# Subcritical Gaussian multiplicative chaos in the Wiener space: construction,
moments and volume decay

Rodrigo Bazaes111Universität Münster<EMAIL_ADDRESS>Isabel
Lammers222Universität Münster<EMAIL_ADDRESS>Chiranjib
Mukherjee333Universität Münster<EMAIL_ADDRESS>
††AMS Subject Classification: 60K35, 60G15, 60G57,60K37.††Keywords: Continuous
directed polymer, weak disorder, Gaussian multiplicative chaos, Liouville
quantum gravity, thick points, Girsanov transformation, moments, Hölder
exponents, small ball probability, multifractal spectrum

Universität Münster

November 16, 2022

> Abstract:  We construct and study properties of an infinite dimensional
> analog of Kahane’s theory of Gaussian multiplicative chaos [14]. Namely, we
> consider a random field defined with respect to space-time white noise
> integrated w.r.t. Brownian paths in $d\geq 3$ and construct the infinite
> volume limit of the normalized exponential of this field, weighted w.r.t.
> the Wiener measure, in the entire weak disorder (subcritical) regime.
> Moreover, we show that this infinite volume measure, which we call the
> subcritical GMC on the Wiener space, is unique w.r.t. the mollification
> scheme in the sense of Shamov [29] and determine its support by identifying
> its thick points. This in turn implies that almost surely, the subcritical
> GMC on the Wiener space is singular w.r.t. the Wiener measure. We also
> prove, in the subcritical regime, existence of negative and positive
> ($L^{p}$ for $p>1$) moments of the total mass of the limiting GMC, and
> deduce its Hölder exponents (small ball probabilities) explicitly. While the
> uniform Hölder exponent (the upper bound) and the pointwise scaling exponent
> (the lower bound) differ for a fixed disorder, we show that, as the disorder
> goes to zero, the two exponents agree, coinciding with the scaling exponent
> of the Wiener measure.

##  1\. Introduction

In this article, we construct and study properties of an infinite dimensional
analog of the Gaussian multiplicative chaos (GMC) measures, namely, the
measures

$\displaystyle\mu_{\gamma}({\rm d}\omega)=\lim_{T\to\infty}\mu_{\gamma,T}({\rm
d}\omega),\qquad\mbox{where}$ (1.1) $\displaystyle\mu_{\gamma,T}({\rm
d}\omega)=\exp\bigg{(}\gamma
H_{T}(\omega)-\frac{\gamma^{2}}{2}\mathbf{E}[H_{T}^{2}(\omega)]\bigg{)}\mathbb{P}_{0}({\rm
d}\omega).$

Here, $\mathbb{P}_{0}$ is the Wiener measure corresponding to Brownian paths
$\omega:[0,\infty)\to\mathbb{R}^{d}$, and the Gaussian process
$\\{H_{T}(\omega)\\}_{\omega}$, indexed by Brownian paths, is driven by a
Gaussian space-time white noise $\dot{B}$ (under the probability measure
$\mathbf{P}$) integrated w.r.t. the Brownian path:

$H_{T}(\omega)=\int_{\mathbb{R}^{d}}\int_{0}^{T}\phi(\omega_{s}-y)\dot{B}(s,y){\rm
d}s{\rm d}y,$ (1.2)

with $\phi$ being a normalized mollifier. Developing this framework for GMC
measures is quite natural and important in the field due to the relations to
the continuous directed polymers as well as the multiplicative noise
stochastic heat equation in $d\geq 3$ [23].

Our main goal is to develop an infinite dimensional analog of Kahane’s theory
[14] of GMC (see below) in this setup. For this purpose, the first task would
be to construct the infinite-volume measure (1.1). This is the content of our
first main result, stated in Theorem 2.1, where it is shown that, for any
$d\geq 3$ and in the entire uniform integrability (or weak disorder) regime
$\gamma\in(0,\gamma_{c})$, the infinite volume GMC measure

$\mu_{\gamma}({\rm d}\omega):=\lim_{T\to\infty}\mu_{\gamma,T}({\rm d}\omega)$
(1.3)

exists, is non-trivial and non-atomic $\mathbf{P}$-a.s. The limit above is
taken with respect to the topology of weak convergence. We then identify the
support of this limit; in Theorem 2.2 we will show that in this regime, for
almost every realization of $\dot{B}$ and for any Brownian path $\omega$
sampled according to the normalized limit $\widehat{\mu}_{\gamma}$, the value
of the field is atypically large – that is,

$H_{T}(\omega)\sim\gamma T(\phi\star\phi)(0)\qquad\mbox{a.s.
w.r.t.}\,\,\widehat{\mu}_{\gamma},\quad\mbox{and w.r.t.}\,\,\mathbf{P}.$ (1.4)

In other words, $\widehat{\mu}_{\gamma}$ is supported on high values of the
field $H_{T}(\omega)$ and every path $\omega$ is “$\gamma$-thick” w.r.t.
$\widehat{\mu}_{\gamma}$. Consequently, $\widehat{\mu}_{\gamma}$ is also
almost surely singular w.r.t. the Wiener measure $\mathbb{P}_{0}$.

We next investigate the universality of the limit (1.3) by determining the
role of the cut-off $\phi$. Following Kahane, one expects that a well defined
limiting object would not depend so much on the choice of the mollifier. In
this vein, denote by $P=\mathbf{P}\dot{B}^{-1}$ the law of the white noise (a
probability measure on tempered distributions
$\mathcal{S}^{\prime}(\mathbb{R}_{+}\times\mathbb{R}^{d})$). We show in
Theorem 2.5 that $\mu_{\gamma}$ is the unique measure such that the
distribution of $H_{T}$ under $\mu_{\gamma}({\rm d}\dot{B},{\rm
d}\omega)P({\rm d}\dot{B})$ is the same as the distribution of
$H_{T}+T(\phi\star\phi)(0)$ under $P\otimes\mathbb{P}_{0}$. That is, the only
way to perturb linearly the distribution $\dot{B}$ with the test function

$(s,y)\mapsto\phi(\omega_{s}-y)$

is by using the limiting GMC measure $\mu_{\gamma}$. This result is
reminiscent of Shamov’s characterization [29] of GMC for log-correlated
fields, see Remark 2.6.

We then deduce some salient properties of $\mu_{\gamma}$. In Theorem 2.7, we
prove the existence of positive and negative moments of its total mass in the
entire weak disorder regime:

$\displaystyle\mu_{\gamma}(\Omega)\in L^{p}(\mathbf{P})\hskip
5.69046pt\quad\mbox{for some }p>1,\quad\mbox{and}$
$\displaystyle\mu_{\gamma}(\Omega)\in L^{-q}(\mathbf{P})\quad\mbox{for
some}\,\,q>0.$

Moreover, for $\gamma$ even smaller (in the so-called $L^{2}$-regime), it
holds that $\mu_{\gamma}(\Omega)$ has negative moments of all order. Finally,
we deduce the volume decay or (uniform) Hölder exponents of the normalized GMC
measure $\widehat{\mu}_{\gamma}$ – namely, in Theorem 2.9 we show that, for
weak disorder, and $\mathbf{P}$-almost surely,

$\liminf_{\varepsilon\downarrow
0}\varepsilon^{2}\log\widehat{\mu}_{\gamma}\big{(}\|\omega\|<\varepsilon\big{)}\geq-
C_{2},\qquad\limsup_{\varepsilon\downarrow
0}\sup_{\eta\in\Omega_{0}}\varepsilon^{2}\log\widehat{\mu}_{\gamma}\big{(}\|\omega-\eta\|<\varepsilon\big{)}\leq-
C_{1}.$ (1.5)

Here $\Omega_{0}$ is a subset of the paths carrying a (weighted) norm
$\|\cdot\|$ that makes $\Omega_{0}$ a Banach space with
$\mathbb{P}_{0}(\Omega_{0})=1$, and $C_{1}$, $C_{2}$ are explicit constants
for weak disorder. While for a fixed $\gamma\in(0,\gamma_{c})$, the constants
$C_{1}$ and $C_{2}$ do not match (and, given their nature, they should not
match, as will be explained below), the bounds given in (1.5) agree in the
limit $\gamma\to 0$ and coincide with the scaling exponents of the Wiener
measure $\mathbb{P}_{0}$ as well. This will also be shown in Theorem 2.9. We
also remark that except for Theorem 2.2 and Theorem 2.5, the above results
also hold for continuous directed polymers in non-Gaussian environments with
exponential moments. We refer to Section 2.5 for the main ideas of proof.

In order to draw analogies, let us briefly recall Kahane’s theory of GMC for
log-correlated fields on finite dimensional spaces. Given a domain
$D\subset\mathbb{R}^{d}$, a GMC is a rigorously defined version of the
measures

$m_{\gamma,h}({\rm d}x)=\exp\big{[}\gamma
h(x)-\frac{\gamma^{2}}{2}\mathbf{E}[h^{2}(x)]\big{]}{\rm d}x.$

Here, $\\{h(x)\\}_{x\in D}$ is a log-correlated centered Gaussian field with
$\mathbf{E}[h(x)h(y)]=-\log|x-y|+O(1)$. The logarithmic divergence along the
diagonal prevents to define $h$ pointwise, and a regularization process
becomes necessary to define $h$, and consequently, $m_{\gamma,h}$, in a
precise sense. Since the work of Kahane [14], there have been very important
works in the field by Robert and Vargas [26], Duplantier and Sheffield [11],
Shamov [29] and by Berestycki [2], who showed that if
$(h_{\varepsilon})_{\varepsilon\in(0,1)}$ is a suitable approximation of $h$,
then as long as $\gamma\in(0,\sqrt{2d})$, which is known as the the uniform
integrability (or the subcritical) regime,

$\lim_{\varepsilon\to 0}m_{\gamma,h_{\varepsilon}}({\rm d}x)=m_{\gamma,h}({\rm
d}x)\quad\mbox{weakly and in $\mathbf{P}$-probability.}$

Moreover, $m_{\gamma,h}$ is non-trivial, non-atomic and a.e. $x\in D$ sampled
according to $m_{\gamma,h}$ is $\gamma$-thick (that is, for such $x$,
$h_{\varepsilon}(x)\sim\gamma\log\big{(}\frac{1}{\varepsilon}\big{)}$ a.s.
w.r.t. $\mathbf{P}$). Consequently, $m_{\gamma,h}$ is singular w.r.t. the
Lebesgue measure. We underline the analogies of these statements to (1.3) and
(1.4). There have been notable instances where, using the scale-invariance of
the logarithmic correlations, studying the positive and negative moments of
the total mass $m_{\gamma,h}(D)$ have been instrumental (see [12, 28] and [3,
Ch. 3.7-3.9]). Also, an important geometric behavior of $m_{\gamma,h}$ is
captured by the asymptotic behavior of the volume decay $\log
m_{\gamma,h}(B_{\varepsilon}(x))$ as $\varepsilon\downarrow 0$. This is known
as the (uniform) Hölder exponents of $m_{\gamma,h}$ and is closely related to
its multifractal behavior. In fact our bounds (1.5) for the infinite-volume
GMC $\mu_{\gamma}$ resemble a similar behavior of the GMC measures
$m_{\gamma,h}$. Indeed, recall that (see [27, Sec. 4.1] for precise
definitions), for $m_{\gamma,h}$, the uniform Hölder exponent is given by
$d(1-\frac{\gamma}{\sqrt{2d}})^{2}$, while its pointwise Hölder exponent is
$d+{\gamma^{2}}/2$. However, none of these exponents fully captures its
multifractal spectrum, which roughly says that if $x$ is $a$-thick, then

$m_{\gamma,h}(B_{\varepsilon}(x))\sim
C\varepsilon^{d+\frac{\gamma^{2}}{2}-a\gamma}.$

Thus, the pointwise (resp. uniform) Hölder scaling exponents are the extremal
values of the multifractal spectrum, and these therefore do not match for a
fixed $\gamma$ (but do so as $\gamma\to 0$). We refer to the discussion below
(1.5) again to underline the analogy to the Hölder exponents $C_{1},C_{2}$ in
our set up.

In summary, it is for the first time, to the best of our knowledge, that the
existence and the properties of $\mu_{\gamma}$ in (1.3) have been established
in the infinite-dimensional set up, in the entire sub-critical regime. While
GMC for log-correlated fields are not formally related to the fields studied
here, we find the point of view of the former quite appealing for studying the
latter. By now there is also a rich theory of log-correlated GMC in the
critical [10, 25] and supercritical regime [22, 6]. With this background, we
expect the present results and the GMC point of view to be useful for further
applications to the continuous directed polymer in $d\geq 3$.

###  1.1. Setup and notation.

For a fixed dimension $d\geq 1$, let
$\Omega:={C}\left([0,\infty),\mathbb{R}^{d}\right)$ be the space of continuous
functions from $[0,\infty)$ to $\mathbb{R}^{d}$, endowed with the topology of
uniform convergence on compact sets. We equip this space with the Wiener
measure denoted by $\mathbb{P}_{0}$, so a typical path
$\omega=(\omega_{s})_{s\in[0,\infty)}\in\Omega$ corresponds to a realization
of a $\mathbb{R}^{d}$-valued Brownian motion starting at $0$. Similarly, we
denote by $\mathbb{P}_{x}$ the Wiener measure corresponding to a Brownian
motion starting at $x\in\mathbb{R}^{d}$. Let $(\mathcal{E},\mathcal{F},\bf P)$
be a complete probability space, so that $\dot{B}$ is a space-time white noise
which is independent of the Wiener measure. More precisely, denote by
$\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ the space of rapidly
decaying Schwartz functions on $\mathbb{R}_{+}\times\mathbb{R}^{d}$. Then
$\dot{B}={\dot{B}}(f)_{f\in\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d})}$
is a Gaussian process with mean zero and covariance

$\displaystyle\mathbf{E}\big{[}{\dot{B}}(f){\dot{B}}(g)\big{]}$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(s,y)g(s,y)\,{\rm
d}y\,{\rm d}s$ (1.6) $\displaystyle=\langle
f,g\rangle_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})},\quad
f,g\in\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d}).$

Here, we use the notation

${\dot{B}}(f)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(s,y){\dot{B}}(s,y)\,{\rm
d}y\,{\rm d}s\qquad\mbox{for any}\quad
f\in\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d}).$

We can extend the integral to $f\in L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$
via approximation. Indeed, if
$(f_{n})_{n\in\mathbb{N}}\subset\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d})$
approximates $f$ in $L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ (such a
sequence exists since $\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ is
dense in $L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$), then the sequence
$({\dot{B}}(f_{n}))_{n}$ is Cauchy in $L^{2}(\bf P)$, since we have

$\mathbf{E}\big{[}({\dot{B}}(f_{n})-{\dot{B}}(f_{m}))^{2}\big{]}=\mathbf{E}\big{[}{\dot{B}}(f_{n}-f_{m})^{2}\big{]}=\|f_{n}-f_{m}\|_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})}.$

Therefore, for every $f\in L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$, we can
define the $L^{2}(\bf P)$-limit

$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(s,y){\dot{B}}(s,y)\,{\rm d}y\,{\rm
d}s:=\lim_{n\to\infty}{\dot{B}}(f_{n}).$

For $f\in L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$ and $T>0$, we will write

$\int_{0}^{T}\int_{\mathbb{R}^{d}}f(s,y){\dot{B}}(s,y)\,{\rm d}y\,{\rm
d}s:=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{([0,T]\times\mathbb{R}^{d})}(s,y)f(s,y){\dot{B}}(s,y)\,{\rm
d}y\,{\rm d}s.$

By construction, for $f\in L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$, the
random variable
$\int_{0}^{\infty}\int_{\mathbb{R}^{d}}f(s,y){\dot{B}}(s,y)\,{\rm d}y\,{\rm
d}s$ is Gaussian distributed with mean zero and variance
$\|f\|_{L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})}$ and for $f,g\in
L^{2}(\mathbb{R}_{+}\times\mathbb{R}^{d})$, the covariance
$\mathbf{E}[\dot{B}(f)\dot{B}(g)]$ is also given by (1.6). We also define the
family of space-time shifts $\\{\theta_{t,x}:t>0,x\in\mathbb{R}^{d}\\}$ acting
on the white-noise environment (i.e.,
$\theta_{t,x}(\dot{B}(s,y))=\dot{B}(t+s,x+y)$), and remind the reader that
$\dot{B}$ is stationary under this action.

Next, let $\phi$ be a mollifier – that is, $\phi$ is a smooth, non-negative,
spherically symmetric and compactly supported function
$\phi\colon\mathbb{R}^{d}\to\mathbb{R}$ such that
$\int_{\mathbb{R}^{d}}\phi(x){\rm d}x=1$. We define the Gaussian field
$\left({H}_{T}(\omega)\right)_{\omega\in\Omega}$ as the Itô integral

${H}_{T}(\omega):=\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi(\omega_{s}-y){\dot{B}}(s,y)\,{\rm
d}y\,{\rm d}s,\qquad\omega\in\Omega.$ (1.7)

In particular, $({H}_{T}(\omega))_{\omega\in\Omega}$ is also a Gaussian
process with mean zero and covariance

$\mathbf{E}\left[{H}_{T}(\omega){H}_{T}(\omega^{\prime})\right]=\int_{0}^{T}\int_{\mathbb{R}^{d}}\phi\left(\omega_{s}-y\right)\phi\left(\omega^{\prime}_{s}-y\right)\,{\rm
d}y\,{\rm
d}s=\int_{0}^{T}(\phi\star\phi)\left(\omega_{s}-\omega^{\prime}_{s}\right){\rm
d}s,$ (1.8)

where $(f\star g)(x)=\int_{\mathbb{R}^{d}}f(x-y)g(y){\rm d}y$. Note that, for
any $\omega\in\Omega$, $\mathrm{Var}[H_{T}(\omega)]=T(\phi\star\phi)(0)$ –
that is, for $T\gg 1$ large, the covariance of the field
$(H_{T}(\omega))_{\omega\in\Omega}$ diverges along the diagonal.

