{a\gamma^{2}}\mathbb{E}_{B}[T^{1+a}]<\infty\,,$

where the last inequality follows from the fact that $T$ has exponentially
decaying tail. Let us now assume that $\varepsilon\in(0,1/2]$. Then it exists
a unique $n\in\mathbb{N}$ such that $2^{-n-1}<\varepsilon\leq 2^{-n}$. For
$x\in\mathbb{R}^{d}$ and $k\in\\{0,1,\dots,n\\}$, we let $Q_{k}(x)$ be the
open box of side length $2^{-k+1}$ centred at $x$, i.e.
$Q_{k}(x)=(-2^{-k},2^{-k})+x$. Then by sub-additivity of the function
$x\mapsto x^{a}$ for $a\in(0,1)$, we have that

$\displaystyle\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\leq\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}\mathbbm{1}_{\\{B_{r}\in
Q_{n}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{k=1}^{n}\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}\mathbbm{1}_{\\{B_{r}\in
Q_{k-1}(B_{s})\setminus Q_{k}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\leq
2^{na\gamma^{2}}\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r}^{2}]}}{(2^{n}|B_{r}-B_{s}|+2^{n}\varepsilon)^{\gamma^{2}}}\mathbbm{1}_{\\{B_{r}\in
Q_{n}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{k=1}^{n}2^{ka\gamma^{2}}\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}\mathbbm{1}_{\\{B_{r}\in
Q_{k-1}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\leq
2^{na\gamma^{2}}\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}\mathbbm{1}_{\\{B_{r}\in
Q_{n}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad\qquad\qquad\qquad+\sum_{k=1}^{n}2^{ka\gamma^{2}}\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}\mathbbm{1}_{\\{B_{r}\in
Q_{k-1}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]\,.$ (B.2)

Now let $k\in\\{0,1,\dots,n\\}$, then we want to estimate the following
quantity

$\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}\mathbbm{1}_{\\{B_{r}\in
Q_{k}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]\,.$

Using the translation and scaling invariance of the field $X_{\varepsilon}$,
Jensen’s inequality, and Fubini’s theorem, we can proceed as follows

$\displaystyle\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}\mathbbm{1}_{\\{B_{r}\in
Q_{k}(B_{s})\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad=\mathbb{E}_{X}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(2^{-k}(2^{k}(B_{r}-B_{s}))+B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(2^{-k}(2^{k}(B_{r}-B_{s}))+B_{s})^{2}]}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}\right]$
$\displaystyle\qquad=\mathbb{E}_{X}\left[e^{\gamma
a\Omega_{2^{-k}}-\frac{1}{2}\gamma^{2}a\mathbb{E}[\Omega_{2^{-k}}^{2}]}\left(\int_{0}^{T}e^{\gamma
X_{2^{k}\varepsilon}(2^{k}(B_{r}-B_{s}))-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{2^{k}\varepsilon}(2^{k}(B_{r}-B_{s}))^{2}]}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}\right]$ $\displaystyle\qquad\leq
2^{k(\frac{1}{2}\gamma^{2}a^{2}-\frac{1}{2}\gamma^{2}a)}\mathbb{E}_{X}\left[\int_{0}^{T}e^{\gamma
X_{2^{k}\varepsilon}(2^{k}(B_{r}-B_{s}))-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{2^{k}\varepsilon}(2^{k}(B_{r}-B_{s}))^{2}]}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right]^{a}$
$\displaystyle\qquad=2^{k(\frac{1}{2}\gamma^{2}a^{2}-\frac{1}{2}\gamma^{2}a)}\left(\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}\,,$ (B.3)

where we recall that $\Omega_{2^{-k}}$ denotes an independent centred Gaussian
random variable with variance $\log(2^{k})$. Therefore, plugging estimate (B)
into (B), we obtain that

$\mathbb{E}_{X}\left[\left(\int_{0}^{T}\frac{e^{\gamma
X_{\varepsilon}(B_{r})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{r})^{2}]}}{(|B_{r}-B_{s}|+\varepsilon)^{\gamma^{2}}}{\mathrm{d}}r\right)^{a}\right]\lesssim\sum_{k=0}^{n}2^{k(\frac{1}{2}\gamma^{2}a^{2}+\frac{1}{2}\gamma^{2}a)}\left(\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}\,.$

Hence, integrating the previous expression over $s\in[0,T]$ and then taking
expectation with respect to $B$, recalling inequality (B), we have that

$\displaystyle\mathbb{E}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s\right)^{q}\right]$
$\displaystyle\qquad\qquad\lesssim\sum_{k=0}^{n}2^{k(\frac{1}{2}\gamma^{2}a^{2}+\frac{1}{2}\gamma^{2}a)}\mathbb{E}_{B}\left[\int_{0}^{T}\left(\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}{\mathrm{d}}s\right]\,.$ (B.4)

Let us now estimate the expectation with respect to $B$ on the right-hand side
of the above expression. For all $k\in\\{0,1,\dots,n\\}$, applying Jensen’s
inequality two times, we obtain that

