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

Abstract

We construct and study properties of an infinite dimensional analog of Kahane’s theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if $H_T(\omega )$ is a random field defined w.r.t. space-time white noise $\dot{B}$ and integrated w.r.t. Brownian paths in $d\ge 3$, we consider the renormalized exponential $\mu _{\gamma ,T}$, weighted w.r.t. the Wiener measure $\mathbb {P}_0(\textrm{d}\omega )$. We construct the almost sure limit $\mu _\gamma = \lim _{T\rightarrow \infty } \mu _{\gamma ,T}$ in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that

$$\begin{aligned} \mu _\gamma \Big \{\omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T(\phi \star \phi )(0)} \ne \gamma \Big \}=0 \qquad \text{ almost } \text{ surely, } \end{aligned}$$

meaning that $\mu _\gamma$ is supported almost surely only on $\gamma$-thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit $\mu _\gamma$ w.r.t. the mollification scheme $\phi$ in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of $\dot{B}$ under the random rooted measure $\mathbb Q_{\mu _\gamma }(\textrm{d}\dot{B}\textrm{d}\omega )= \mu _\gamma (\textrm{d}\omega ,\dot{B})P(\textrm{d}\dot{B})$ is the same as the law of the distribution $f\mapsto \dot{B}(f)+ \gamma \int _0^\infty \int _{\mathbb {R}^d} f(s,y) \phi (\omega _s-y) \textrm{d}s \textrm{d}y$ under $P \otimes \mathbb {P}_0$. We then determine the fractal properties of the measure around $\gamma$-thick paths: $-C_2 \le \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \log {\widehat{\mu }}_\gamma (\Vert \omega \Vert< \varepsilon ) \le \limsup _{\varepsilon \downarrow 0}\sup _\eta \varepsilon ^2 \log {\widehat{\mu }}_\gamma (\Vert \omega -\eta \Vert < \varepsilon ) \le -C_1$ w.r.t a weighted norm $\Vert \cdot \Vert$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) Hölder exponents which are explicit in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and $L^p$ ($p>1$) moments for the total mass of $\mu _\gamma$ in the weak disorder regime.

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

$$\begin{aligned} \begin{aligned}&\mu _\gamma (\textrm{d}\omega )=\lim _{T\rightarrow \infty } \mu _{\gamma ,T}(\textrm{d}\omega ), \qquad \text{ where }\\&\mu _{\gamma ,T}(\textrm{d}\omega )= \exp \bigg (\gamma H_T(\omega ) - \frac{\gamma ^2}{2} \textbf{E}[H_T^2(\omega )]\bigg ) \mathbb {P}_0(\textrm{d}\omega ). \end{aligned} \end{aligned}$$

(1.1)

Here, $\mathbb {P}_0$ stands for the Wiener measure corresponding to Brownian paths $\omega : [0,\infty ) \mapsto \mathbb {R}^d$, and the Gaussian process $\{H_T(\omega )\}_{\omega \in \Omega }$, indexed by Brownian paths, is driven by a Gaussian space-time white noise $\dot{B}$ (under the probability measure $\textbf{P}$) integrated w.r.t. the Brownian path:

$$\begin{aligned} H_T(\omega )=H_T(\phi ,\dot{B},\omega )= \int _{\mathbb {R}^d}\int _0^T \phi (\omega _s- y) \dot{B}(s,y) \textrm{d}s \textrm{d}y, \end{aligned}$$

(1.2)

Here, $\phi$ is 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 $d\ge 3$ [10, 25]. For this case, it is known that there is a non-trivial constant $\gamma _c\in (0,\infty )$ such that the weak disorder or sub-critical phase is characterized by the uniform integrability of the martingale $\{\mu _{\gamma ,T}(\Omega )\}_{T>0}$ when its almost sure limit $\lim _{T\rightarrow \infty } \mu _{\gamma ,T}(\Omega )$ remains strictly positive. We note that the above definition of $\gamma _c$ is rather implicit and that makes the weak disorder phase $(0,\gamma _c)$ harder to analyze.¹ Given this background, the following questions arise naturally:

• Does the limiting measure $\lim _{T\rightarrow \infty } \mu _{\gamma ,T}=\mu _\gamma$ exist, almost surely w.r.t. $\dot{B}$, and in the entire weak disorder regime $(0,\gamma _c)$?