Given any $\gamma>0$ and $T>0$, we define the random measure on the path space
$\Omega$

$\displaystyle\mu_{\gamma,T}({\rm d}\omega)$
$\displaystyle:=\exp\bigg{(}\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}\mathrm{Var}(H_{T}(\omega))\bigg{)}\mathbb{P}_{0}({\rm
d}\omega)$ (1.9)
$\displaystyle=\exp\bigg{(}\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\bigg{)}\mathbb{P}_{0}({\rm
d}\omega),$

and in the sequel, we will refer to $\mu_{\gamma,T}$ as the finite-volume
Gaussian multiplicative chaos on $\Omega$ or the continuous directed polymer.
Also, we will write

$\widehat{\mu}_{\gamma,T}(A):=\frac{\mu_{\gamma,T}(A)}{\mu_{\gamma,T}(\Omega)},\qquad
A\subset\Omega$ (1.10)

for its normalized counterpart. Let $\mathcal{F}_{T}$ be the $\sigma$-algebra
generated by the noise $\dot{B}$ until time $T$. Since $\dot{B}$ is smoothened
only in space, the total mass, or the partition function,

$\mu_{\gamma,T}(\Omega)=\int_{\Omega}\exp\bigg{(}\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\bigg{)}\mathbb{P}_{0}({\rm
d}\omega)$

is adapted to the filtration $(\mathcal{F}_{T})_{T}$ and is a martingale. Our
main results will hold in the uniform integrability or the entire weak-
disorder phase $\gamma\in(0,\gamma_{c})$ of the martingale
$(\mu_{\gamma,T}(\Omega))_{T}$ determined by the critical value
$\gamma_{c}\in(0,\infty)$ for $d\geq 3$ defined in (3.3). We are now ready to
state our main results.

##  2\. Main results.

###  2.1. Infinite volume GMC on $\Omega$: existence, support and singularity.

Our first main result provides the existence of the infinite-volume measure
$\lim_{T\to\infty}\mu_{\gamma,T}(\cdot)$ on the path space.

######  Theorem 2.1 (Existence of the GMC on Wiener space).

Fix $d\geq 3$ and $\gamma\in(0,\gamma_{c})$. Then there exists a non-trivial
measure $\mu_{\gamma}$ on $\Omega$ such that $\mathbf{P}$-a.s.,
$\mu_{\gamma,T}$ converges weakly to $\mu_{\gamma}$ as $T\to\infty$. Moreover,
for any fixed Borel set $A\subset\Omega$ with $\mathbb{P}_{0}(A)>0$,
$\mu_{\gamma}(A)>0$ $\mathbf{P}$-a.s. Similarly, there exists a probability
measure $\widehat{\mu}_{\gamma}$ on $\Omega$ such that $\mathbf{P}$-a.s., the
normalized approximations $\widehat{\mu}_{\gamma,T}$ converge weakly to
$\widehat{\mu}_{\gamma}$. Finally, the same conclusion holds if the noise
$\dot{B}$ is replaced by any other random environment with finite exponential
moments.

The next main result identifies the support of the limiting measure
$\widehat{\mu}_{\gamma}$ in $\Omega$. For any fixed realization of $\dot{B}$
and $a>0$, let

$\mathcal{T}_{a}:=\bigg{\\{}\omega\in\Omega:\lim_{T\to\infty}\frac{H_{T}(\omega)}{T(\phi\star\phi)(0)}=a\bigg{\\}}$

be the (random) set of $a$-thick paths $\omega\in\Omega$. The next result
shows that, for any $\gamma\in(0,\gamma_{c})$, the infinite volume GMC
$\mu_{\gamma}$ is supported only on $\gamma$-thick paths $\omega\in\Omega$:

######  Theorem 2.2 (Thick points and singularity of GMC).

Fix $d\geq 3$ and $\gamma\in(0,\gamma_{c})$. Let $\widehat{\mu}_{\gamma}$ be
the measure defined in Theorem 2.1. Then we have

$\widehat{\mu}_{\gamma}(\mathcal{T}_{\gamma})=1\qquad\bf P\text{-}{\rm a.s.}$
(2.1)

In particular,

$\widehat{\mu}_{\gamma}(\mathcal{T}_{\gamma}^{c})=0\qquad\mathbf{P}\text{-}{\rm
a.s.,}$

and consequently, for $\mathbf{P}$-a.e. realization of $\dot{B}$,
$\widehat{\mu}_{\gamma}$ is singular w.r.t. the Wiener measure
$\mathbb{P}_{0}$.

###  2.2. Uniqueness.

We turn to the issue of determining to what extent the limiting measure
$\mu_{\gamma}$ depends on the choice of the mollification $\phi$. To emphasize
this dependence, we will write $\mu_{\gamma,\phi,T}=\mu_{\gamma,T}$ for the
approximating measure defined in (1.9) and
$\mu_{\gamma,\phi}=\lim_{T\to\infty}\mu_{\gamma,\phi,T}$ for its infinite
volume limit constructed in Theorem 2.1. Before addressing the aforementioned
issue, let us note the following immediate consequence of Theorem 2.2.

######  Corollary 2.3.

Let $\phi$ and $\phi^{\prime}$ be two mollifiers. Then the limiting measures
$\mu_{\gamma,\phi}$ and $\mu_{\gamma,\phi^{\prime}}$ are singular unless
$(\phi\star\phi)(0)=(\phi^{\prime}\star\phi^{\prime})(0)$.

The next result shows that, even if
$(\phi\star\phi)(0)=(\phi^{\prime}\star\phi^{\prime})(0)$, the two measures
$\mu_{\gamma,\phi}$ and $\mu_{\gamma,\phi^{\prime}}$ are different unless
$\phi=\phi^{\prime}$.

######  Proposition 2.4.

Let $\phi$ and $\phi^{\prime}$ be two mollifiers. If
$\phi(\cdot)\not\equiv\phi^{\prime}(\cdot)$, then
$\mu_{\gamma,\phi}\neq\mu_{\gamma,\phi^{\prime}}$.

While the limiting measures may not be exactly the same for different
mollifiers, the following result identifies the limit $\mu_{\gamma,\phi}$ in a
unique manner and determines to what extent it depends on the mollification
scheme $\phi$. We introduce the following notation. Let
$P=\mathbf{P}\circ\dot{B}^{-1}$ be the law of the white noise. For a measure
$\nu$ on $\Omega$, which may depend on $\dot{B}$, set

$\mathbb{Q}_{\nu}({\rm d}\dot{B},{\rm d}\omega):=\nu({\rm
d}\omega,\dot{B})P({\rm d}\dot{B}).$ (2.2)

######  Theorem 2.5 (Uniqueness).

For a fixed $\gamma>0$ and $\phi$ as above, the (unnormalized) Liouville
measure $\mu_{\gamma,\phi}$ is the unique measure such that the law of
${\dot{B}}$ under $\mathbb{Q}_{\mu_{\gamma,\phi}}$ is the same as the law of
the (Schwartz) distribution

${{\dot{B}}}_{\phi}(f)={\dot{B}}(f)+\gamma\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}f(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y,\qquad
f\in\mathcal{S}^{\prime}(\mathbb{R}_{+}\times\mathbb{R}^{d})$

under $P\otimes\mathbb{P}_{0}$. In other words, $\mu_{\gamma,\phi}$ is the
unique measure satisfying

$\mathbb{E}^{P}\bigg{[}\int_{\Omega}\mu_{\gamma,\phi}({\rm
d}\omega)F(\dot{B},\omega)\bigg{]}=\mathbb{E}^{P\times\mathbb{P}_{0}}\big{[}F(\dot{B}_{\phi},\omega)\big{]}.$
(2.3)

for any bounded measurable function $F:\Omega\times\mathcal{E}\to\mathbb{R}$.

######  Remark 2.6.

The theorem above asserts that the only way to perturb linearly the
distribution $\dot{B}$ with the test function

$(s,y)\mapsto\phi(\omega_{s}-y)$

is by using the measure $\mu_{\gamma,\phi}$. In particular,
$\mu_{\gamma,\phi}$ is the unique measure such that the distribution of
$H_{T}$ under $\mathbb{Q}_{\mu_{\gamma,\phi}}$ is the same as the distribution
of $H_{T}+T(\phi\star\phi)(0)$ under $P\otimes\mathbb{P}_{0}$. Note that the
uniqueness of $\mu_{\gamma}$ provided by Theorem 2.5 is reminiscent of
Shamov’s definition of a subcritical GMC [29]. Under this setup, a GMC over
the (generalized) Gaussian field $H$ is a random measure $\mu=\mu_{H}$
(measurable w.r.t. $H$) on a measure space $(X,\mathcal{G})$ if for all
deterministic $v:X\to\mathbb{R}$ such that the law of $H+v$ is absolutely
continuous w.r.t. that of $H$, then $\mu_{H+v}(dx)={\rm e}^{v(x)}\mu_{H}(dx)$.
The construction of such measure goes via approximating $H$ by a sequence of
fields $(H_{n})$ such that $H_{n}\to H$ in a suitable sense. A difference to
the present infinite dimensional setup is that the limiting field does depend
on the mollifier $\phi$, since for $\omega\neq\omega^{\prime}\in\Omega$,

$\lim_{T\to\infty}\mathbf{E}[H_{T}(\omega)H_{T}(\omega^{\prime})]=\int_{0}^{\infty}(\phi\star\phi)(\omega_{s}-\omega^{\prime}_{s}){\rm
d}s<\infty\qquad\mathbb{P}^{\otimes 2}\text{-a.s.}$

###  2.3. Moments of the GMC $\mu_{\gamma}$.

By the martingale convergence theorem, we always have
$\mathbf{E}[\mu_{\gamma}(\Omega)]=1$ for all $\gamma\in(0,\gamma_{c})$.
Moreover, we also have

######  Theorem 2.7 (Positive and negative moment).

Fix $d\geq 3$ and $\gamma\in(0,\gamma_{c})$.

  1. (i)

There is some $p>1$ such that

$\displaystyle\mu_{\gamma}(\Omega)\in L^{p}(\mathbf{P}).$ (2.4)

  2. (ii)

There is some $q>0$ such that

$\displaystyle\mu_{\gamma}(\Omega)\in L^{-q}(\mathbf{P}).$ (2.5)

Moreover, if $\gamma\in(0,\gamma_{c})$ is chosen so that the martingale
$(\mu_{\gamma,T}(\Omega))_{T}$ is bounded in $L^{2}(\mathbf{P})$, then

$\mu_{\gamma}(\Omega)\in L^{-q}(\mathbf{P})\qquad\forall q\in(0,\infty).$
(2.6)

######  Remark 2.8.

We can show (2.4) and (2.5) exactly in the same manner if $\dot{B}$ is
replaced by a random environment with finite exponential moments. In [8], it
was shown (using Talagrand’s concentration inequality and for Gaussian random
environment) that in the “$L^{2}$ region” (i.e., when
$\gamma\in(0,\gamma_{c})$ is restricted so that the martingale
$(\mu_{\gamma,T}(\Omega))_{T}$ remains $L^{2}(\mathbf{P})$ bounded) and for
all $q\in(-\infty,0)$, $(\mu_{\gamma,T}(\Omega))_{T}$ is $L^{q}(\mathbf{P})$
bounded. It remains an open problem to extend (2.6) in the full weak disorder
regime $\gamma\in(0,\gamma_{c})$. For negative moments of the total mass of
finite dimensional GMC (defined w.r.t. log-correlated or scale-invariant
fields), we refer to [26, 12].

###  2.4. Volume decay of $\widehat{\mu}_{\gamma}$.

We now turn to the volume decay rate of GMC-balls, i.e., we determine
asymptotic behavior of the quantities

$\log\widehat{\mu}_{\gamma}(\omega\in\Omega\colon\|\omega\|<\varepsilon)\qquad\mbox{as}\,\,\varepsilon\downarrow
0.$ (2.7)

Note that $\Omega=C([0,\infty);\mathbb{R}^{d})$ is not a Banach space. To
define a suitable norm, we will consider a subset $\Omega_{0}\subset\Omega$
with a norm $\|\cdot\|$ such that $\mathbb{P}_{0}(\Omega_{0})=1$ and that
$(\Omega_{0},\|\cdot\|)$ is a Banach space. For this purpose, we consider a
class of maps $g:(0,\infty)\mapsto(0,\infty]$ satisfying

$\left(\inf_{0<t<\infty}g(t)\right)\wedge\left(\inf_{0<t<\infty}\frac{g(t)}{t}\right)>0.$
(2.8)

For a fixed $g$ as above, we set

$\Omega_{0}=\Omega_{0}(g):=\left\\{\omega\in\Omega:\lim_{t\to\infty}\frac{|\omega_{t}|}{g(t)}=0\right\\}.$
(2.9)

Since
$\mathbb{P}_{0}\left(\lim_{t\to\infty}\frac{|\omega_{t}|}{t}=0\right)=1$,
condition (2.8) assures that $\mathbb{P}_{0}(\Omega_{0})=1$. Moreover, under
the norm $\|\cdot\|$, defined as

$\|\omega\|:=\sup_{t>0}\frac{|\omega_{t}|}{g(t)}\qquad(\text{finite for all
}\omega\in\Omega_{0}\text{ by }\eqref{eq:g-cond}),$ (2.10)

$\Omega_{0}$ is a separable Banach space. From now on, we consider an
arbitrary $g$ satisfying (2.8).

The next theorem consists of two parts. In the first one, we will determine
the decay exponents (upper and lower estimates) for (2.7) valid in the whole
weak disorder region $\gamma\in(0,\gamma_{c})$. In the second one, we will
show that, as $\gamma\to 0$, the upper and lower estimates coincide.

######  Theorem 2.9 (Volume decay).

Fix $d\geq 3$.

  1. (i)

Given $\gamma\in(0,\gamma_{c})$ and $g$ satisfying (2.8), there exists
$r_{0}>0$ such that for all $r\in(0,r_{0})$, there are explicit constants
$0<C_{1}\leq C_{2}<\infty$ (defined in (7.12) and (7.18)) fulfilling

$\displaystyle-C_{2}$ $\displaystyle\leq\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\widehat{\mu}_{\gamma}(\|\omega\|<r\varepsilon)$ (2.11)
$\displaystyle\leq\limsup_{\varepsilon\to
0}\varepsilon^{2}\sup_{\eta\in\Omega_{0}}\log\widehat{\mu}_{\gamma}(\|\omega-\eta\|<r\varepsilon)\leq-
C_{1}.$

Moreover, if $\gamma>0$ is small enough, there are $p_{0}>1$ and $q_{0}>0$
such that $C_{1},C_{2}$ can be chosen as

$\displaystyle C_{1}$
$\displaystyle:=\frac{(p_{0}-1)}{4p_{0}r^{2}}j^{2}_{\frac{d-2}{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t-\frac{(p_{0}+1)}{2(p_{0}-1)}\gamma^{2}\phi\star\phi(0),$ (2.12)
$\displaystyle C_{2}$
$\displaystyle:=\left(\frac{q_{0}+1}{q_{0}}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)\frac{q_{0}^{2}+3q_{0}+1}{(q_{0}+1)^{2}}\right),$

where $j_{\frac{d-2}{2}}$ is the smallest positive root of the Bessel function
$J_{\frac{d-2}{2}}$.

  2. (ii)

For every $\delta>0$, there exists $0<\gamma_{\delta}$ such that for all
$0<\gamma<\gamma_{\delta}$, the constants $C_{1},C_{2}$ from part
$\mathrm{(i)}$ can be chosen so that

$C_{2}-C_{1}<\delta.$

######  Remark 2.10.

We first remark that instead of considering weighted norms on $\Omega_{0}$, it
is also possible to obtain analogous estimates as in Theorem 2.9 on $\Omega$
by considering the sets $\\{\omega\in\Omega:\sup_{0\leq t\leq
r}|\omega_{t}|<\varepsilon\\}$ as $\varepsilon\to 0$. Also, we note that the
upper bound in (2.11) is uniform over shifted balls of a given radius, while
the lower bound holds pointwise. In fact, a uniform lower bound over
$\eta\in\Omega_{0}$ in Theorem 2.9 is not expected. Indeed, already for the
Wiener measure $\mathbb{P}_{0}$ (corresponding to the case $\gamma=0$), for
every function in the Cameron-Martin space, defined as the Hilbert space

$H^{1}:=\big{\\{}\eta\in\Omega_{0}:\dot{\eta}\in
L^{2}([0,\infty);\mathbb{R}^{d})\big{\\}}$

with inner product induced by the norm
$\|\eta\|_{H^{1}}:=\|\dot{\eta}\|_{L^{2}([0,\infty);\mathbb{R}^{d})}$, we have

$\mathbb{P}_{0}(\|\omega-\eta\|<r)\geq\mathbb{P}_{0}(\|\omega\|<r){\rm
e}^{-\frac{\|\eta\|^{2}_{H^{1}}}{2}}.$

We refer to (B.2) in Theorem B.1.

###  2.5. Key ideas of the proofs.

We briefly outline the key ideas of the proof of our results. For the proof of
Theorem 2.1, we have drawn inspiration from the simple and very elegant
approach of Berestycki [2] for the construction of GMC for log-correlated
fields on finite dimensional spaces. His route goes over a $L^{2}$ computation
to prove convergence in the $L^{2}$ region, then using Girsanov’s formula to
determine thick points (and show that every Liouville point $x\in D$ is
$\gamma$-thick in the uniform integrability threshold), and finally removing
points that are thicker than this prescribed value to show convergence of the
measure in the entire uniform integrability region. We approach the uniform
integrability phase directly via the martingale properties of
$(\mu_{\gamma,T}(\Omega))_{T\geq 0}$ to show that for a fixed Borel set
$A\subset\Omega$, $\lim_{T\to\infty}\mu_{\gamma,T}(A)$ exists
$\mathbf{P}$-a.s. and in $L^{1}(\mathbf{P})$. Together with a suitable
tightness criteria valid in the Wiener space (see Lemma 4.3), we will deduce
the weak convergence of $\mu_{\gamma,T}$ and also its normalization
$\widehat{\mu}_{\gamma,T}$. It might be worth pointing out that this approach
is robust in the sense that it does not rely on the $L^{2}$ computation, and
it does not need Gaussianity of the environment for the existence of the limit
in Theorem 2.1. On the other hand, the proofs of Theorems 2.2 and 2.5 (which
from our point of view are quite important and of independent interest) do
require the Cameron-Martin-Girsanov theorem (and therefore, the Gaussianity of
the environment is necessary).

To show Theorem 2.7, we take ideas from the model of discrete directed
polymers in $\mathbb{Z}^{d}$. For bounded environments, it has been shown very
recently [13] that the polymer martingale is $L^{p}$ bounded in weak disorder,
for some $p>1$ and in $L^{-q}$ for some $q>0$. We adapt this approach in the
continuum, without requiring any boundedness assumption on the environment and
building on the aforementioned martingale properties in the continuum. This
argument is also robust and can be easily adapted to continuous directed
polymers in environments that are not necessarily Gaussian.

Finally, to show Theorem 2.9, we use our earlier result [5] as a guiding
philosophy for treating finite-volume GMC $\widehat{\mu}_{\gamma,T}$. However,
the Hölder exponents of the infinite-volume limit $\widehat{\mu}_{\gamma}$ is
more subtle. For this purpose, we need to additionally exploit the
aforementioned moment estimates – the $L^{p}$ moment for $p>1$ of
$\mu_{\gamma}(\Omega)$ is used for the upper bound of Theorem 2.9, while the
negative moments are needed for the corresponding lower bound. Using
stationarity of the white noise and and a suitable application of the ergodic
theorem, we then derive Theorem 2.9.