$\displaystyle\mathbb{E}_{B}\left[\int_{0}^{T}\left(\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r\right)^{a}{\mathrm{d}}s\right]$
$\displaystyle\leq\mathbb{E}_{B}\left[T^{\frac{{1-a}}{a}}\int_{0}^{T}\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r{\mathrm{d}}s\right]^{a}\,.$

In particular, letting $\alpha\in(0,2)$, using Markov’s inequality,
Cauchy–Schwarz’s inequality, and proceeding similarly to the proof of Lemma
5.1, we have that

$\displaystyle\mathbb{E}_{B}\left[T^{\frac{{1-a}}{a}}\int_{0}^{T}\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r{\mathrm{d}}s\right]$
$\displaystyle\qquad\qquad=\sum_{n\in\mathbb{N}}\mathbb{E}_{B}\left[\mathbbm{1}_{\\{n-1\leq
T<n\\}}T^{\frac{{1-a}}{a}}\int_{0}^{T}\int_{0}^{T}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r{\mathrm{d}}s\right]$
$\displaystyle\qquad\qquad\leq\sum_{n\in\mathbb{N}}n^{\frac{{1-a}}{a}}\mathbb{E}_{B}\left[\mathbbm{1}_{\\{n-1\leq
T<n\\}}\int_{0}^{n}\int_{0}^{n}\mathbbm{1}_{\\{2^{k}(B_{r}-B_{s})\in
Q_{0}(0)\\}}{\mathrm{d}}r{\mathrm{d}}s\right]$
$\displaystyle\qquad\qquad\leq\sum_{n\in\mathbb{N}}n^{\frac{{1-a}}{a}}\mathbb{P}_{B}(T\geq
n-1)^{\frac{1}{2}}\int_{0}^{n}\int_{0}^{n}\mathbb{P}_{B}(|B_{r}-B_{s}|<2^{-k+1})^{\frac{1}{2}}{\mathrm{d}}r{\mathrm{d}}s$
$\displaystyle\qquad\qquad\leq
2^{-\alpha(k-1)}\mathbb{E}_{B}[|B_{1}|^{-2\alpha}]^{\frac{1}{2}}\sum_{n\in\mathbb{N}}2n^{\frac{{1-a}}{a}+1}\mathbb{P}_{B}(T\geq
n-1)^{\frac{1}{2}}\int_{0}^{n}r^{-\frac{\alpha}{2}}{\mathrm{d}}r\,,$

and the latter series is obviously finite since we chose $\alpha\in(0,2)$ and
$T$ has exponentially decaying tail. Therefore, going back to (B), we have
that

$\mathbb{E}\left[\left(\int_{0}^{T}e^{\gamma
X_{\varepsilon}(B_{s})-\frac{1}{2}\gamma^{2}\mathbb{E}[X_{\varepsilon}(B_{s})^{2}]}{\mathrm{d}}s\right)^{q}\right]\lesssim\sum_{k=0}^{n}2^{ka(\frac{1}{2}\gamma^{2}a+\frac{1}{2}\gamma^{2}-\alpha)}\,,$

where the implicit constant does not depend on $n$. To conclude it is
sufficient to notice that for $a<4/\gamma^{2}-1$, there exists
$\alpha\in(0,2)$ such that $\gamma^{2}a/2+\gamma^{2}/2-\alpha<0$.

As a final remark, we emphasize that, if $\gamma^{2}<2$, we only proved the
existence of moments of order $q\in(0,2)$. However, the strategy of the proof
could be adapted to treat the general case $q\in(0,4/\gamma^{2})$. In this
case, one has to choose $n\in\mathbb{N}$ such that $n\leq q<n+1$. Then
applying Girsanov’s theorem $n$ times in (B), one obtain an expression similar
to the last line of (B) except that we get an integral with $n$ singularities.
Then, we can reproduce the argument up to modifications that are obvious but
notationally heavy. ∎

##  Appendix C Gaussian toolbox

We collect here some well-known results on Gaussian fields. In all the
subsequent lemmas, we assume $D$ to be a bounded domain of $\mathbb{R}^{d}$,
$d\geq 1$.

######  Lemma C.1 ([RV16, Theorem 2.1]).

Consider an almost surely continuous centred Gaussian field $(X(x))_{x\in D}$
and a Gaussian random variable $Z$ which belongs to the $L^{2}$ closure of the
vector space spanned by $(X(x))_{x\in D}$. Let $F:C(D)\to\mathbb{R}$ be a
bounded continuos functional. Then the following equality holds

$\mathbb{E}\left[e^{Z-\frac{\mathbb{E}[Z^{2}]}{2}}F\left(X(\cdot)\right)\right]=\mathbb{E}\left[F\left(X(\cdot)+\mathbb{E}[X(\cdot)Z]\right)\right]\,.$

Let us observe that Lemma C.1 is equivalent to the fact that under the
probability measure $e^{Z-\mathbb{E}[Z^{2}]/2}{\mathrm{d}}\mathbb{P}$ the
field $(X(x))_{x\in D}$ has the same law of the shifted field