• Can we identify the support of the limit $\mu _\gamma$ on the path space, and characterize its dependence explicitly in terms of the disorder $\gamma \in (0,\gamma _c)$?

• Is there a way to characterize this limit in terms of the mollification scheme $\phi$?

• How does the measure $\mu _\gamma$ look like in a neighborhood of its support? Phrased differently, can the above construction yield quantitative and tractable (in terms of $\gamma \in (0,\gamma _c)$ and $\phi$) information about the local fractal geometry of $\mu _\gamma$ on “points" close to its support?

The goal of the present article is to answer the above questions. More precisely, our first main result in Theorem 2.1 shows that, for any $d\ge 3$ and in the entire weak disorder regime $\gamma \in (0,\gamma _c)$, the infinite volume GMC measure

$$\begin{aligned} \mu _{\gamma }(\textrm{d}\omega ):=\lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\textrm{d}\omega ) \qquad \qquad \textbf{P}\text{-a.s. } \end{aligned}$$

(1.3)

exists, is non-trivial and non-atomic. This limit, which is taken w.r.t. the topology of weak convergence, also exists if the Gaussian noise $\dot{B}$ is replaced by a random environment with finite exponential moment. Next, also in Theorem 2.1, we identify the support of this limit and show that

$$\begin{aligned} \mu _\gamma \bigg (\omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T(\phi \star \phi )(0)} \ne \gamma \bigg )=0\qquad \textbf{P}\,\text{-a.s. } \end{aligned}$$

(1.4)

That is, for almost every realization of $\dot{B}$ and for any Brownian path $\omega$ sampled according to infinite-volume limit $\mu _\gamma (\cdot )$, the value of the underlying field $H_T(\omega )$ is atypically large – or, for any $\gamma \in (0,\gamma _c)$, every path $\omega$ is $\gamma$-thick w.r.t. $\mu _\gamma$. Consequently, the normalized probability measure ${\widehat{\mu }}_\gamma =\mu _\gamma /\mu _\gamma (\Omega )$ is almost surely singular w.r.t. the base measure $\mathbb {P}_0$.

We next investigate the universality of the limit (1.3) and determine the role of the cut-off $\phi$. Following Kahane’s construction of log-correlated GMC, one expects that, in the current infinite dimensional setup, a well-defined limiting object should not depend so much on the choice of the mollifier. For this purpose, and to emphasize the role of $\phi$, let us write $\mu _{\scriptscriptstyle {\gamma ,H(\phi )}}$ for the infinite-volume limit (which is almost surely a functional of the field $H(\phi )$). In this vein, we first show that a strict uniqueness cannot hold in the current infinite dimensional setup – Proposition 3.12 implies that $\mu _{\scriptscriptstyle {\gamma ,H(\phi )}}(\cdot )\ne \mu _{\scriptscriptstyle {\gamma ,H(\phi ^\prime )}}(\cdot )$ unless $\phi$ and $\phi ^\prime$ are identically equal. Thus, at very small scales, the infinite-volume limit still remembers how the field H was regularized (and in infinite dimensions, such a small dependence is conceivable). Then the question naturally arises if one can determine to what degree the limit $\mu _{\gamma ,H(\phi )}$ depends on the mollifying scheme $\phi$. In this regard, denote by $P= \textbf{P}\dot{B}^{-1}$ the law of the white noise (a probability measure on tempered distributions ${\mathcal {S}}^\prime (\mathbb {R}_+\times \mathbb {R}^d)$). In the third part of Theorem 2.1 we show that $\mu _{\scriptscriptstyle {\gamma ,H(\phi )}}$ is the unique measure such that the distribution of $H_T(\phi )$ under $\mu _\gamma (\textrm{d}\dot{B},\textrm{d}\omega ) P(\textrm{d}\dot{B})$ is the same as the distribution of $H_T(\phi )+T(\phi \star \phi )(0)$ under $P\otimes \mathbb {P}_0$. In other words, the only way to perturb linearly the distribution $\dot{B}$ with the test function

$$\begin{aligned} (s,y)\mapsto \phi (\omega _s-y) \end{aligned}$$

is by using the limiting GMC measure $\mu _{\gamma ,H(\phi )}$. That is, the limit satisfies a “Cameron-Martin equation"

$$\begin{aligned} \mu _{\scriptscriptstyle {\gamma , H(\phi ) + v}} (\textrm{d}\omega )= \textrm{e} ^{v(\omega )} \mu _{\scriptscriptstyle {\gamma ,H(\phi )}}(\textrm{d}\omega ) \end{aligned}$$