Organization of the article: The rest of the article is organized as follows.
Sections 3 and 4 constitute the proof of Theorem 2.1. Theorems 2.2 and 2.5
will be shown in Section 5. The moment estimates in Theorem 2.7 are shown in
Section 6, while the Hölder exponents of the GMC measures will be derived in
Section 7.

##  3\. Martingale arguments.

######  Lemma 3.1.

Let $(\mathcal{F}_{T})$ be the $\sigma$-algebra generated by the noise up to
time $T$. Then the following statements hold with respect to the filtration
$(\mathcal{F}_{T})_{T>0}$:

  * (a)

For every $\omega\in\Omega$, the process $H(\omega)=({H}_{T}(\omega))_{T>0}$
is a martingale. Its quadratic variation is given by

$\langle
H(\omega)\rangle_{T}=\mathbf{E}[{H}_{T}(\omega)^{2}]=\int_{0}^{T}(\phi\star\phi)(0){\rm
d}s=T(\phi\star\phi)(0).$

  * (b)

For every $\omega\in\Omega$ and $\gamma>0$, the process
$\left(\exp\left\\{\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}\right)_{T>0}$
is a martingale.

  * (c)

For every Borel set $A\subset\Omega$ and $\gamma>0$,

$\left(\int_{A}\exp\left\\{\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}{\rm
d}\mathbb{P}_{0}(\omega)\right)_{T>0}$

is a martingale.

###### Proof.

Parts (a) and (b) are well-known. The statement in (c) follows by the lemma
below together with part (b). ∎

######  Lemma 3.2.

Let $X\in L^{1}(\mathbf{P}\otimes\mathbb{P}_{0})$ and
$\mathcal{G}\subset\mathcal{F}$ be a $\sigma$-algebra. Then

$\mathbf{E}\bigg{[}\,\int_{\Omega}X(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\,\bigg{|}\,\mathcal{G}\,\bigg{]}=\int_{\Omega}\mathbf{E}\big{[}X(\omega,\cdot)\,|\,\mathcal{G}\big{]}{\rm
d}\mathbb{P}_{0}(\omega).$ (3.1)

###### Proof.

Let

$\mathcal{H}{:=}\bigg{\\{}X\in
L^{1}(\mathbf{P}\otimes\mathbb{P}_{0})\,:\,\text{equation (\ref{eq}) is
satisfied for any $\sigma$-algebra $\mathcal{G}\subset\mathcal{F}$
}\,\bigg{\\}}.$

Step 1: We will first show that $\mathcal{H}$ is a monotone class:

  * -

Let $X,Y\in\mathcal{H}$, and $a>0$. Then

$\displaystyle\mathbf{E}\left[\ \int_{\Omega}\alpha
X(\omega,\cdot)+Y(\omega,\cdot){\rm d}\mathbb{P}_{0}(\omega)\ \middle|\
\mathcal{G}\ \right]$ $\displaystyle=$
$\displaystyle\alpha\mathbf{E}\left[\,\int_{\Omega}X(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\,|\,\mathcal{G}\,\right]+\mathbf{E}\left[\,\int_{\Omega}Y(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\,|\,\mathcal{G}\,\right]$ $\displaystyle=$
$\displaystyle\alpha\int_{\Omega}\mathbf{E}\left[X(\omega,\cdot)\,|\,\mathcal{G}\right]{\rm
d}\mathbb{P}_{0}(\omega)+\int_{\Omega}\mathbf{E}\left[Y(\omega,\cdot)\,|\,\mathcal{G}\right]{\rm
d}\mathbb{P}_{0}(\omega)$ $\displaystyle=$
$\displaystyle\int_{\Omega}\mathbf{E}\left[\alpha
X(\omega,\cdot)+Y(\omega,\cdot)\,|\,\mathcal{G}\right]{\rm
d}\mathbb{P}_{0}(\omega),$

hence $\alpha X+Y\in\mathcal{H}$.

  * -

Let $X_{1},X_{2},\ldots\in\mathcal{H}$ be monotonically increasing, i.e. for
every $n\in\mathbb{N}$ and every $\omega\in\Omega$, we have
$X_{n}(\omega,\cdot)\leq X_{n+1}(\omega,\cdot)$. Then for every
$\zeta\in\mathcal{E}$, by monotone convergence

$\int_{\Omega}\lim_{n\to\infty}X_{n}(\omega,\zeta){\rm
d}\mathbb{P}_{0}(\omega)=\lim_{n\to\infty}\int_{\Omega}X_{n}(\omega,\zeta){\rm
d}\mathbb{P}_{0}(\omega).$

On the other hand, for every $\omega\in\Omega$, by monotone convergence of the
conditional expectation, we have

$\mathbf{E}\left[\,\lim_{n\to\infty}X_{n}(\omega,\cdot)\,\middle|\,\mathcal{G}\,\right]=\lim_{n\to\infty}\mathbf{E}\left[\,X_{n}(\omega,\cdot)\,\middle|\,\mathcal{G}\,\right].$

These two observations and further usage of monotone convergence plus the fact
that $X_{n}\in\mathcal{H}$ for all $n\in\mathbb{N}$ yield

$\displaystyle\mathbf{E}\left[\,\int_{\Omega}\lim_{n\to\infty}X_{n}(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\,\middle|\,\mathcal{G}\,\right]$
$\displaystyle=\lim_{n\to\infty}\mathbf{E}\left[\,\int_{\Omega}X_{n}(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\ \middle|\,\mathcal{G}\,\right]$
$\displaystyle=\lim_{n\to\infty}\int_{\Omega}\mathbf{E}\left[\,X_{n}(\omega,\cdot)\,\middle|\,\mathcal{G}\,\right]{\rm
d}\mathbb{P}_{0}(\omega)$
$\displaystyle=\int_{\Omega}\mathbf{E}\left[\,\lim_{n\to\infty}X_{n}(\omega,\cdot)\,\middle|\,\mathcal{G}\,\right]{\rm
d}\mathbb{P}_{0}(\omega).$

Hence, $\mathcal{H}$ is a monotone class.

Step 2: Next, we will show that $\mathcal{H}$ contains all indicator functions
of the form
${\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}_{A\times
B}$ for $A\in\mathcal{B}$ and $B\in\mathcal{F}$. To that end, fix such sets
$A$ and $B$. Then

$\mathbf{E}\left[\,\int_{\Omega}{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{A\times
B}(\omega,\cdot){\rm
d}\mathbb{P}_{0}(\omega)\,|\,\mathcal{G}\,\right]=\mathbf{E}\left[\,\mathbb{P}_{0}(A){{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{B}(\cdot)\,|\,\mathcal{G}\,\right]=\mathbb{P}_{0}(A)\mathbf{P}(B\,|\,\mathcal{G}).$

Similarly and by using independence, we obtain

$\int_{\Omega}\mathbf{E}\left[\,{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{A\times
B}(\omega,\cdot)\,|\,\mathcal{G}\,\right]{\rm
d}\mathbb{P}_{0}(\omega)=\int_{\Omega}{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{A}(\omega)\mathbf{E}\left[\,{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{B}(\cdot)\,|\,\mathcal{G}\,\right]{\rm
d}\mathbb{P}_{0}(\omega)\mathbb{P}_{0}(A)\mathbf{P}(B\,|\,\mathcal{G}).$

Therefore,
${{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{A\times
B}\in\mathcal{H}$.

Step 3: The same conclusion as Step 2 is true for the indicator function of
$(A\times B)^{c}=(A^{c}\times B^{c})\dot{\cup}(A^{c}\times
B)\dot{\cup}(A\times B^{c})$ (here, $\dot{\cup}$ means that the union is
disjoint). By the monotone class argument, it follows that $\mathcal{H}$
contains all positive measurable functions and by linearity, also all
$L^{1}(\mathbf{P}\otimes\mathbb{P}_{0})$-functions. ∎

###  3.1. Uniform integrability.

######  Lemma 3.3.

If for some $\gamma>0$, $(\mu_{\gamma,T}(\Omega))_{T>0}$ is uniformly
integrable, then the same holds for all $\gamma^{\prime}<\gamma$.

###### Proof.

Fix $\gamma>0$ such that $(\mu_{\gamma,T}(\Omega))_{T\geq 0}$ is uniformly
integrable. Let $\dot{B},\dot{B}^{\prime}$ be independent copies of the noise,
and let $\gamma^{\prime}<\gamma$, so that $\gamma^{\prime}=c\gamma$ for some
$0<c<1$. To avoid ambiguities, we write
$\mu_{\gamma,T}(\Omega)=\mu_{\gamma,T}(\Omega,\dot{B})$. Note that

$\mu_{\gamma^{\prime},T}(\Omega,\dot{B})=\mu_{c\gamma,T}(\Omega,\dot{B})=\mathbf{E}\left[\mu_{\gamma,T}(\Omega,c\dot{B}+\sqrt{1-c^{2}}\dot{B}^{\prime})|\dot{B}\right].$
(3.2)

Since $(\mu_{\gamma,T}(\Omega,\dot{B}))_{T\geq 0}$ is uniformly integrable,
then by de La Vallée-Poussin’s theorem, there exists a function
$f:[0,\infty)\to[0,\infty)$ convex and increasing such that

$\lim_{x\to\infty}\frac{f(x)}{x}=\infty,\qquad\mbox{and}\qquad\sup_{T}\mathbf{E}[f(\mu_{\gamma,T}(\Omega,\dot{B}))]<\infty.$

By (3.2) and Jensen’s inequality,

$\displaystyle\mathbf{E}[f(\mu_{\gamma^{\prime},T}(\Omega,\dot{B}))]$
$\displaystyle=\mathbf{E}\left[f\left(\mathbf{E}\left[\mu_{\gamma,T}(\Omega,c\dot{B}+\sqrt{1-c^{2}}\dot{B}^{\prime})|\dot{B}\right]\right)\right]$
$\displaystyle\leq\mathbf{E}\left[f\left(\mu_{\gamma,T}(\Omega,c\dot{B}+\sqrt{1-c^{2}}\dot{B}^{\prime})\right)\right]$
$\displaystyle=\mathbf{E}[f(\mu_{\gamma,T}(\Omega,\dot{B}))].$

It follows that
$\sup_{T}\mathbf{E}[f(\mu_{\gamma,T}(\Omega,\dot{B}))]<\infty$, which in turn
implies the uniform integrability of
$(\mu_{\gamma^{\prime},T}(\Omega,\dot{B}))_{T\geq 0}$. ∎

The last lemma allows us to define the critical parameter

$\gamma_{c}=\gamma_{c}(d)=\sup\bigg{\\{}\gamma:\mbox{the martingale
}\mu_{\gamma,T}(\Omega)\mbox{ is uniformly integrable}\bigg{\\}}.$ (3.3)

In particular, if $\gamma<\gamma_{c}$, then $(\mu_{\gamma,T}(\Omega))_{T\geq
0}$ is uniformly integrable, while for $\gamma>\gamma_{c}$,
$(\mu_{\gamma,T}(\Omega))_{T\geq 0}$ is not uniformly integrable. Since we
will be dealing also with the normalized probability measure
$\widehat{\mu}_{\gamma,T}$ from (1.9), we need to know whether the denominator
vanishes or not at the limit. The next lemma tell us that in the uniform
integrable phase, the limit $\lim_{T\to\infty}\mu_{\gamma,T}(\Omega)$ is
positive $\bf P$-a.s:

######  Lemma 3.4.

If $(\mu_{\gamma,T}(\Omega))_{T\geq 0}$ is uniformly integrable, then
$\lim_{T\to\infty}\mu_{\gamma,T}(\Omega)>0$ $\bf P$-a.s.

###### Proof.

First we note that $\mathbf{E}[\mu_{\gamma,T}(\Omega)]=1$, so that
$\mathbf{P}(\lim_{T\to\infty}\mu_{\gamma,T}(\Omega)>0)>0$. Moreover, the event

$A_{\gamma}:=\\{\mu_{\gamma}(\Omega)>0\\}$ (3.4)

has probability 0 or 1 in virtue of Kolmogorov’s 0-1 law. ∎

The non-triviality of $\gamma_{c}$, i.e., that $\gamma_{c}>0$ for $d\geq 3$ is
implied by the existence of the so-called $L^{2}$-phase:

######  Lemma 3.5.

For $d\geq 3$ and $\gamma>0$ small enough, the martingale
$(\mu_{\gamma,T}(\Omega))_{T\geq 0}$ is bounded in $L^{2}(\bf P)$. In
particular, $\gamma_{c}>0$.

###### Proof.

Let $\omega^{\prime}$ be an independent copy of $\omega$ under
$\mathbb{P}_{0}$. Denote by $\mathbb{P}_{0}^{\otimes 2}$ the product measure
of $\mathbb{P}_{0}$. Then by definition and Fubini’s theorem,

$\displaystyle\mathbf{E}\left[\mu_{\gamma,T}(\Omega)^{2}\right]$
$\displaystyle=\mathbf{E}\left[\int_{\Omega^{2}}{\rm
e}^{\gamma(H_{T}(\omega)+H_{T}(\omega^{\prime}))-\gamma^{2}T(\phi\star\phi)(0)}{\rm
d}\mathbb{P}_{0}^{\otimes 2}(\omega,\omega^{\prime})\right]$
$\displaystyle=\int_{\Omega^{2}}{\rm d}\mathbb{P}_{0}^{\otimes
2}(\omega,\omega^{\prime})\mathbf{E}\left[{\rm
e}^{\gamma(H_{T}(\omega)+H_{T}(\omega^{\prime}))-\gamma^{2}T(\phi\star\phi)(0)}\right].$

Recall that for $\omega,\omega^{\prime}\in\Omega$,
$H_{T}(\omega)+H_{T}(\omega^{\prime})$ is a Gaussian random variable with mean
0 and variance

$2T(\phi\star\phi)(0)+2\int_{0}^{T}(\phi\star\phi)(\omega_{s}-\omega^{\prime}_{s}){\rm
d}s,$

and so

$\mathbf{E}\left[{\rm
e}^{\gamma(H_{T}(\omega)+H_{T}(\omega^{\prime}))-\gamma^{2}T(\phi\star\phi)(0)}\right]={\rm
e}^{\gamma^{2}\int_{0}^{T}(\phi\star\phi)(\omega_{s}-\omega^{\prime}_{s}){\rm
d}s}.$

Thus,

$\displaystyle\mathbf{E}\left[\mu_{\gamma,T}(\Omega)^{2}\right]=\int_{\Omega^{2}}{\rm
d}\mathbb{P}_{0}^{\otimes 2}(\omega,\omega^{\prime}){\rm
e}^{\gamma^{2}\int_{0}^{T}(\phi\star\phi)(\omega_{s}-\omega^{\prime}_{s}){\rm
d}s}$ $\displaystyle=\int_{\Omega}{\rm d}\mathbb{P}_{0}(\omega){\rm
e}^{\gamma^{2}\int_{0}^{T}(\phi\star\phi)(\sqrt{2}\omega_{s}){\rm d}s}$
$\displaystyle\leq\int_{\Omega}{\rm d}\mathbb{P}_{0}(\omega){\rm
e}^{\gamma^{2}\int_{0}^{\infty}(\phi\star\phi)(\sqrt{2}\omega_{s}){\rm d}s}.$

Since $d\geq 3$ and $\phi\star\phi$ is bounded with compact support,

$I(\phi):=\sup_{x\in\mathbb{R}^{d}}\int_{\Omega}{\rm
d}\mathbb{P}_{x}(\omega)\int_{0}^{\infty}(\phi\star\phi)(\sqrt{2}\omega_{s}){\rm
d}s<\infty,$

so that that for $\gamma>0$ small enough,

$\gamma^{2}I(\phi)<1.$

By Kahs’minski’s lemma [17] we deduce that

$\sup_{T}\mathbf{E}\left[\mu_{\gamma,T}(\Omega)^{2}\right]\leq\sup_{x\in\mathbb{R}^{d}}\int_{\Omega}{\rm
d}\mathbb{P}_{x}(\omega){\rm
e}^{\gamma^{2}\int_{0}^{\infty}\phi\star\phi(\sqrt{2}\omega_{s}){\rm
d}s}<\infty.$

∎

######  Remark 3.6.

By an application of Kahane’s inequality [15], in [23] it is proved
additionally that $\gamma_{c}<\infty$, and

$(\mu_{\gamma,T}(\Omega))_{T\geq 0}\text{ is uniformly
integrable}\Longleftrightarrow\lim_{T\to\infty}\mu_{\gamma,T}(\Omega)>0\qquad\mathbf{P}\text{-a.s.,}$

strengthening Lemma 3.4 to an “if and only if” statement. These results are
not required in the sequel.

##  4\. Proof of Theorem 2.1.

The proof of Theorem 2.1 is split into two parts. At first, we will show that
for a fixed Borel set $A\subset\Omega$, the sequence
$(\mu_{\gamma,T}(A))_{T\geq 0}$ converges to some random variable. The second
step will be to deduce that $\mu_{\gamma,T}$ and its normalized version
converge weakly as a measure.

######  Lemma 4.1.

Fix a Borel set $A\subset\Omega$. For $\gamma<\gamma_{c}$, the sequence
$(\mu_{\gamma,T}(A))_{T\geq 0}$ converges in $L^{1}(\mathbf{P})$ and almost
surely to a random variable $\mu_{\gamma}(A)$.

###### Proof.

By Lemma 3.1, for any Borel set $A\subset\Omega$,

$\mu_{\gamma,T}(A)=\int_{A}\exp\left\\{\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}{\rm
d}\mathbb{P}_{0}(\omega)$ (4.1)

is a martingale with respect to the filtration $(\mathcal{F}_{T})_{T}$. By our
choice of $\gamma$, the sequence $(\mu_{\gamma,T}(\Omega))_{T}$ is uniformly
integrable and therefore the same is true for $(\mu_{\gamma,T}(A))_{T}$. By
the martingale convergence theorem, the latter converges in
$L^{1}(\mathbf{P})$ and almost surely to a random variable. We denote the
limit by $\mu_{\gamma}(A)$.

∎

######  Remark 4.2.

By Lemma 3.4, we know that $\mu_{\gamma}(\Omega)>0$ $\bf P$-a.s. Hence, we can
replace $\mu_{\gamma,T}$ by its normalized version while preserving the almost
sure convergence. Indeed, for any $A\in\mathcal{B}$, we have

$\widehat{\mu}_{\gamma,T}(A){:=}\frac{\mu_{\gamma,T}(A)}{\mu_{\gamma,T}(\Omega)}\longrightarrow\frac{\mu(A)}{\mu(\Omega)}\quad\mathbf{P}\text{-a.s.}$

The next lemma provides a tightness criterion for a sequence of probability
measures on the Wiener space.

######  Lemma 4.3.

[18, Theorem. 18.13] A sequence $P_{n}$ of probability measures on
$\left(\Omega,\mathcal{B}\right)$ is tight if and only if the following two
conditions hold:

  * (a)

For any $T>0$ and $\eta>0$, there is $a>0$ such that

$P_{n}\left(\left\\{\omega\,:\,\sup_{0\leq t\leq
T}|\omega(t)|>a\right\\}\right)<\eta,\quad n\geq 1.$

  * (b)

For any $T>0,\eta>0$ and $\varepsilon>0$, there is $\delta>0$ such that

$P_{n}\left(\left\\{\omega\,:\,m^{T}(\omega,\delta)>\varepsilon\right\\}\right)<\eta,\quad
n\geq 1,$

where $m^{T}(\omega,\delta){:=}\sup_{0\leq s,t\leq
T,|t-s|<\delta}|\omega(t)-\omega(s)|$.

######  Lemma 4.4.

Let $\mu_{\gamma}$ be defined as in Lemma 4.1. If
$(A_{n})_{n}\subset\mathcal{B}$ is a sequence of subsets of $\Omega$ that is
increasing in the sense that $A_{n}\subset A_{n+1}$ for all $n\geq 1$, then
$\mathbf{P}$-a.s. it is true that

$\mu_{\gamma}\left(\bigcup_{n}A_{n}\right)=\lim_{n\to\infty}\mu_{\gamma}(A_{n}).$
(4.2)

An analogous statement holds for decreasing events.

###### Proof.

First note that the limit on the right hand side in (4.2) exists since the
sequence $(\mu_{\gamma}(A_{n}))_{n}$ is non-decreasing (by monotonicity of the
limit and of $\mu_{\gamma,T}$ as a measure) and it is bounded by

$\mu_{\gamma}\left(\bigcup_{n}A_{n}\right)\geq\lim_{n\to\infty}\mu_{\gamma}(A_{n}).$

To prove almost sure equality, it suffices to show that their expectations
coincide. By $L^{1}(\mathbf{P})$-convergence, the definition of the measure
$\mu_{\gamma,T}$ and Fubini’s theorem, we obtain

$\displaystyle\mathbf{E}\left[\mu_{\gamma}\left(\bigcup_{n}A_{n}\right)\right]$
$\displaystyle=\lim_{T\to\infty}\mathbf{E}\left[\mu_{\gamma,T}\left(\bigcup_{n}A_{n}\right)\right]$
$\displaystyle=\lim_{T\to\infty}\mathbf{E}\left[\int_{\bigcup_{n}A_{n}}\exp\left\\{\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}{\rm
d}\mathbb{P}_{0}(\omega)\right]$
$\displaystyle=\lim_{T\to\infty}\int_{\bigcup_{n}A_{n}}\mathbf{E}\left[\exp\left\\{\gamma{H}_{T}(\omega)-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}\right]{\rm
d}\mathbb{P}_{0}(\omega)$
$\displaystyle=\lim_{T\to\infty}\mathbb{P}_{0}\left(\bigcup_{n}A_{n}\right)=\mathbb{P}_{0}\left(\bigcup_{n}A_{n}\right)=\lim_{n\to\infty}\mathbb{P}_{0}(A_{n}).$

On the other hand, by monotone convergence, $L^{1}(\mathbf{P})$-convergence
and the same argument as above, we obtain

$\displaystyle\mathbf{E}\left[\lim_{n\to\infty}\mu_{\gamma}(A_{n})\right]$
$\displaystyle=\lim_{n\to\infty}\mathbf{E}\left[\mu_{\gamma}(A_{n})\right]=\lim_{n\to\infty}\mathbf{E}\left[\lim_{T\to\infty}\mu_{\gamma,T}(A_{n})\right]$
$\displaystyle\lim_{n\to\infty}\lim_{T\to\infty}\mathbf{E}\left[\mu_{\gamma,T}(A_{n})\right]=\lim_{n\to\infty}\mathbb{P}_{0}(A_{n}).$

This concludes the proof.

∎

######  Remark 4.5.

If we normalize $\mu_{\gamma}(A)$, i.e. if we consider the random variables
given by $\frac{\mu_{\gamma}(A)}{\mu_{\gamma}(\Omega)}$ for $A\in\mathcal{B}$,
then clearly the statement in Lemma 4.4 remains true.

######  Proposition 4.6.

Suppose that for each Borel set $A\subset\Omega$, the sequence
$\mu_{\gamma,T}(A)$ converges in $L^{1}(\mathbf{P})$ and almost surely to some
random variable $\mu_{\gamma}(A)$ as $T\to\infty$. Then the sequence
$(\mu_{\gamma,T})_{T\geq 0}$ converges weakly $\mathbf{P}$-a.s. to a random
measure $\mu_{\gamma}$. Similarly, the sequence of normalized measures
$(\widehat{\mu}_{\gamma,T})_{T\geq 0}$ converges weakly $\mathbf{P}$-a.s. to a
random probability measure $\widehat{\mu}_{\gamma}$.

###### Proof.

First we will introduce some notation. Define the sets

$\displaystyle\mathcal{A}_{o}{:=}\\{(x_{1},y_{1})\times\ldots\times(x_{d},y_{d})\,:\,x_{i},y_{i}\in\mathbb{Q}\\},$
(4.3)
$\displaystyle\mathcal{A}_{c}{:=}\\{[x_{1},y_{1}]\times\ldots\times[x_{d},y_{d}]\,:\,x_{i},y_{i}\in\mathbb{Q}\\},$
$\displaystyle\mathcal{A}_{s}{:=}\\{[x_{1},y_{1})\times\ldots\times[x_{d},y_{d})\,:\,x_{i},y_{i}\in\mathbb{Q}\\},$

and

$\mathcal{X}_{o}^{1}{:=}\bigg{\\{}\\{\omega:\omega(t_{1})\in
A_{1},\ldots,\omega(t_{n})\in
A_{n}\\}:n\in\mathbb{N},t_{1}<\ldots<t_{n}\in\mathbb{Q}^{+},A_{1},\ldots,A_{n}\in\mathcal{A}_{o}\bigg{\\}}.$
(4.4)

Analogously define $\mathcal{X}_{c}^{1}$ and $\mathcal{X}_{s}^{1}$ and set
$\mathcal{X}_{1}{:=}\mathcal{X}_{o}^{1}\cup\mathcal{X}_{c}^{1}\cup\mathcal{X}_{s}^{1}$.
Further, define

$\displaystyle\mathcal{X}_{2}:=\Omega\cup\Bigg{\\{}\bigg{\\{}\omega:\sup_{0\leq
t\leq
T}|\omega(t)|>a\bigg{\\}}:T,a\in\mathbb{Q}^{+}\Bigg{\\}}\cup\bigg{\\{}\big{\\{}\omega:m^{T}(\omega,\delta)>\varepsilon\big{\\}}:T,\delta,\varepsilon\in\mathbb{Q}^{+}\bigg{\\}},\quad\mbox{and}$
(4.5) $\displaystyle\mathcal{X}:=\mathcal{X}_{1}\cup\mathcal{X}_{2}.$

Then $\mathcal{X}$ is a countable subset of the Borel $\sigma$-algebra on
$\Omega$. Since $\mathcal{X}$ is countable, by Lemma 4.1 (or more precisely,
by the subsequent remark),

$\widehat{\mu}_{\gamma,T}(A)\overset{T\to\infty}{\longrightarrow}\nicefrac{{\mu_{\gamma}(A)}}{{m_{\gamma}(\Omega)}}$

almost surely for all $A\in\mathcal{X}$ (simultaneously). We will use this
observation to show that

$(\widehat{\mu}_{\gamma,T})_{T\geq 0}\,\,\,\,\mbox{is $\mathbf{P}${\rm-a.s.}
tight}.$

In order to show tightness, we will verify that the conditions of Lemma 4.3
are satisfied, for which we will first check condition (a) thereof. Note that
for a fixed $s\in\mathbb{Q}^{+}$, the events

$\mathbb{Q}^{+}\ni a\mapsto\left\\{\omega:\sup_{0\leq t\leq
s}|\omega(t)|>a\right\\}$

are decreasing. Hence, by Lemma 4.4 (and its the subsequent remark), we deduce
that

$\lim_{a\to\infty,a\in\mathbb{Q}^{+}}\frac{\mu_{\gamma}\left(\omega:\sup_{0\leq
t\leq
s}|\omega(t)|>a\right)}{\mu_{\gamma}(\Omega)}=0\qquad\mathbf{P}\text{-a.s.}$

In particular, for fixed $s,\eta\in\mathbb{Q}^{+}$, one can find an
$a\in\mathbb{Q}^{+}$ such that $\mathbf{P}$-a.s. we have

$\frac{\mu_{\gamma}\left(\\{\omega:\sup_{0\leq t\leq
s}|\omega(t)|>a\\}\right)}{\mu_{\gamma}(\Omega)}\leq\frac{\eta}{2}.$

Since

$\lim_{T\to\infty}\widehat{\mu}_{\gamma,T}\left(\left\\{\omega:\sup_{0\leq
t\leq
s}|\omega(t)|>a\right\\}\right)=\frac{\mu_{\gamma}\left(\left\\{\omega:\sup_{0\leq
t\leq
s}|\omega(t)|>a\right\\}\right)}{\mu_{\gamma}(\Omega)}\leq\frac{\eta}{2}\qquad\mathbf{P}\text{-a.s.,}$

we have

$\widehat{\mu}_{\gamma,T}\left(\left\\{\omega:\sup_{0\leq t\leq
s}|\omega(t)|>a\right\\}\right)\leq\eta,\qquad\mathbf{P}\mbox{-a.s.}$

for some $T>0$ large enough. Hence, we can choose $a$ large enough such that
we can assure that the inequality holds for all $T>0$. Therefore, condition
(a) of Lemma 4.3 is satisfied. Condition (b) follows with a similar
argumentation. Indeed, for fixed $\ell,\eta,\varepsilon\in\mathbb{Q}^{+}$, the
events

$\mathbb{Q}^{+}\ni\delta\mapsto\left\\{\omega:\sup_{0\leq
s,t\leq\ell,|t-s|<\delta}|\omega(t)-\omega(s)|>\varepsilon\right\\}$

are decreasing as $\delta\searrow 0$. Again by Lemma 4.4, it holds that

$\lim_{\delta\to
0,\delta\in\mathbb{Q}^{+}}\frac{\mu_{\gamma}\left(\left\\{\omega:m^{\ell}(\omega,\delta)>\varepsilon\right\\}\right)}{\mu_{\gamma}(\Omega)}=0\quad\mathbf{P}\text{-a.s.}$

Thus, there exists a $\delta\in\mathbb{Q}^{+}$ such that

$\lim_{T\to\infty}\widehat{\mu}_{\gamma,T}\left(\left\\{\omega:m^{\ell}(\omega,\delta)>\varepsilon\right\\}\right)=\frac{\mu_{\gamma}\left(\left\\{\omega:m^{T}(\omega,\delta)>\varepsilon\right\\}\right)}{\mu_{\gamma}(\Omega)}\leq\frac{\eta}{2}\quad\mathbf{P}\text{-a.s.}$

Hence, for some $T$ large enough we have

$\widehat{\mu}_{\gamma,T}\left(\left\\{\omega:m^{\ell}(\omega,\delta)>\varepsilon\right\\}\right)\leq\eta,\qquad\mathbf{P}\mbox{-a.s.}$

If we now choose $\delta$ sufficiently small, we can assure that the
inequality holds for all $k\geq 1$. We have shown that $\mathbf{P}$-a.s.
conditions (a) and (b) of Lemma 4.3 are satisfied and therefore the sequence
$(\widehat{\mu}_{\gamma,T})_{T\geq 0}$ is $\mathbf{P}$-a.s. tight.

Therefore, for each subsequence of $(\widehat{\mu}_{\gamma,T})_{T\geq 0}$
there is a further sub-subsequence converging weakly to some random measure
$\widehat{\mu}_{\gamma}$. It remains to show that the limiting measure
$\widehat{\mu}_{\gamma}$ is uniquely determined (i.e., independent of the
subsequence). We will use Portmanteau’s Theorem to show that for all Borel
sets $A\subset\Omega$, it is true that any weak limit satisfies

$\widehat{\mu}_{\gamma}(A)=\nicefrac{{\mu_{\gamma}(A)}}{{\mu_{\gamma}(\Omega)}}.$

More precisely, it suffices to show that this is true for all sets
$A\in\mathcal{X}_{s}^{1}$ (recall (4.4)). Hence, let
$A\in\mathcal{X}_{s}^{1}$, i.e. $A$ is of the form
$A=\\{\omega:\omega(t_{1})\in A_{1},\ldots,\omega(t_{n})\in A_{n}\\}$ for some
$n\in\mathbb{N},t_{1}<\ldots<t_{n}\in\mathbb{Q}^{+}$ and
$A_{1},\ldots,A_{n}\in\mathcal{A}_{s}$. Then by Lemma 4.4,

$\frac{\mu_{\gamma}(A)}{\mu_{\gamma}(\Omega)}=\sup_{A_{1}^{\prime},\ldots,A_{n}^{\prime}}\frac{\mu_{\gamma}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in
A_{n}^{\prime}\\})}{\mu_{\gamma}(\Omega)}\quad\mathbf{P}\text{-a.s.},$ (4.6)

where the supremum is over
$A_{1}^{\prime},\ldots,A_{n}^{\prime}\in\mathcal{A}_{c}$ (recall (4.3)) such
that $A_{i}^{\prime}\subset A_{i}$ for all $1\leq i\leq n$. Similarly, we have

$\frac{\mu_{\gamma}(A)}{\mu_{\gamma}(\Omega)}=\inf_{A_{1}^{\prime},\ldots,A_{n}^{\prime}}\frac{\mu_{\gamma}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in
A_{n}^{\prime}\\})}{\mu_{\gamma}(\Omega)}\qquad\mathbf{P}\text{-a.s.,}$ (4.7)

where the infimum is over all
$A_{1}^{\prime},\ldots,A_{n}^{\prime}\in\mathcal{A}_{o}$ such that
$\bar{A_{i}}\subset A_{i}^{\prime}$ for all $1\leq i\leq n$. By Lemma 4.4 and
Portmanteau’s Theorem, for any $A\in\mathcal{X}_{s}^{1}$ and any
$A_{1}^{\prime},\ldots,A_{n}^{\prime}\in\mathcal{A}_{o}$ with
$\bar{A_{i}}\subset A_{i}^{\prime}$ for all $1\leq i\leq n$,
$\mathbf{P}$-a.s., we have

$\displaystyle\widehat{\mu}_{\gamma}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in A_{n}^{\prime}\\})$ $\displaystyle\leq$
$\displaystyle\liminf_{T\to\infty}\frac{\mu_{\gamma,T}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in
A_{n}^{\prime}\\})}{\mu_{\gamma,T}(\Omega)}$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{\mu_{\gamma,T}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in
A_{n}^{\prime}\\})}{\mu_{\gamma,T}(\Omega)}$ $\displaystyle=$
$\displaystyle\frac{\mu_{\gamma}(\\{\omega:\omega(t_{1})\in
A_{1}^{\prime},\ldots,\omega(t_{n})\in
A_{n}^{\prime}\\})}{\mu_{\gamma}(\Omega)}.$

Since $A_{1}^{\prime},\ldots,A_{n}^{\prime}$ are arbitrary, by taking the
infimum and using (4.7), we can deduce that $\mathbf{P}$-a.s.,
$\widehat{\mu}_{\gamma}(A)\leq\nicefrac{{\mu_{\gamma}(A)}}{{\mu_{\gamma}(\Omega)}}$.
Proceeding analogously with sets
$A_{1}^{\prime},\ldots,A_{n}^{\prime}\in\mathcal{A}_{c}$ such that
$A_{i}^{\prime}\subset A_{i}$ for all $1\leq i\leq n$ and using (4.6), we
obtain $\mathbf{P}$-a.s.,
$\widehat{\mu}_{\gamma}(A)\geq\nicefrac{{\mu_{\gamma}(A)}}{{\mu_{\gamma}(\Omega)}}$.
Therefore,
$\widehat{\mu}_{\gamma}(A)=\nicefrac{{\mu_{\gamma}(A)}}{{\mu_{\gamma}(\Omega)}}$
$\mathbf{P}$-a.s. as desired. The proof of the weak convergence for the
unnormalized measures can be performed by the same methods, by noting that
$\mu_{\gamma}(\Omega)>0$ $\mathbf{P}$-a.s. ∎

We can now complete the

###### Proof of Theorem 2.1..

By Lemma 4.1, for each Borel set $A\subset\Omega$, the sequence
$(\widehat{\mu}_{\gamma,T}(A))$ converges $\mathbf{P}$-a.s. to
$\nicefrac{{\mu_{\gamma}(A)}}{{\mu_{\gamma}(\Omega)}}$ as $T\to\infty$. Hence,
by Proposition 4.6, $\widehat{\mu}_{\gamma,T}$ converges weakly
$\mathbf{P}$-a.s. to the measure $\widehat{\mu}_{\gamma}$.

∎

######  Remark 4.7.

For discrete directed polymers, Comets and Yosida [9] construct a limiting
measure $\mu_{\infty}$ on the $\sigma$-algebra generated by finite paths, such
that, on this $\sigma$-algebra, $\mu_{\infty}(A)=\lim_{n\to\infty}\mu_{n}(A)$
(here, $\mu_{n}$ is the normalized polymer measure at time $n$). Indeed, the
limiting measure can be identified explicitly as the law of a random, time-
inhomogeneous Markov chain, and therefore the martingale convergence theorem
guarantees the weak convergence of $\mu_{n}$ to $\mu_{\infty}$. In our setup,
such limiting measure does not exist a priori, so that showing convergence
over fixed sets does not guarantee weak convergence of
$(\widehat{\mu}_{\gamma,T})_{T\geq 0}$.

##  5\. Uniqueness: Proofs of Theorem 2.2 and Theorem 2.5.

We will now prove Theorem 2.5, which will imply Theorem 2.2. We conclude this
section by providing the proof of Proposition 2.4.