(1.5)

for all deterministic $v: \Omega \mapsto \mathbb {R}$ so that the law of $H(\phi )+v$ is absolutely continuous w.r.t. that of $H(\phi )$. In other words, this limiting measure can be thought as a family $\{\mu _{\gamma ,H(\phi )}\}_\phi$, where each member verifies the Cameron-Martin equation (1.5) for a fixed $\phi$, as the field H (being a function of the noise $\dot{B}$) varies. This Cameron-Martin characterization (i.e., validity of (1.5)) is reminiscent of Shamov’s definition [33] of finite-dimensional GMC, see Remark 1. Shamov’s argument shows that, in finite-dimensions and for log-correlated fields, the solution to the Cameron-Martin equation is unique (see also the book by Berestycki and Powell [3, Sec. 3.4] where Shamov’s argument is revisited in a simpler way).

We subsequently deduce fractal properties of $\mu _\gamma$ and deduce the volume decay or Hölder exponents of the normalized GMC measure ${\widehat{\mu }}_\gamma$ – namely, in Theorem 2.2 we show that, for weak disorder, and $\textbf{P}$-almost surely,

$$\begin{aligned} \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \log {\widehat{\mu }}_\gamma \big (\Vert \omega \Vert<\varepsilon \big ) \ge - C_2, \qquad \limsup _{\varepsilon \downarrow 0} \sup _{\eta \in \Omega _0} \varepsilon ^2 \log {\widehat{\mu }}_\gamma \big (\Vert \omega -\eta \Vert <\varepsilon \big ) \le - C_1. \end{aligned}$$

(1.6)

Here, $\Omega _0$ is a subset of the paths carrying a (weighted) norm $\Vert \cdot \Vert$ 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 (1.6) agree in the limit $\gamma \rightarrow 0$ and coincide with the scaling exponents of the Wiener measure $\mathbb {P}_0$ as well. This will also be shown in Theorem 2.2. In Theorem 2.3, we also prove the existence of positive and negative moments of its total mass in the entire weak disorder regime, showing that for all $\gamma \in (0,\gamma _c)$, there is some $p>1$ such that $\mu _\gamma (\Omega )\in L^p(\textbf{P})$ and also $\mu _\gamma (\Omega ) \in L^{-q}$ for some $q>0$.² Moreover, for $\gamma$ even smaller (in the so-called $L^2$-regime), it holds that $\mu _\gamma (\Omega )$ has negative moments of all order. Let us finally mention that, as in the first part of Theorem 2.1, Theorem 2.2 and Theorem 2.3 hold also for continuous directed polymers in random environments with finite exponential moments (i.e., the environment is not required to be Gaussian), while for the second and third parts of Theorem 2.1 we need the Gaussianity of the environment. We also refer to Sect. 2.4 for the main ideas of the 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

$$\begin{aligned} m_{\gamma ,h}(\textrm{d}x)= \exp \bigg [\gamma h(x) - \frac{\gamma ^2}{2} \textbf{E}[h^2(x)]\bigg ]\textrm{d}x. \end{aligned}$$

Here, $\{h(x)\}_{x\in D}$ is a log-correlated centered Gaussian field with $\textbf{E}[h(x) h(y)]=-\log |x-y| + O(1)$. The logarithmic divergence along the diagonal prevents to define h pointwise, and a regularization process becomes necessary to define h, and consequently, the measures $m_{\gamma ,h}$, in a precise sense. Since the work of Kahane [18], there have been very important works in the field by Robert and Vargas [29], Duplantier and Sheffield [15], Shamov [33] and Berestycki [2], who showed that if $(h_\varepsilon )_{\varepsilon \in (0,1)}$ is a suitable approximation of h, then as long as $\gamma \in (0,\sqrt{2d})$,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}m_{\gamma ,h_\varepsilon }(\textrm{d}x) = m_{\gamma ,h}(\textrm{d}x) \quad \text{ weakly } \text{ and } \text{ in }\, \textbf{P}\text{-probability. } \end{aligned}$$

This is the so-called subcritical regime. In the critical/supercritical phase, (i.e.,$\gamma \ge \sqrt{2}$), the measure cannot be constructed as above, but still one can make sense of it; we refer to the works of Duplantier-Rhodes-Sheffield-Vargas [13, 14], Powell [27, 28], Madaule-Rhodes-Vargas [24] and Biskup-Louidor [7] for the theory of critical and supercritical GMC, which is outside the scope of this work.