###  5.1. Proof of Theorem 2.5.

Recall that ${H}_{T}(\omega)={H}_{T}(\omega,{\dot{B}})$ where ${\dot{B}}$ is
the space-time white noise. Similarly, we write $\mu_{\gamma}({\rm
d}\omega)=\mu_{\gamma}({\rm d}\omega,{\dot{B}})$. Given $T>0$, set (recall the
notation from (2.2))

$\mathbb{Q}_{\mu_{\gamma,T}}({\rm d}{\dot{B}},{\rm
d}\omega){=}\exp\left\\{\gamma{H}_{T}(\omega,{\dot{B}})-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}\mathbb{P}_{0}({\rm
d}\omega)P({\rm d}{\dot{B}}).$

This is a probability measure since
$E\left[\exp\left\\{\gamma{H}_{T}(\omega,{\dot{B}})-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}\right]=1$.
Similarly define $\mathbb{Q}_{\mu_{\gamma}}({\rm d}{\dot{B}},{\rm
d}\omega){=}\mu_{\gamma}({\rm d}\omega,{\dot{B}})P({\rm d}{\dot{B}})$. Before
continuing, we show that

######  Lemma 5.1.

$\mathbb{Q}_{\mu_{\gamma,T}}\to\mathbb{Q}_{\mu_{\gamma}}$ weakly as
$T\to\infty$.

###### Proof.

The proof follows a similar line of arguments as that of Theorem 2.1. We prove
first that $(\mathbb{Q}_{\mu_{\gamma,T}})_{T\geq 0}$ is tight. It is enough to
verify that the marginals
$\mathbb{Q}_{\mu_{\gamma,T}}^{1}(\cdot):=\mathbb{Q}_{\mu_{\gamma,T}}(\cdot\times\mathcal{S}^{\prime})$
and
$\mathbb{Q}_{\mu_{\gamma,T}}^{2}(\cdot):=\mathbb{Q}_{\mu_{\gamma,T}}(\Omega\times\cdot)=P(\cdot)$
are tight. We can again check the tightness of
$\mathbb{Q}_{\mu_{\gamma,T}}^{1}$ by applying Lemma 4.3 as in Proposition 4.6.
On the other hand, since $\mathcal{S}^{\prime}$ is $\sigma$-compact (by the
Banach-Alaoglu theorem), then $P$ is also tight. ∎

By the previous lemma, if
$n\in\mathbb{N},f_{1},\ldots,f_{n}\in\mathcal{S}(\mathbb{R}_{+}\times\mathbb{R}^{d})$
and $g:\Omega\to\mathbb{R}$ is bounded and continuous, i.e the map
$(\omega,{\dot{B}})\mapsto
g(\omega){\dot{B}}(f_{1})\cdot\ldots\cdot{\dot{B}}(f_{n})$ is a bounded
continuous function on $\Omega\times\mathcal{S}^{\prime}$, then

$\displaystyle\lim_{T\to\infty}E\left[{\dot{B}}(f_{1})\cdot\ldots\cdot{\dot{B}}(f_{n})\int_{\Omega}g(\omega)\exp\left\\{\gamma{H}_{T}(\omega,{\dot{B}})-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}\mathbb{P}_{0}({\rm
d}\omega)\right]$ (5.1) $\displaystyle=$ $\displaystyle
E\left[{\dot{B}}(f_{1})\cdot\ldots\cdot{\dot{B}}(f_{n})\int_{\Omega}g(\omega)\mu_{\gamma}({\rm
d}\omega,{\dot{B}})\right].$

Now, conditioning $\mathbb{Q}_{\mu_{\gamma,T}}({\rm d}{\dot{B}},{\rm
d}\omega)$ on $\omega\in\Omega$, we obtain

$\mathbb{Q}_{\mu_{\gamma,T}}({\rm
d}{\dot{B}}\,|\,\omega)=\exp\left\\{\gamma{H}_{T}(\omega,{\dot{B}})-\frac{\gamma^{2}}{2}T(\phi\star\phi)(0)\right\\}P({\rm
d}{\dot{B}}).$ (5.2)

By the Cameron-Martin-Girsanov Theorem, we know that under the measure
$\mathbb{Q}_{\mu_{\gamma,T}}({\rm d}{\dot{B}}\,|\,\omega)$,
$({\dot{B}}(f))_{f}$ is a Gaussian process with the same covariance structure
as in (1.6) and mean given by

$\int{\dot{B}}(f){\rm d}\mathbb{Q}_{\mu_{\gamma,T}}({\rm
d}{\dot{B}}\,|\,\omega)=\gamma\operatorname{Cov}\left({\dot{B}}(f),{H}_{T}(\omega)\right)=\gamma\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{[0,T]}(s)f(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y.$

Hence, the expression in (5.1) is equal to

$\displaystyle\lim_{T\to\infty}\int_{\Omega}g(\omega)\int{\dot{B}}(f_{1})\cdot\ldots\cdot{\dot{B}}(f_{n}){\rm
d}\mathbb{Q}_{\mu_{\gamma,T}}({\rm d}{\dot{B}},\omega)\mathbb{P}_{0}({\rm
d}\omega)$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\int_{\Omega}g(\omega)\int\prod_{i=1}^{n}\left({\dot{B}}(f_{i})+\gamma\operatorname{Cov}\left({\dot{B}}(f_{i}),{H}_{T}(\omega)\right)\right)P({\rm
d}{\dot{B}})\mathbb{P}_{0}({\rm d}\omega)$ $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\int_{\Omega}g(\omega)\int\prod_{i=1}^{n}\left({\dot{B}}(f_{i})+\gamma\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}{{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}}_{[0,T]}(s)f_{i}(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y\right)P({\rm d}{\dot{B}})\mathbb{P}_{0}({\rm d}\omega).$

Writing $\mathbb{Q}_{\mathbb{P}_{0}}({\rm d}{\dot{B}},{\rm
d}\omega){=}\mathbb{P}_{0}({\rm d}\omega)P({\rm d}{\dot{B}})$, and using
(5.1), we deduce that

$\displaystyle\int
g(\omega)\prod_{i=1}^{n}\left({\dot{B}}(f_{i})+\gamma\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}f_{i}(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y\right)\mathbb{Q}_{\mathbb{P}_{0}}({\rm d}{\dot{B}},{\rm d}\omega)$
$\displaystyle=$ $\displaystyle\int
g(\omega)\prod_{i=1}^{n}{\dot{B}}(f_{i})\mathbb{Q}({\rm d}{\dot{B}},{\rm
d}\omega).$

Since $g,f_{1},\ldots,f_{n}$ are arbitrary, we conclude that the law of
${\dot{B}}$ under $\mathbb{Q}$ is the same as the law of
$\widetilde{{\dot{B}}}$ defined by
$\widetilde{{\dot{B}}}(f){:=}{\dot{B}}(f)+\gamma\int_{R_{+}\times\mathbb{R}^{d}}f(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y$ under $\mathbb{Q}_{\mathbb{P}_{0}}$. The uniqueness is immediate
from this construction. ∎

###  5.2. Proof of Theorem 2.2.

By Theorem 2.5, we know that the law of ${H}_{T}$ under the measure
$\mathbb{Q}_{\mu_{\gamma}}({\rm d}{\dot{B}},{\rm d}\omega)=\mu_{\gamma}({\rm
d}\omega,{\dot{B}})P(d{\dot{B}})$ is the same as the law of ${H}_{T}+\gamma
T(\phi\star\phi)(0)$ under $P\otimes\mathbb{P}_{0}$. Furthermore, for every
$\omega\in\Omega$, we have

$\lim_{T\to\infty}\frac{{H}_{T}(\omega)}{T}=0\quad\mathbf{P}\text{-a.s.}$

(since the stochastic integral is a martingale and its quadratic variation is
equal to $T(\phi\star\phi)(0)$ [21, Chapter 2, §6]), so that

$\lim_{T\to\infty}\frac{1}{T}({H}_{T}+T\gamma(\phi\star\phi)(0))=\gamma(\phi\star\phi)(0)\quad
P\otimes\mathbb{P}_{0}\text{-a.s.}$

Hence,

$\lim_{T\to\infty}\frac{{H}_{T}}{T}=\gamma(\phi\star\phi)(0)\quad\mathbb{Q}\text{-a.s.},$

so that
$\mu_{\gamma}\left(\lim_{T\to\infty}\frac{{H}_{T}}{T}\neq\gamma(\phi\star\phi)(0)\right)=0$
$\mathbf{P}$-a.s.∎

Proof of Proposition 2.4: Since for any $A\subset\Omega$ Borel measurable,
$(\mu_{\gamma,T,\phi}(A))_{T\geq 0}$ and
$(\mu_{\gamma,T,\phi^{\prime}}(A))_{T\geq 0}$ are uniformly integrable
martingales converging respectively to $\mu_{\gamma,\phi}(A)$ and
$\mu_{\gamma,\phi^{\prime}}(A)$, we have $\mathbf{P}$-a.s. the following
identities:

$\displaystyle\mathbf{E}[\mu_{\gamma,\phi}(A)|\mathcal{F}_{T}]$
$\displaystyle=\mu_{\gamma,T,\phi}(A),$
$\displaystyle\mathbf{E}[\mu_{\gamma,\phi^{\prime}}(A)|\mathcal{F}_{T}]$
$\displaystyle=\mu_{\gamma,T,\phi^{\prime}}(A).$

Thus, if $\mu_{\gamma,\phi}(A)=\mu_{\gamma,\phi^{\prime}}(A)$
$\mathbf{P}$-a.s., then we deduce that for all $T\geq 0$,
$\mu_{\gamma,T,\phi}(A)=\mu_{\gamma,T,\phi^{\prime}}(A)$ $\mathbf{P}$-a.s. By
choosing an appropriate countable collection of Borel sets $A\in\Omega$ (such
as the sets $\mathcal{X}^{1}_{s}$ that appear in the proof of Proposition 4.6,
one can deduce that $\mathbf{P}\otimes\mathbb{P}_{0}$-a.s. for all
$T\in\mathbb{Q}^{+}$ and assuming that
$\phi\star\phi(0)=\phi^{\prime}\star\phi^{\prime}(0)$,

${\rm e}^{\gamma H_{T,\phi}-\frac{\gamma^{2}}{2}T\phi\star\phi(0)}={\rm
e}^{\gamma
H_{T,\phi^{\prime}}-\frac{\gamma^{2}}{2}T\phi^{\prime}\star\phi^{\prime}(0)}={\rm
e}^{\gamma H_{T,\phi^{\prime}}-\frac{\gamma^{2}}{2}T\phi\star\phi(0)},$

concluding that $\mathbf{P}\otimes\mathbb{P}_{0}$-a.s.,
$H_{T,\phi}=H_{T,\phi^{\prime}}$. Note that that for $\omega\in\Omega$, the
quadratic variation of $H_{T,\phi}(\omega)-H_{T,\phi^{\prime}}(\omega)$ is
equal to

$2T\phi\star\phi(0)-2T\int_{\mathbb{R}^{d}}\phi(y)\phi^{\prime}(y){\rm
d}y=2T\left(\int_{\mathbb{R}^{d}}\phi(y)^{2}{\rm
d}y-\int_{\mathbb{R}^{d}}\phi(y)\phi^{\prime}(y){\rm d}y\right).$

By Cauchy-Schwarz inequality, we know that the last display is equal to zero
if and only if $\phi=\lambda\phi^{\prime}$ for some $\lambda>0$, but using
that $\int_{\mathbb{R}^{d}}\phi(y){\rm
d}y=\int_{\mathbb{R}^{d}}\phi^{\prime}(y){\rm d}y=1$, we deduce that
$\lambda=1$, so $\phi=\phi^{\prime}$.

######  Remark 5.2.

One may wonder if, using Theorem 2.2, for a fixed $\phi$, $\mu_{\gamma,\phi}$
is the unique measure such that

$\mu_{\gamma,\phi}\left(\lim_{T\to\infty}\frac{{H}_{T}(\omega)}{T}\neq\gamma(\phi\star\phi)(0)\right)=0\qquad\bf
P\text{-a.s.}$ (5.3)

However, this is false: by perturbing $\mathbb{P}_{0}$ by ${\rm
e}^{\gamma(H_{T}+H_{1})-\frac{\gamma^{2}}{2}\phi\star\phi(0)(T+1+2(T\wedge
1))}$ instead of by ${\rm e}^{\gamma
H_{T}-\frac{\gamma^{2}}{2}T\phi\star\phi(0)}$ leads to a measure
$\tilde{\mu}_{\gamma,\phi}$ (at least for $\gamma$ small enough so that the
limiting measure also exists) such that (5.3) still holds. Indeed, one can
follow the proof of Theorem 2.5 and notice that the law of $\dot{B}$ under
$\mathbb{Q}_{\tilde{\mu}_{\gamma,\phi}}$ is the same as the law of

${{\dot{B}}}_{\phi}(f)={\dot{B}}(f)+\gamma\int_{\mathbb{R}_{+}\times\mathbb{R}^{d}}f(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y+\gamma\int_{[0,1]\times\mathbb{R}^{d}}f(s,y)\phi(\omega_{s}-y){\rm
d}s{\rm d}y,\qquad
f\in\mathcal{S}^{\prime}(\mathbb{R}_{+}\times\mathbb{R}^{d})$

under $P\otimes\mathbb{P}_{0}$. In particular,
$\mu_{\gamma,\phi}\neq\tilde{\mu}_{\gamma,\phi}$. Nevertheless, the
distribution of $(H_{T})_{T\geq 1}$ under
$\mathbb{Q}_{\tilde{\mu}_{\gamma,\phi}}$ is the same as the distribution of
$(H_{T}+\gamma(T+1)\phi\star\phi(0))_{T\geq 1}$ under under
$P\otimes\mathbb{P}_{0}$. Thus, $\tilde{\mu}_{\gamma,\phi}$ also satisfies
(5.3).

##  6\. Proof of Theorem 2.7

The proof is consequence of the following result:

######  Lemma 6.1.

Given $T>0$ and $\gamma<\gamma_{c}$ let $M_{T}:=\sup_{0\leq s\leq
T}\mu_{\gamma,s}(\Omega)$ and $M_{\infty}:=\lim_{T\to\infty}M_{T}$. Then
$\mathbf{E}[M_{\infty}]<\infty$.

Before proving the lemma, we introduce, for $u>0$, the stopping time

$\tau=\tau_{u}:=\inf\\{T\geq 0:\mu_{\gamma,T}(\Omega)=u\\}.$ (6.1)

######  Lemma 6.2.

For every convex function $f:[0,\infty)\to\mathbb{R}$ and $\gamma,T>0$,

$\mathbf{E}\bigg{[}f\left(\frac{\mu_{\gamma,T}(\Omega)}{\mu_{\gamma,\tau}(\Omega)}\right),\tau\leq
T\bigg{]}\leq\mathbf{P}(\tau\leq T)\mathbf{E}[f(\mu_{\gamma,T}(\Omega))].$
(6.2)

###### Proof.

Let $(\tau_{n})_{n}$ be a discrete approximation of $\tau$ such that
$\tau_{n}\searrow\tau$. If (6.2) holds for $\tau_{n}$, for each $n$, then by
Fatou’s lemma we can deduce (6.2) for $\tau$. Thus, we can assume that $\tau$
takes values in a discrete, countable set $\\{t_{i}\\}_{n\in\mathbb{N}}$
(which we may assume to be ordered in increasing order).

By the Markov property, if $s\leq T$,

$\displaystyle\frac{\mu_{\gamma,T}(\Omega)}{\mu_{\gamma,s}(\Omega)}$
$\displaystyle=\frac{\mathbb{E}_{0}\Big{[}{\rm e}^{\gamma
H_{s}(\omega)-\frac{\gamma^{2}s}{2}\phi\star\phi(0)}\mu_{\gamma,T-s}(\Omega)\circ\theta_{s,\omega_{s}}\Big{]}}{\mathbb{E}_{0}\Big{[}{\rm
e}^{\gamma H_{s}(\omega)-\frac{\gamma^{2}s}{2}\phi\star\phi(0)}\Big{]}}$
$\displaystyle=:\mathbb{E}_{0,s}\Big{[}\mu_{\gamma,T-s}(\Omega)\circ\theta_{s,\omega_{s}}\Big{]},$

where we remind the reader that $\theta_{t,x}$ is the space-time shift in the
environment.

If $[0,T]\cap\\{t_{i}\\}_{n\in\mathbb{N}}=\\{t_{1},\cdots,t_{n}\\}$, then
using the convexity of $f$ and Jensen’s inequality,

$\displaystyle\mathbf{E}\bigg{[}f\left(\frac{\mu_{\gamma,T}(\Omega)}{\mu_{\gamma,\tau}(\Omega)}\right),\tau\leq
T\bigg{]}$
$\displaystyle\leq\mathbf{E}\bigg{[}\mathbb{E}_{0,\tau}\bigg{(}f\left(\mu_{\gamma,T-\tau}(\Omega)\circ\theta_{\tau,\omega_{\tau}}\right)\bigg{)},\tau\leq
T\bigg{]}$ $\displaystyle=\mathbf{E}\bigg{[}\mathbb{E}_{0}\bigg{(}\frac{{\rm
e}^{\gamma
H_{\tau}(\omega)-\frac{\gamma^{2}}{2}\tau\phi\star\phi(0)}f\left(\mu_{\gamma,T-\tau}(\Omega)\circ\theta_{\tau,\omega_{\tau}}\right)}{\mathbb{E}_{0}[{\rm
e}^{\gamma
H_{\tau}(\omega)-\frac{\gamma^{2}}{2}\tau\phi\star\phi(0)}]}\bigg{)},\tau\leq
T\bigg{]}$
$\displaystyle=\sum_{i=1}^{n}\mathbf{E}\bigg{[}\mathbb{E}_{0}\bigg{(}\frac{{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}f\left(\mu_{\gamma,T-t_{i}}(\Omega)\circ\theta_{t_{i},\omega_{t_{i}}}\right)}{\mathbb{E}_{0}[{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}]}\bigg{)},\tau=t_{i}\bigg{]}$
$\displaystyle=\sum_{i=1}^{n}\mathbf{E}\bigg{[}\mathbf{E}\bigg{(}\mathbb{E}_{0}\bigg{[}\frac{{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}f\left(\mu_{\gamma,T-t_{i}}(\Omega)\circ\theta_{t_{i},\omega_{t_{i}}}\right)}{\mathbb{E}_{0}[{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}]}\bigg{]}\bigg{|}\mathcal{F}_{t_{i}}\bigg{)},\tau=t_{i}\bigg{]}$

By Lemma 3.2, the last expression is equal to

$\displaystyle\sum_{i=1}^{n}\mathbf{E}\bigg{[}\mathbb{E}_{0}\bigg{(}\mathbf{E}\bigg{[}\frac{{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}f\left(\mu_{\gamma,T-t_{i}}(\Omega)\circ\theta_{t_{i},\omega_{t_{i}}}\right)}{\mathbb{E}_{0}[{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}]}\bigg{|}\mathcal{F}_{t_{i}}\bigg{)}\bigg{]},\tau=t_{i}\bigg{]}$
$\displaystyle=\sum_{i=1}^{n}\mathbf{E}\bigg{[}\mathbb{E}_{0}\bigg{(}\frac{{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}}{\mathbb{E}_{0}[{\rm
e}^{\gamma
H_{t_{i}}(\omega)-\frac{\gamma^{2}}{2}t_{i}\phi\star\phi(0)}]}\mathbf{E}\bigg{[}f\left(\mu_{\gamma,T-t_{i}}(\Omega)\circ\theta_{t_{i},\omega_{t_{i}}}\right)\big{|}\mathcal{F}_{t_{i}}\bigg{]}\bigg{)},\tau=t_{i}\bigg{]}$

Note that $f(\mu_{\gamma,T-t_{i}}(\Omega)\circ\theta_{t_{i},\omega_{t_{i}}})$
is independent on $\mathcal{F}_{t_{i}}$, and so the last sum reduces to

$\sum_{i=1}^{n}\mathbf{P}\left[\tau=t_{i}\right]\mathbf{E}\left[f\left(\mu_{\gamma,T-t_{i}}(\Omega)\right)\right]\leq\mathbf{P}(\tau\leq
T)\mathbf{E}\left[f\left(\mu_{\gamma,T}(\Omega)\right)\right],$

where we used that $f$ is convex, so that $(f(\mu_{\gamma,T}(\Omega)))_{T\geq
0}$ is a submartingale. ∎

###### Proof of Lemma 6.1.

Given $\varepsilon>0$, let

$f_{\varepsilon}(x):=\left(\frac{x}{\varepsilon}-1\right)\wedge 1.$

We note that $f_{\varepsilon}$ is concave and for all $x\geq 0$,

$\mathbbm{1}_{[\varepsilon,\infty)}(x)\geq
f_{\varepsilon}(x)\geq\mathbbm{1}(x)_{[2\varepsilon,\infty)}-\mathbbm{1}_{[0,\varepsilon]}(x).$
(6.3)

The proof is complete once we can find $\varepsilon>0$ such that for all $T>0$
and $u>1$,

$\mathbf{P}(M_{T}>u)\leq 2\mathbf{P}(\mu_{\gamma,T}(\Omega)>u\varepsilon).$
(6.4)

Indeed,

$\mathbf{E}[M_{T}]=\int_{0}^{1}\mathbf{P}(M_{T}>u){\rm
d}u+\int_{1}^{\infty}\mathbf{P}(M_{T}>u){\rm d}u\leq
1+\frac{2}{\varepsilon}\mathbf{E}[\mu_{\gamma,T}(\Omega)]=1+\frac{2}{\varepsilon}.$

For a fixed $u>1$, recall the definition of $\tau=\tau_{u}$ from (6.1), so
that $\mu_{\gamma,\tau}(\Omega)=u$. Hence, by Lemma 6.2 and Eq. (6.3)

$\displaystyle\mathbf{P}(\mu_{\gamma,T}(\Omega)>u\varepsilon)$
$\displaystyle\geq\mathbf{P}\left(\tau\leq
T,\frac{\mu_{\gamma,T}(\Omega)}{\mu_{\gamma,\tau}(\Omega)}>\varepsilon\right)$
(6.5)
$\displaystyle\geq\mathbf{E}\left[f_{\varepsilon}\left(\frac{\mu_{\gamma,T}(\Omega)}{\mu_{\gamma,\tau}(\Omega)}\right),\tau\leq
T\right]$ $\displaystyle\geq\mathbf{P}(\tau\leq
T)\mathbf{E}\left[f_{\varepsilon}(\mu_{\gamma,T}(\Omega))\right]$
$\displaystyle\geq\mathbf{P}(\tau\leq T)\inf_{T\geq
0}\mathbf{E}\left[f_{\varepsilon}(\mu_{\gamma,T}(\Omega))\right]$
$\displaystyle\geq\mathbf{P}(\tau\leq T)\mathbf{E}\left[\inf_{T\geq
0}f_{\varepsilon}(\mu_{\gamma,T}(\Omega))\right].$

We use again (6.3) to deduce

$\displaystyle\mathbf{E}\left[\inf_{T\geq
0}f_{\varepsilon}(\mu_{\gamma,T}(\Omega))\right]$
$\displaystyle\geq\mathbf{E}\left[\inf_{T\geq
0}\mathbbm{1}_{\mu_{\gamma,T}(\Omega)\geq
2\varepsilon}\right]-\mathbf{E}\left[\sup_{T\geq
0}\mathbbm{1}_{\mu_{\gamma,T}(\Omega)\leq\varepsilon}\right]$
$\displaystyle=\mathbf{P}\left(\inf_{T\geq 0}\mu_{\gamma,T}(\Omega)\geq
2\varepsilon\right)-\mathbf{P}\left(\inf_{T\geq
0}\mu_{\gamma,T}(\Omega)\leq\varepsilon\right).$

To see the last equality, note that

$\inf_{T\geq 0}\mathbbm{1}_{\mu_{\gamma,T}(\Omega)\geq
2\varepsilon}=1\quad\mbox{if and only if}\quad\inf_{T\geq
0}\mu_{\gamma,T}(\Omega)\geq 2\varepsilon$

and

$\sup_{T\geq
0}\mathbbm{1}_{\mu_{\gamma,T}(\Omega)\leq\varepsilon}=1\quad\mbox{if and only
if}\quad\inf_{T\geq 0}\mu_{\gamma,T}(\Omega)\leq\varepsilon.$

Letting $\varepsilon\to 0$ in the last display and noting that
$\mathbf{P}(\inf_{T\geq
0}\mu_{\gamma,T}(\Omega)=0)=\mathbf{P}(\mu_{\gamma}(\Omega)=0)=0$, we conclude
that

$\liminf_{\varepsilon\to 0}\mathbf{E}\left[\inf_{T\geq
0}f_{\varepsilon}(\mu_{\gamma,T}(\Omega))\right]\geq\mathbf{P}\left(\inf_{T\geq
0}\mu_{\gamma,T}(\Omega)\geq 0\right)-\mathbf{P}\left(\inf_{T\geq
0}\mu_{\gamma,T}(\Omega)=0\right)=1.$

Thus, for $\varepsilon>0$ small enough, and recalling the computations from
(6.5), we conclude that

$\mathbf{P}(\mu_{\gamma,T}(\Omega)>u\varepsilon)\geq\frac{1}{2}\mathbf{P}(\tau\leq
T)=\frac{1}{2}\mathbf{P}(M_{T}>u),$

which is (6.4). ∎

Now we are ready to give the

###### Proof of Theorem 2.7.

Set $\gamma<\gamma_{c}$. For some fixed $u>1$, to be determined later, we use
again the stopping time $\tau=\tau_{u}$ defined in (6.1). For any $p>1$ and
$T>0$,

$\displaystyle\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p}]$
$\displaystyle=\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p},\tau>T]+\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p},\tau\leq
T]$ (6.6) $\displaystyle\leq
u^{p}+u^{p}\mathbf{E}\left[\left(\frac{\mu_{\gamma,T(\Omega)}}{\mu_{\gamma,\tau}(\Omega)}\right)^{p},\tau\leq
T\right]$ $\displaystyle\leq u^{p}+u^{p}\mathbf{P}(\tau\leq
T)\mathbf{E}\left[\mu_{\gamma,T}(\Omega)^{p}\right],$

where in the last line we used Lemma 6.2. By Lemma 6.1,

$\mathbf{E}[M_{\infty}]=\int_{0}^{\infty}\mathbf{P}(M_{\infty}>u){\rm
d}u=1+\int_{1}^{\infty}\mathbf{P}(M_{\infty}>u){\rm d}u<\infty,$

so that there exists some $u>1$ satisfying
$\mathbf{P}(M_{\infty}>u)\leq\frac{1}{2u}$. Since

$\mathbf{P}(\tau\leq
T)=\mathbf{P}(M_{T}>u)\leq\mathbf{P}(M_{\infty}>u)\leq\frac{1}{2u},$

we deduce from (6.6) the upper bound

$\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p}]\leq
u^{p}+\frac{u^{p-1}}{2}\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p}].$

If we choose $p>1$ satisfying $u^{p-1}<2$, we conclude that for all $T>0$,

$\mathbf{E}[\mu_{\gamma,T}(\Omega)^{p}]\leq\frac{2u^{p}}{(2-u^{p-1})}.$

We turn to the proof of (2.5) and (2.6). To show (2.5), we use the stopping
time $\tau=\tau_{1/u}$ for $u>1$ (recall (6.1)). Proceeding as in (6.6),
noting that $x\mapsto x^{-q}$ is convex on $(0,\infty)$ and using Lemma 6.2,
we have

$\displaystyle\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q}]$
$\displaystyle=\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q},\tau>T]+\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q},\tau\leq
T]$ (6.7) $\displaystyle\leq
u^{q}+u^{q}\mathbf{E}\left[\left(\frac{\mu_{\gamma,T(\Omega)}}{\mu_{\gamma,\tau}(\Omega)}\right)^{-q},\tau\leq
T\right]$ $\displaystyle\leq u^{q}+u^{q}\mathbf{P}(\tau\leq
T)\mathbf{E}\left[\mu_{\gamma,T}(\Omega)^{-q}\right].$

Since $\mathbf{P}(\tau\leq T)\leq\mathbf{P}(\inf_{T\geq 0}\mu_{\gamma,T}\leq
u^{-1})\to 0$ as $u\to\infty$, we infer

$\sup_{T\geq 0}\mathbf{P}(\tau\leq T)\leq\frac{1}{4}$

for $u>1$ large enough. If $q\in(0,1)$ is chosen so that $u^{q}\leq 2$, from
(6.7) we conclude that

$\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q}]\leq
2+\frac{\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q}]}{2},$

and hence

$\sup_{T\geq 0}\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q}]\leq 4.$

This finishes the proof of (2.5). To show (2.6), we appeal to [8, Theorem
1.3], where it was shown that in the “$L^{2}$-region” (i.e., when the
martingale $(\mu_{\gamma,T}(\Omega))_{T}$ is bounded in $L^{2}(\mathbf{P})$),
for all $q\in(0,\infty)$,

$\sup_{T\geq 0}\mathbf{E}[\mu_{\gamma,T}(\Omega)^{-q}]<\infty.$

Since Theorem 2.1 holds in the entire weak disorder regime (and therefore, in
particular in the $L^{2}$ region), the above estimate also implies (2.6) for
all negative $q$ for $L^{2}$ disorder. ∎

##  7\. Proof of Theorem 2.9

Recall the definition of $(\Omega_{0},\|\cdot\|)$ defined in Section 2.4. We
will need the following estimate valid for the Wiener measure $\mathbb{P}_{0}$
on $\Omega_{0}$:

######  Proposition 7.1.

[19, Theorem 1.4] Let $j_{\frac{d-2}{2}}$ be the smallest positive root of the
Bessel function $J_{\frac{d-2}{2}}$. If $g$ satisfies (2.8), then

$\lim_{\varepsilon\to
0}\varepsilon^{2}\log\mathbb{P}_{0}\left(\|\omega\|<\varepsilon\right)=-\frac{j^{2}_{\frac{d-2}{2}}}{2}\int_{0}^{\infty}g^{-2}(t){\rm
d}t.$ (7.1)

Additionally, we need an explicit formula for the free energy
$\lim_{T\to\infty}\frac{1}{T}\log\mu_{\gamma,T}(\Omega)$ for all $\gamma>0$.
To describe this formula, we need some further definitions. Denote by
${\mathcal{M}}_{1}={{\mathcal{M}}_{1}}(\mathbb{R}^{d})$ (resp.
${\mathcal{M}}_{\leq 1}$) the space of probability (resp. sub-probability)
measures on $\mathbb{R}^{d}$, which acts as an additive group of translations
on these spaces. Let
$\widetilde{\mathcal{M}}_{1}={\mathcal{M}}_{1}\big{/}\sim$ be the quotient
space of ${\mathcal{M}}_{1}$ under this action, that is, for any
$\mu\in{\mathcal{M}}_{1}$, its orbit is defined by
$\widetilde{\mu}=\\{\mu\star\delta_{x}\colon\,x\in\mathbb{R}^{d}\\}\in\widetilde{\mathcal{M}}_{1}$.
The quotient space $\widetilde{\mathcal{M}}_{1}$ can be embedded in a larger
space

$\widetilde{\mathcal{X}}:=\bigg{\\{}\xi:\xi=\\{\widetilde{\alpha}_{i}\\}_{i\in
I},\alpha_{i}\in\mathcal{M}_{\leq 1},\sum_{i\in
I}\alpha_{i}(\mathbb{R}^{d})\leq 1\bigg{\\}},$

which consists of all empty, finite or countable collections of orbits from
$\widetilde{\mathcal{M}}_{\leq 1}$ whose masses add up to at most one. The
space $\widetilde{\mathcal{X}}$ and a metric structure there was introduced in
[24], and it was shown that under that metric,
$\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ is densely embedded in
$\widetilde{\mathcal{X}}$ and any sequence in
$\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ converges along some subsequence
to an element $\xi$ of $\widetilde{\mathcal{X}}$ – that is,
$\widetilde{\mathcal{X}}$ is the compactification of the quotient space
$\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$, see Appendix A for details. On
the space $\widetilde{\mathcal{X}}$ we define an energy functional

$\displaystyle
F_{\gamma}(\xi)=\frac{\gamma^{2}}{2}\sum_{\widetilde{\alpha}\in\xi}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}(\phi\star\phi)(x_{1}-x_{2})\prod_{j=1}^{2}\alpha(\mathrm{d}x_{j})\quad\forall\,\xi\in\widetilde{\mathcal{X}},\quad\mbox{and}$
(7.2)
$\displaystyle{\mathcal{E}}_{F_{\gamma}}(\vartheta)=\int_{\widetilde{\mathcal{X}}}F_{\gamma}(\xi)\,\vartheta({\rm
d}\xi)\qquad\vartheta\in{\mathcal{M}}_{1}(\widetilde{\mathcal{X}}).$

Here, ${\mathcal{M}}_{1}(\widetilde{\mathcal{X}})$ denotes the space of
probability measures on the space $\widetilde{\mathcal{X}}$. There is an
interesting connection between the structure of the space
$\widetilde{\mathcal{X}}$ and the solution of the variational problem
$\sup_{\vartheta\in{\mathcal{M}}_{1}(\widetilde{\mathcal{X}})}{\mathcal{E}}_{F_{\gamma}}(\vartheta)$:
indeed, (i) there is a non-empty, compact subset
${\mathfrak{m}_{\gamma}}\subset{\mathcal{M}}_{1}(\widetilde{\mathcal{X}})$
consisting of the maximizer(s) of the variational problem
$\sup_{\vartheta\in{{\mathfrak{m}_{\gamma}}}}{\mathcal{E}}_{F_{\gamma}}(\vartheta)$,(ii)
the maximizing set is a singleton
$\delta_{\tilde{0}}\in\widetilde{\mathcal{X}}$ if $d\geq 3$ and small enough
$\gamma$ (in particular, this is true if $\gamma\in(0,\gamma_{c})$) and (iii)
any maximizer assigns positive mass only to those elements of the
compactification $\widetilde{\mathcal{X}}$ whose total mass add up to one, see
Proposition A.4 for details.

######  Theorem 7.2.

[5, Eq. (4.12)] For any $d\in\mathbb{N}$ and $\gamma>0$, it holds that

$\lim_{T\to\infty}\frac{1}{T}\log\mu_{\gamma,T}(\Omega)=\lambda(\gamma):=-\sup_{\vartheta\in{{\mathfrak{m}_{\gamma}}}}{\mathcal{E}}_{F_{\gamma}}(\vartheta),$
(7.3)

where $\mathfrak{m}_{\gamma}$ is the compact subset of
$\mathcal{M}_{1}(\widetilde{\mathcal{X}})$ defined in (A.8). Furthermore,
$\lambda(\gamma)=0$ if $d\geq 3$ and $\gamma\in(0,\gamma_{c})$.

######  Remark 7.3.

It is known that for any $d\geq 1$, $\gamma\mapsto\lambda(\gamma)$ is non-
decreasing and continuous in $(0,\infty)$. Moreover, $\lambda(\gamma)>0$
implies that $\lim_{T\to\infty}\mu_{\gamma,T}(\Omega)=0$. Thus, if we set
$\gamma_{1}:=\inf\\{\gamma>0\colon\lambda(\gamma)>0\\}$, then
$\gamma_{1}\geq\gamma_{c}$ (recall Remark 3.6), and in fact equality is
conjectured to be true.

###### Proof of Theorem 2.9 .

Proof of Part (i): since $\gamma<\gamma_{c}$, we know that $\mathbf{P}$-a.s.,

$\varepsilon^{2}\log\widehat{\mu}_{\gamma}(\|\omega-\eta\|<r\varepsilon)=\varepsilon^{2}\log\mu_{\gamma}(\|\omega-\eta\|<r\varepsilon)-\varepsilon^{2}\log\mu_{\gamma}(\Omega)\qquad\forall\eta\in\Omega_{0}.$

Hence, when letting $\varepsilon\to 0$, the second term vanishes, so we
consider from now on only the asymptotic behavior of the first term

$\varepsilon^{2}\log\mu_{\gamma}(\|\omega-\eta\|<r\varepsilon).$

Upper bound: We start with the upper bound, uniform on $\eta\in\Omega_{0}$. By
the Markov property,

$\displaystyle\log\mu_{\gamma}(\|\omega-\eta\|<r\varepsilon)$
$\displaystyle=\log\lim_{s\to\infty}\mu_{\gamma,s+\varepsilon^{-2}}(\|\omega-\eta\|<r\varepsilon)$
(7.4) $\displaystyle\leq\log\lim_{s\to\infty}\mathbb{E}_{0}\left[{\rm
e}^{\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{\gamma^{2}}{2}\varepsilon^{-2}\phi\star\phi(0)}\mu_{\gamma,s}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\mathbbm{1}_{\|\omega-\eta\|_{\varepsilon}<r\varepsilon}\right]$
$\displaystyle\leq\log\mathbb{E}_{0}\left[{\rm e}^{\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{\gamma^{2}}{2}\varepsilon^{-2}\phi\star\phi(0)}\mu_{\gamma}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\mathbbm{1}_{\|\omega-\eta\|_{\varepsilon}<r\varepsilon}\right],$

where

$\|f\|_{\varepsilon}:=\sup_{0<t<\varepsilon^{-2}}\frac{|f(t)}{g(t)}.$ (7.5)

Since $\gamma<\gamma_{c}$, by Part (i) of Theorem 2.7,

$\mu_{\gamma}(\Omega)\in L^{p_{0}}(\mathbf{P})\qquad\mbox{for some }p_{0}>1.$
(7.6)

For any $1<p\leq p_{0}$, we apply Hölder’s inequality to (7.4), so that

$\displaystyle\log\mu_{\gamma}\big{(}\|\omega-\eta\|<r\varepsilon\big{)}$
$\displaystyle\leq\frac{p-1}{2p}\log\mathbb{E}_{0}\left[{\rm
e}^{\frac{2p}{p-1}\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{p}{p-1}\gamma^{2}\varepsilon^{-2}\phi\star\phi(0)}\right]$
(7.7)
$\displaystyle\qquad+\frac{p-1}{2p}\log\mathbb{P}_{0}(\|\omega-\eta\|_{\varepsilon}<r\varepsilon)$
$\displaystyle\qquad+\frac{1}{p}\log\mathbb{E}_{0}\left[\mu_{\gamma}^{p}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]$
$\displaystyle=\frac{p-1}{2p}\log\mathbb{E}_{0}\left[{\rm
e}^{\frac{2p}{p-1}\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{2p^{2}}{(p-1)^{2}}\gamma^{2}{\varepsilon^{-2}}\phi\star\phi(0)}\right]$
$\displaystyle\qquad+\frac{p-1}{2p}\log\mathbb{P}_{0}(\|\omega-\eta\|_{\varepsilon}<r\varepsilon)$
$\displaystyle\qquad+\frac{1}{p}\log\mathbb{E}_{0}\left[\mu_{\gamma}^{p}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]+\frac{(p+1)}{2(p-1)}\gamma^{2}\varepsilon^{-2}\phi\star\phi(0).$

By Theorem 7.2, we can handle the first term in the last display on the right
hand side above: indeed, for any $\gamma>0$ and $\mathbf{P}$-a.s. we have

$\frac{p-1}{2p}\,\,\lim_{\varepsilon\to
0}\varepsilon^{2}\log\mathbb{E}_{0}\left[{\rm e}^{\frac{2p}{p-1}\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{2p^{2}}{(p-1)^{2}}\gamma^{2}{\varepsilon^{-2}}\phi\star\phi(0)}\right]=\frac{p-1}{2p}\lambda\left(\frac{2p}{p-1}\gamma\right).$
(7.8)

We bound the second term in the last display of the right hand side of (7.7)
as follows. By Anderson’s inequality (see (B.1) in Theorem B.1), for any
$\eta\in\Omega_{0}$,

$\mathbb{P}_{0}(\|\omega-\eta\|_{\varepsilon}<r\varepsilon)\leq\mathbb{P}_{0}(\|\omega\|_{\varepsilon}<r\varepsilon).$

Thus, an application of Proposition 7.1 leads to

$\limsup_{\varepsilon\to
0}\sup_{\eta\in\Omega_{0}}\frac{p-1}{2p}\varepsilon^{2}\log\mathbb{P}_{0}(\|\omega-\eta\|_{\varepsilon}<r\varepsilon)\leq-\frac{(p-1)}{4pr^{2}}j^{2}_{\frac{d-2}{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t.$ (7.9)

Thus, we concentrate on the third term of the last display of the (r.h.s.) of
(7.7). Set

$f:=\mathbb{E}_{0}\big{[}\mu_{\gamma}^{p}(\Omega)\circ\theta_{0,\omega_{\varepsilon^{-2}}}\big{]},\quad\mbox{so
that}\quad\mathbb{E}_{0}\big{[}\mu_{\gamma}^{p}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\big{]}=f\circ\theta_{\varepsilon^{-2},0}.$
(7.10)

Since $\dot{B}$ is stationary with respect to space-time shifts
$(\theta_{t,x})_{t>0,x\in\mathbb{R}^{d}}$ and
$\mathbf{E}[f]=\mathbf{E}[\mu_{\gamma}(\Omega)^{p}]<\infty$, we deduce by the
ergodic theorem for stationary processes1 ††1Here we invoke the ergodic
theorem for general stationary processes which implies that the time averages
of a stationary process evaluated w.r.t. a $L^{1}$ function $f$ converge
almost surely to the conditional expectation of $f$ w.r.t. the invariant
$\sigma$-algebra. In the present context, the function (7.10) is in
$L^{1}(\mathbf{P})$, therefore the conditional expectation
$C(\dot{B})<+\infty$ almost surely. Moreover, for $\gamma\in(0,\gamma_{c})$,
$\mu_{\gamma}(\Omega)>0$ almost surely w.r.t. $\mathbf{P}$, therefore
$C(\dot{B})>0$ almost surely, hence $\log C(\dot{B})<\infty$, implying (7.11).
that there is a (possibly random) $C=C(\dot{B})$ such that for all
$\varepsilon>0$ sufficiently small,

$\mathbb{E}_{0}\left[\mu_{\gamma}^{p}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]\leq
C\varepsilon^{-2}\qquad\mathbf{P}\,\textrm{-a.s.},$

and consequently,

$\limsup_{\varepsilon\to
0}\varepsilon^{2}\log\mathbb{E}_{0}\left[\mu_{\gamma}^{p}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]\leq
0\qquad\mathbf{P}\textrm{-a.s.}$ (7.11)

Combining (7.7)-(7.11), we obtain

$\displaystyle\limsup_{\varepsilon\to
0}\varepsilon^{2}\log\widehat{\mu}_{\gamma}(\|\omega\|<r\varepsilon)$ (7.12)
$\displaystyle\leq\inf_{1<p\leq
p_{0}}\left[\frac{(p+1)}{2(p-1)}\gamma^{2}\phi\star\phi(0)-\frac{(p-1)}{4pr^{2}}j^{2}_{\frac{d-2}{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{p-1}{2p}\lambda\left(\frac{2p}{p-1}\gamma\right)\right]$
$\displaystyle=:-C_{1}(d,\gamma,g,r),$

and observe that $C_{1}>0$ if we choose $r$ small enough.

Lower bound: Recall the norm $\|\cdot\|_{\varepsilon}$ from (7.5). For a fixed
$\gamma<\gamma_{c}$, we apply part (ii) of Theorem 2.7 to find some $q_{0}>0$
such that for all $q>q_{0}$,

$\mathbf{E}[\mu_{\gamma}(\Omega)^{-q}]<\infty.$ (7.13)

By the Markov property and the reverse Hölder’s inequality,2††2 Recall that
Hölder’s inequality also implies that for $\theta>1$ and measurable functions
$F,G$ satisfying $\|FG\|_{1}<\infty$ and $\|G\|_{-\frac{1}{\theta-1}}<\infty$,
then $\|FG\|_{1}\geq\|F\|_{\frac{1}{\theta}}\|G\|_{-\frac{1}{\theta-1}}$. To
deduce (7.14), we apply this inequality for $\theta=1+q$.

$\displaystyle\mu_{\gamma}(\|\omega\|_{\varepsilon}<r\varepsilon)$
$\displaystyle=\mathbb{E}_{0}\left[\left({\rm e}^{\gamma
H_{\varepsilon^{-2}}-\frac{\gamma^{2}}{2}\varepsilon^{-2}\phi\star\phi(0)}\mathbbm{1}_{\|\omega\|_{\varepsilon}<r\varepsilon}\right)~{}\left(\mu_{\gamma}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right)\right]$
(7.14) $\displaystyle\geq\mathbb{E}_{0}\left[{\rm
e}^{\frac{q\gamma}{q+1}H_{\varepsilon^{-2}}-\frac{\gamma^{2}q}{2(q+1)}\varepsilon^{-2}\phi\star\phi(0)}\mathbbm{1}_{\|\omega\|_{\varepsilon}<r\varepsilon}\right]^{1+\frac{1}{q}}\mathbb{E}_{0}\left[\mu_{\gamma}(\Omega)^{-q}\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]^{-\frac{1}{q}}.$

Thus,

$\displaystyle\varepsilon^{2}\log\mu_{\gamma}(\|\omega\|_{\varepsilon}<r\varepsilon)$
$\displaystyle\geq\varepsilon^{2}\left(\frac{q+1}{q}\right)\log\mathbb{E}_{0}\left[{\rm
e}^{\frac{q\gamma}{q+1}H_{\varepsilon^{-2}}-\frac{q\gamma^{2}}{2(q+1)}\varepsilon^{-2}\phi\star\phi(0)}\mathbbm{1}_{\|\omega\|_{\varepsilon}<r\varepsilon}\right]$
(7.15)
$\displaystyle-\frac{1}{q}\varepsilon^{2}\log\mathbb{E}_{0}\left[\mu_{\gamma}(\Omega)^{-q}\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]$
$\displaystyle=-\frac{\gamma^{2}}{2(q+1)}\phi\star\phi(0)+\varepsilon^{2}\left(\frac{q+1}{q}\right)\log\mu_{\frac{q\gamma}{q+1},{\varepsilon^{-2}}}(\|\omega\|_{\varepsilon}<r\varepsilon)$
$\displaystyle\qquad-\frac{1}{q}\varepsilon^{2}\log\mathbb{E}_{0}\left[\mu_{\gamma}(\Omega)^{-q}\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right].$

In [5, Theorem 2.2], it was shown that for all $\gamma>0$,

$\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma,\varepsilon^{-2}}(\|\omega\|_{\varepsilon}<r\varepsilon)\geq-\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}(\phi\star\phi)(0)+\lambda(\gamma)\right).$

In particular,

$\displaystyle\liminf_{\varepsilon\to
0}\varepsilon^{2}\left(\frac{q+1}{q}\right)\log\mu_{\frac{q\gamma}{q+1},{\varepsilon^{-2}}}(\|\omega\|_{\varepsilon}<r\varepsilon)$
(7.16)
$\displaystyle\geq-\left(\frac{q+1}{q}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)+\lambda\left(\frac{\gamma
q}{q+1}\right)\right)$

Thus, using (7.13), the ergodic theorem and following exactly the same
argument as for deducing (7.11), we obtain that

$\limsup_{\varepsilon\to
0}\frac{1}{q}\varepsilon^{2}\log\mathbb{E}_{0}\left[\mu_{\gamma}(\Omega)^{-q}\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]\leq
0.$ (7.17)

Finally, combining (7.15)-(7.17), it holds $\mathbf{P}$-a.s.

$\displaystyle\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma}(\|\omega\|_{\varepsilon}<r\varepsilon)$
(7.18)
$\displaystyle\geq-\left(\frac{q+1}{q}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)\frac{q^{2}+3q+1}{(q+1)^{2}}+\lambda\left(\frac{\gamma
q}{q+1}\right)\right)$
$\displaystyle\geq-\inf_{q>q_{0}}\left[\left(\frac{q+1}{q}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)\frac{q^{2}+3q+1}{(q+1)^{2}}+\lambda\left(\frac{\gamma
q}{q+1}\right)\right)\right]$ $\displaystyle=:-C_{2}(d,\gamma,g,r).$

Clearly, $C_{2}<\infty$. To extend the argument to the norm $\|\cdot\|$, we
can apply again the reverse Hölder’s inequality to deduce, for any $p>1$,

$\displaystyle\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma}(\|\omega\|<\varepsilon$ $\displaystyle)\geq
p\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma}(\|\omega\|_{\varepsilon}<\varepsilon)-(p-1)\limsup_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma}\left(\sup_{t>\varepsilon^{-2}}\frac{|\omega(t)|}{g(t)}<\varepsilon\right)$
$\displaystyle\geq-pC_{2}-(p-1)\limsup_{\varepsilon\to
0}\varepsilon^{2}\log\mu_{\gamma}\left(\Omega\right)$ $\displaystyle=-pC_{2}.$

As $p>1$ is arbitrary, we conclude the proof of (2.11). To deduce the
constants in (2.12), note that for $\gamma>0$ small enough, one can choose
$p_{0}$ in (7.6) such that $\gamma\frac{2p_{0}}{p_{0}-1}<\gamma_{c}$. In
particular, $\lambda(\gamma\frac{2p_{0}}{p_{0}-1})=0$ and hence, the
calculations that lead to (7.12) imply

$\limsup_{\varepsilon\to
0}\varepsilon^{2}\log\widehat{\mu}_{\gamma}(\|\omega\|<r\varepsilon)\leq\frac{(p_{0}+1)}{2(p_{0}-1)}\gamma^{2}\phi\star\phi(0)-\frac{(p_{0}-1)}{4p_{0}r^{2}}j^{2}_{\frac{d-2}{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t.$ (7.19)

Similarly, for any $\gamma<\gamma_{c}$, if $q_{0}>0$ satisfies $\frac{\gamma
q_{0}}{q_{0}+1}<\gamma_{c}$, then $\lambda(\frac{\gamma q_{0}}{q_{0}+1})=0$
and, the same arguments that led to (7.18) give the lower bound

$\liminf_{\varepsilon\to
0}\varepsilon^{2}\log\widehat{\mu}_{\gamma}(\|\omega\|<r\varepsilon)\geq-\left(\frac{q_{0}+1}{q_{0}}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)\frac{q_{0}^{2}+3q_{0}+1}{(q_{0}+1)^{2}}\right).$
(7.20)

Eqs. (7.19)-(7.20) imply (2.12), finishing the proof of part (i).

Proof of Part (ii): fix $\delta>0$. We explain briefly how to adapt the
previous arguments to obtain the decay estimates that match as $\gamma\to 0$.
For a fixed $r>0$, the lower bound is the same as in (7.18). To estimate the
upper bound, we choose $p>1$ and $q>0$ large enough in such a way that

$\left(\frac{q+1}{q}-\frac{1}{p}\right)\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t<\frac{\delta}{2},$ (7.21)

and let $\gamma^{\prime}$ small enough so that

$(\mu_{\gamma,T}(\Omega))_{T\geq 0}\in
L^{\frac{2p}{p-1}}(\mathbf{P})\quad\mbox{for all }\gamma<\gamma^{\prime}.$

By Hölder’s inequality,

$\displaystyle\varepsilon^{2}\sup_{\eta\in\Omega_{0}}\log\mu_{\gamma}(\|\omega-\eta\|<\varepsilon)$
$\displaystyle\leq\varepsilon^{2}\frac{p-1}{2p}\log\mathbb{E}_{0}\left[{\rm
e}^{\frac{2p}{p-1}\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{2p^{2}}{(p-1)^{2}}\gamma^{2}{\varepsilon^{-2}}\phi\star\phi(0)}\right]$
(7.22)
$\displaystyle\quad+\frac{\varepsilon^{2}}{p}\sup_{\eta\in\Omega_{0}}\log\mathbb{P}_{0}(\|\omega-\eta\|<r\varepsilon)$
$\displaystyle\quad+\varepsilon^{2}\frac{p-1}{2p}\log\mathbb{E}_{0}\left[\mu_{\gamma}^{\frac{2p}{p-1}}(\Omega)\circ\theta_{\varepsilon^{-2},\omega_{\varepsilon^{-2}}}\right]+\frac{(p+1)}{2(p-1)}\gamma^{2}\phi\star\phi(0).$

Next, choose $\gamma^{\prime\prime}$ small enough so that

$\displaystyle\lim_{\varepsilon\to
0}\varepsilon^{2}\frac{p-1}{2p}\log\mathbb{E}_{0}\left[{\rm
e}^{\frac{2p}{p-1}\gamma
H_{\varepsilon^{-2}}(\omega)-\frac{2p^{2}}{(p-1)^{2}}\gamma^{2}{\varepsilon^{-2}}\phi\star\phi(0)}\right]$
$\displaystyle=\frac{p-1}{2p}\lambda\left(\frac{2p}{p-1}\gamma\right)=0$

as well as $\lambda\left(\frac{\gamma q}{q+1}\right)=0$ for all
$\gamma<\gamma^{\prime\prime}$ (such $\gamma^{\prime\prime}$ exists due to
Theorem 7.2). According to (7.9), (7.11) and (7.22), we infer that

$\limsup_{\varepsilon\to
0}\sup_{\eta\in\Omega_{0}}\varepsilon^{2}\sup_{\eta\in\Omega_{0}}\log\mu_{\gamma}(\|\omega\|<\varepsilon)\leq\frac{(p+1)}{2(p-1)}\gamma^{2}\phi\star\phi(0)-\frac{1}{2p}\frac{j^{2}_{\frac{d-2}{2}}}{r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t.$ (7.23)

Thus, if we choose $\gamma^{\prime\prime\prime}$ small enough satisfying

$\frac{(\gamma^{\prime\prime\prime})^{2}}{2}\phi\star\phi(0)\left[\frac{p+1}{p-1}+\frac{q^{2}+3q+1}{q(q+1)}\right]<\frac{\delta}{2},$

and then setting
$\gamma_{\delta}:=\gamma^{\prime}\wedge\gamma^{\prime\prime}\wedge\gamma^{\prime\prime\prime}$,
we deduce from (7.21) and (7.23) that

$\displaystyle C_{2}-C_{1}$
$\displaystyle\leq\left(\frac{q+1}{q}\right)\left(\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{\gamma^{2}}{2}\phi\star\phi(0)\frac{q^{2}+3q+1}{(q+1)^{2}}\right)$
$\displaystyle+\frac{(p+1)}{2(p-1)}\gamma^{2}\phi\star\phi(0)-\frac{1}{2p}j^{2}_{\frac{d-2}{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t$
$\displaystyle=\left(\frac{q+1}{q}-\frac{1}{p}\right)\frac{j^{2}_{\frac{d-2}{2}}}{2r^{2}}\int_{0}^{\infty}g^{-2}(t){\rm
d}t+\frac{(\gamma^{\prime\prime\prime})^{2}}{2}\phi\star\phi(0)\left[\frac{p+1}{p-1}+\frac{q^{2}+3q+1}{q(q+1)}\right]$
$\displaystyle<\delta$

for all $\gamma<\gamma_{\delta}$, concluding the proof of the theorem.

∎

##  Appendix A The space $\widetilde{\mathcal{X}}$ and its energy functionals

We denote by ${\mathcal{M}}_{1}={{\mathcal{M}}_{1}}(\mathbb{R}^{d})$ (resp.,
${\mathcal{M}}_{\leq 1}$) the space of probability (resp., subprobability)
distributions on $\mathbb{R}^{d}$ and by
$\widetilde{\mathcal{M}}_{1}={\mathcal{M}}_{1}\big{/}\sim$ the quotient space
of ${\mathcal{M}}_{1}$ under the action of $\mathbb{R}^{d}$ (as an additive
group on ${\mathcal{M}}_{1}$), that is, for any $\mu\in{\mathcal{M}}_{1}$, its
orbit is defined by
$\widetilde{\mu}=\\{\mu\star\delta_{x}\colon\,x\in\mathbb{R}^{d}\\}\in\widetilde{\mathcal{M}}_{1}$.
Then we define

$\widetilde{\mathcal{X}}=\Big{\\{}\xi:\xi=\\{\widetilde{\alpha}_{i}\\}_{i\in
I},\alpha_{i}\in\mathcal{M}_{\leq 1},\sum_{i\in
I}\alpha_{i}(\mathbb{R}^{d})\leq 1\Big{\\}}$ (A.1)

to be the space of all empty, finite or countable collections of orbits of
subprobability measures with total masses bounded by 1. Note that the quotient
space $\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ is embedded in
$\widetilde{\mathcal{X}}$ – that is, for any
$\mu\in{\mathcal{M}}_{1}(\mathbb{R}^{d})$,
$\widetilde{\mu}\in\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ and the single
orbit element $\\{\widetilde{\mu}\\}\in\widetilde{\mathcal{X}}$ belongs to
$\widetilde{\mathcal{X}}$ (in this context, sometimes we will write
$\widetilde{\mu}\in\widetilde{\mathcal{X}}$ for
$\\{\widetilde{\mu}\\}\in\widetilde{\mathcal{X}}$).

The space $\widetilde{\mathcal{X}}$ also comes with a metric structure. If for
any $k\geq 2$, $\mathcal{H}_{k}$ is the space of functions
$h:\left(\mathbb{R}^{d}\right)^{k}\rightarrow\mathbb{R}$ which are invariant
under rigid translations and which vanish at infinity, we define, for any
$h\in\mathcal{H}=\bigcup_{k\geq 2}\mathcal{H}_{k}$, the functionals

$\mathscr{J}(h,\xi)=\sum_{\widetilde{\alpha}\in\xi}\int_{(\mathbb{R}^{d})^{k}}h(x_{1},\ldots,x_{k})\alpha({\rm
d}x_{1})\cdots\alpha({\rm d}x_{k}).$ (A.2)

A sequence $\xi_{n}$ is said to converge to $\xi$ in the space
$\widetilde{\mathcal{X}}$ if

$\mathscr{J}(h,\xi_{n})\to\mathscr{J}(h,\xi)\qquad\forall\,\,h\in\mathcal{H}.$

This leads to the following definition of the metric $\mathbf{D}$ on
$\widetilde{\mathcal{X}}$. For any
$\xi_{1},\xi_{2}\in\widetilde{\mathcal{X}}$, set

$\displaystyle\mathbf{D}(\xi_{1},\xi_{2})$
$\displaystyle=\sum_{r=1}^{\infty}\frac{1}{2^{r}}\frac{1}{1+\lVert
h_{r}\rVert_{\infty}}\bigg{|}\mathscr{J}(h_{r},\xi_{1})-\mathscr{J}(h_{r},\xi_{2})\bigg{|}$
$\displaystyle=\sum_{r=1}^{\infty}\frac{1}{2^{r}}\frac{1}{1+\lVert
h_{r}\rVert_{\infty}}\bigg{|}\sum_{\widetilde{\alpha}\in\xi_{1}}\int
h_{r}(x_{1},...,x_{k_{r}})\prod_{i=1}^{k_{r}}\alpha(\mathrm{d}x_{i})-\sum_{\widetilde{\alpha}\in\xi_{2}}\int
h_{r}(x_{1},...,x_{k_{r}})\prod_{i=1}^{k_{r}}\alpha(\mathrm{d}x_{i})\bigg{|}.$

The following result was proved in [24, Theorem 3.1-3.2].

######  Theorem A.1.

We have the following properties of the space $\widetilde{\mathcal{X}}$.

  * •

$\mathbf{D}$ is a metric on $\widetilde{\mathcal{X}}$ and the space
$\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ is dense in
$(\widetilde{\mathcal{X}},\mathbf{D})$.

  * •

Any sequence in $\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ has a convergent
subsequence with a limit point in $\widetilde{\mathcal{X}}$. Thus,
$\widetilde{\mathcal{X}}$ is the completion and the compactification of the
totally bounded metric space $\widetilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$
under $\mathbf{D}$.

  * •

Let a sequence $(\xi_{n})_{n}$ in $\widetilde{\mathcal{X}}$ consist of a
single orbit $\widetilde{\gamma}_{n}$ and $\mathbf{D}(\xi_{n},\xi)\to 0$ where
$\xi=(\widetilde{\alpha}_{i})_{i}\in\widetilde{\mathcal{X}}$ such that
$\alpha_{1}(\mathbb{R}^{d})\geq\alpha_{2}(\mathbb{R}^{d})\geq\dots$. Then
given any $\varepsilon>0$, we can find $k\in\mathbb{N}$ such that
$\sum_{i>k}\alpha_{i}(\mathbb{R}^{d})<\varepsilon$ and we can write
$\gamma_{n}=\sum_{i=1}^{k}\alpha_{n,i}+\beta_{n}$, such that

    * –

for any $i=1,\dots,k$, there is a sequence
$(a_{n,i})_{n}\subset\mathbb{R}^{d}$ satisfying

$\displaystyle\alpha_{n,i}\star\delta_{a_{n,i}}\Rightarrow\alpha_{i}\quad\mbox{with}\quad\lim_{n\to\infty}\,\inf_{i\neq
j}\,\,|a_{n,i}-a_{n,j}|=\infty.$

    * –

The sequence $\beta_{n}$ totally disintegrates, meaning that for any $r>0$,
$\sup_{x\in\mathbb{R}^{d}}\beta_{n}\big{(}B_{r}(x)\big{)}\to 0$.∎

Recall the definition of
$F_{\gamma}:\widetilde{\mathcal{X}}\rightarrow\mathbb{R}$ from (7.2):

$\displaystyle F_{\gamma}(\xi)=\frac{\gamma^{2}}{2}\sum_{i\in
I}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}V(x_{1}-x_{2})\prod_{j=1}^{2}\alpha_{i}(\mathrm{d}x_{j}),\qquad\xi=(\widetilde{\alpha}_{i})_{i\in
I}.$ (A.3)

Because of shift-invariance of the integrand in (A.3), $F_{\gamma}$ is well-
defined on $\widetilde{\mathcal{X}}$. Moreover, we have

######  Lemma A.2.

$F_{\gamma}$ is continuous and non-negative on $\widetilde{\mathcal{X}}$, and
$F_{\gamma}(\cdot)\leq\frac{\gamma^{2}}{2}V(0)$.

###### Proof.

For the continuity of $F_{\gamma}$, we refer to [24, Corollary 3.3]. Recall
that $V=\phi\star\phi$ and $\phi$ is rotationally symmetric. Hence, for any
$\alpha\in{\mathcal{M}}_{\leq 1}(\mathbb{R}^{d})$, by Cauchy-Schwarz
inequality,

$\displaystyle\int_{\mathbb{R}^{2d}}$ $\displaystyle
V(x_{1}-x_{2})\,\alpha({\rm d}x_{1})\,\alpha({\rm
d}x_{2})=\int_{\mathbb{R}^{2d}}\alpha({\rm d}x_{1})\alpha({\rm
d}x_{2})\,\int_{\mathbb{R}^{d}}{\rm d}z\,\phi(x_{1}-z)\,\phi(x_{2}-z)$ (A.4)
$\displaystyle\leq\int_{\mathbb{R}^{2d}}\alpha({\rm d}x_{1})\alpha({\rm
d}x_{2})\,\bigg{[}\int_{\mathbb{R}^{d}}{\rm
d}z\phi^{2}(x_{1}-z)\bigg{]}^{1/2}\,\,\bigg{[}\int_{\mathbb{R}^{d}}{\rm
d}z\phi^{2}(x_{2}-z)\bigg{]}^{1/2}\leq\alpha\big{(}\mathbb{R}^{d}\big{)}^{2}\|\phi\|_{2}^{2}.$

Thus, $F_{\gamma}(\xi)\leq\frac{\gamma^{2}(\phi\star\phi)(0)}{2}\sum_{i\in
I}(\alpha_{i}(\mathbb{R}^{d}))^{2}\leq\frac{\gamma^{2}(\phi\star\phi)(0)}{2}$
since, for $\xi=(\widetilde{\alpha}_{i})_{i\in I}\in\widetilde{\mathcal{X}}$,
we have $\sum_{i\in I}\alpha_{i}(\mathbb{R}^{d})\leq 1$. Moreover, since
$V=\phi\star\phi$ is non-negative, also ${F_{\gamma}}(\cdot)\geq 0$. ∎

For any $\alpha\in{\mathcal{M}}_{\leq}(\mathbb{R}^{d})$, let

$\mathscr{G}_{t}(\alpha)=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\alpha(\mathrm{d}z)\mathbb{E}_{z}\bigg{[}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}{\\{\omega_{t}\in\mathrm{d}x\\}}\,\exp\bigg{\\{}\gamma\mathscr{H}_{t}(\omega)-\frac{\gamma^{2}}{2}tV(0)\bigg{\\}}\bigg{]},$

and note that for any $a\in\mathbb{R}^{d}$ and $t>0$,
$\mathscr{G}_{t}(\alpha_{i})\overset{{\scriptscriptstyle{({d}})}}{=}\mathscr{G}_{t}(\alpha_{i}\star\delta_{a})$.
Hence, we may define
$\mathscr{G}_{t},\overline{\mathscr{G}}_{t}:\widetilde{\mathcal{X}}\to\mathbb{R}$
as

$\displaystyle\begin{split}&\qquad\mathscr{G}_{t}(\xi)=\sum_{i}\mathscr{G}_{t}(\alpha_{i}),\qquad\quad\overline{\mathscr{G}}_{t}(\xi)=\mathscr{G}_{t}(\xi)+\mathbb{E}\big{[}\mathscr{Z}_{t}-\mathscr{G}_{t}(\xi)\big{]}\qquad\forall\xi=(\widetilde{\alpha}_{i})_{i\in
I}\in\widetilde{\mathcal{X}}.\\\ \end{split}$ (A.5)

Next, for any $t>0$, and for
$\xi=(\widetilde{\alpha}_{i})_{i}\in\widetilde{\mathcal{X}}$, we set

$\displaystyle\alpha_{i}^{{\scriptscriptstyle{({t}})}}(\mathrm{d}x):=\frac{1}{\overline{\mathscr{G}}_{t}(\xi)}\int_{\mathbb{R}^{d}}\alpha_{i}(\mathrm{d}z)\mathbb{E}_{z}\bigg{[}{\mathchoice{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.0mu\mathrm{l}}{1\mskip-4.5mu\mathrm{l}}{1\mskip-5.0mu\mathrm{l}}}{\\{\omega_{t}\in\mathrm{d}x\\}}\,\exp\big{\\{}\gamma\mathscr{H}_{t}(\omega)-\frac{\gamma^{2}}{2}tV(0)\big{\\}}\bigg{]},$
(A.6)
$\displaystyle\xi^{{\scriptscriptstyle{({t}})}}:=\big{(}\widetilde{\alpha}_{i}^{{\scriptscriptstyle{({t}})}}\big{)}_{i\in
I}\in\widetilde{\mathcal{X}}.$

Recall that
$\mathscr{G}_{t}(\alpha_{i})\overset{{\scriptscriptstyle{({d}})}}{=}\mathscr{G}_{t}(\alpha_{i}\star\delta_{a})$
and likewise,
$(\alpha_{i}\star\delta_{a})^{{\scriptscriptstyle{({t}})}}(\mathrm{d}x)\overset{(d)}{=}(\alpha_{i}^{{\scriptscriptstyle{({t}})}}\star\delta_{a})(\mathrm{d}x)$.
For any $\vartheta\in{\mathcal{M}}_{1}(\widetilde{\mathcal{X}})$, then (A.6)
further defines a transition kernel

$\displaystyle\Pi_{t}(\vartheta,{\rm
d}\xi^{\prime})=\int_{\widetilde{\mathcal{X}}}\pi_{t}(\xi,{\rm
d}\xi^{\prime})\vartheta({\rm
d}\xi)\qquad\mbox{where}\quad\pi_{t}(\xi,\mathrm{d}\xi^{\prime})=\mathbb{P}\big{[}\xi^{{\scriptscriptstyle{({t}})}}\in\mathrm{d}\xi^{\prime}|\xi\big{]}\in{\mathcal{M}}_{1}(\widetilde{\mathcal{X}}).$
(A.7)

A discrete version of this transition kernel was constructed in in
$\mathbb{Z}^{d}$ (for discrete time random walks) in [1] on the compactified
space $\widetilde{\mathcal{X}}$.

######  Lemma A.3.

The set

$\mathfrak{m}_{\gamma}=\big{\\{}\vartheta\in\mathcal{M}_{1}(\widetilde{\mathcal{X}}):\Pi_{t}\,\vartheta=\vartheta\text{
for all }t>0\big{\\}}$ (A.8)

of fixed points of $\Pi_{t}$ is a non-empty, compact subset of
${\mathcal{M}}_{1}(\widetilde{\mathcal{X}})$.

###### Proof.

Note that ${\mathfrak{m}_{\gamma}}\neq\emptyset$, because
$\delta_{\widetilde{0}}\in{\mathfrak{m}_{\gamma}}$. Moreover, by the
definition of the metric $\mathbf{D}$ on $\widetilde{\mathcal{X}}$ and by the
resulting convergence criterion determined by Theorem A.1, the map
$\widetilde{\mathcal{X}}\ni\xi\mapsto\pi_{t}(\xi,\cdot)$ is continuous. This
property, together with the compactness of $\widetilde{\mathcal{X}}$ (and
therefore also that of ${\mathcal{M}}_{1}(\widetilde{\mathcal{X}})$), we have
that
${\mathcal{M}}_{1}(\widetilde{\mathcal{X}})\ni\vartheta\mapsto\Pi_{t}(\vartheta,\cdot)$
is continuous too for any $t>0$. It follows that ${\mathfrak{m}_{\gamma}}$ is
a closed subset of the compact metric space
$\mathcal{M}_{1}(\widetilde{\mathcal{X}})$, implying the compactness of
${\mathfrak{m}_{\gamma}}$. ∎

The following proposition provides additional information about the maximizers
of the continuous map $\mathcal{E}_{F_{\gamma}}$, defined in (7.2) in the
compact set $\mathfrak{m}_{\gamma}$. Its proof can be found in [5, Proposition
4.5].

######  Proposition A.4.

Fix $d\in\mathbb{N}$ and $\gamma>0$. Then the supremum (recall (A.8))

$\sup_{\vartheta\in{\mathfrak{m}_{\gamma}}}{\mathcal{E}}_{F_{\gamma}}(\vartheta)=\sup_{\vartheta\in{\mathfrak{m}_{\gamma}}}\,\,\int_{\widetilde{\mathcal{X}}}F_{\gamma}(\xi)\vartheta({\rm
d}\xi)$ (A.9)

is attained, and we always have
$\sup_{{\mathfrak{m}_{\gamma}}}{\mathcal{E}}_{F_{\gamma}}(\cdot)\in[0,\gamma^{2}(\phi\star\phi)(0)/2]$.
Moreover, there exists $\gamma_{1}=\gamma_{1}(d)$ such that $\gamma_{1}>0$ if
$d\geq 3$, and if $\gamma\in(0,\gamma_{1}]$, then
${\mathfrak{m}_{\gamma}}=\\{\delta_{\widetilde{0}}\\}$ is a singleton
consisting of the Dirac measure at $\widetilde{0}\in\widetilde{\mathcal{X}}$.
Consequently, in this regime,
$\sup_{{\mathfrak{m}_{\gamma}}}{\mathcal{E}}_{F_{\gamma}}(\cdot)=0$. If
$\gamma>\gamma_{1}$, then
$\sup_{{\mathfrak{m}_{\gamma}}}{\mathcal{E}}_{F_{\gamma}}>0$. Finally, if
$\vartheta\in{\mathfrak{m}_{\gamma}}$ is a maximizer of
${\mathcal{E}}_{F_{\gamma}}(\cdot)$ and $\vartheta(\xi)>0$ for
$\xi=(\widetilde{\alpha}_{i})_{i\in I}\in\widetilde{\mathcal{X}}$, then
$\sum_{i\in I}\alpha_{i}(\mathbb{R}^{d})=1$ (i.e., any maximizer of (A.9)
assigns positive mass only to those elements of $\widetilde{\mathcal{X}}$
whose total mass add up to one).

##  Appendix B Gaussian inequalities.

######  Theorem B.1.

Let $E$ be a separable Banach space, with $X$ being a $E$-valued centered
Gaussian with law $\mu$. Then the following hold:

  1. (i)

For any symmetric convex subset $A\subset E$ and $x\in E$,

$\mu(A+x)\leq\mu(A).$ (B.1)

  2. (ii)

If $H_{\mu}$ is the Cameron-Martin space of $\mu$,3††3 Let $E^{\star}$ denote
the dual of $E$, with $S:E^{\star}\to E$ being the map defined via the
integral $S\xi=\int_{E}x\xi(x)\mu(dx)$ for $\xi\in E^{\star}$. Then
$H_{\mu}\subset E$ is the Hilbert space which is the completion of the range
of $S$ under the inner product
$\langle\xi,\xi^{\prime}\rangle_{\mu}:=\int\xi(x)\xi^{\prime}(x)\mu({\rm
d}x)$. This inner product induces a norm $\|\cdot||_{\mu}$ on $H_{\mu}$. then
for any $r>0$, and $\eta\in H_{\mu}$,

${\rm e}^{-\frac{1}{2}\|\eta\|_{\mu}^{2}}\mu(\omega\in E:\|\omega\|\leq
r)\leq\mu(\omega\in E\colon\|\omega-\eta\|\leq r)\leq\mu(\omega\in
E:\|\omega\|\leq r).$ (B.2)

###### Proof.

(B.1) is Anderson’s inequality, which follows from log-concavity of Gaussian
measures (see [4, Thm. 2.8.10], or [20, Thm. 2.13]). The upper bound of (B.2)
follows from (B.1), while the lower bound follows from the Cameron-Martin
formula

$\mu(A-\eta)=\int_{A}\exp\big{(}-\frac{1}{2}\|\eta\|_{\mu}^{2}+\langle\omega,\eta\rangle_{\mu}\big{)}\mu({\rm
d}\omega),\qquad A\subset E,\eta\in H_{\mu},$

as well as Hölder’s inequality and the symmetry of
$\langle\omega,\eta\rangle_{\mu}$ on $\\{\omega\in E:\|\omega\|\leq r\\}$, see
[20, Thm. 3.1]. ∎

Acknowledgement: It is a pleasure to thank Nathanael Berestycki for explaining
his approach [2] and sharing many insights. The research of all three authors
is funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s
Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-
Geometry-Structure.

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