In the subcritical phase, $m_{\gamma ,h}$ is non-trivial, non-atomic and $\textbf{P}$-a.s. and for a.e. $x\in D$ sampled according to $m_{\gamma ,h}$, it holds that $h_\varepsilon (x) \sim \gamma \log \big (\frac{1}{\varepsilon }\big )$ – that is, almost every point chosen via $\mu _{\gamma ,h}$ is $\gamma$-thick. 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 [16, 31] and [3, Ch. 3.7$-$3.9]). A related geometric feature of $m_{\gamma ,h}$ is captured by the asymptotic volume decay of $\log m_{\gamma ,h}(B_\varepsilon (x))$ as $\varepsilon \downarrow 0$, which is the analogue of our bounds (1.6). This is known as the (uniform) Hölder exponents of $m_{\gamma ,h}$ and it is closely related to its multifractal behavior. Indeed, recall that for $m_{\gamma ,h}$, the uniform Hölder exponent is given by $d(1-\frac{\gamma }{\sqrt{2d}})^2$, while its pointwise Hölder exponent is $d+ {\gamma ^2}/2$ (see [30, Sec. 4.1] for precise definitions). However, none of these exponents fully captures its multifractal spectrum, which roughly says that if a point $x\in D$ is a-thick, then

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

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 $\gamma \rightarrow 0$). We refer to the discussion below (1.6) again to underline the analogy to the Hölder exponents $C_1, C_2$ in our setup.

In summary, it is for the first time, to the best of our knowledge, that the existence, characterization, thick points and the above fractal properties of $\mu _\gamma$ in (1.3) have been established in the infinite-dimensional setup and in the entire weak disorder regime. As mentioned in the above paragraph, for log-correlated Gaussian fields on ${\mathbb {R}}^d$, critical parameters such as $\gamma _c$ and sharp bounds for the positive moments of $\mu _\gamma (\Omega )$ are well-established. In contrast, in our framework most of these parameters as well as the relevant information are qualitative in the entire weak-disorder regime. We underline that our results on the $\gamma$-thick paths, characterization of the measure and the Hölder exponents $C_1,C_2$ provide tractable and quantitative information for weak disorder $\gamma \in (0,\gamma _c)$. Our objective is to leverage the GMC approach to derive quantifiable estimates that, until now, have only been validated in a limited number of integrable models. Let us now turn to the precise mathematical layout of the model.

1.1 The model and notation

For a fixed dimension $d \ge 3$, 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 $({\mathcal {E}}, \mathcal {F}, \textbf{P})$ be a complete probability space so that $\dot{B}$ is a space-time white noise 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

$$\begin{aligned} \textbf{E} \big [ {\dot{B}} (f){\dot{B}} (g) \big ]= & {} \int _{0}^{\infty }\int _{\mathbb {R}^d} f(s,y)g(s,y)\, \textrm{d}y \, \textrm{d}s\nonumber \\= & {} \langle f,g \rangle _{L^2(\mathbb {R}_+ \times \mathbb {R}^d)}, \quad f,g \in {\mathcal {S}}(\mathbb {R}_+ \times \mathbb {R}^d). \end{aligned}$$

(1.7)

Here, we use the notation

$$\begin{aligned} {\dot{B}} (f) = \int _{0}^{\infty }\int _{\mathbb {R}^d} f(s,y){\dot{B}} (s,y)\,\textrm{d}y \, \textrm{d}s \qquad \text{ for } \text{ any }\quad f \in {\mathcal {S}}(\mathbb {R}_+ \times \mathbb {R}^d). \end{aligned}$$

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 $L^2(\textbf{P})$, since we have

$$\begin{aligned} \textbf{E}\big [ ({\dot{B}} (f_n) - {\dot{B}} (f_m))^2 \big ] = \textbf{E}\big [ {\dot{B}} (f_n - f_m)^2 \big ] = \Vert f_n - f_m \Vert _{L^2(\mathbb {R}_{+}\times \mathbb {R}^d)}. \end{aligned}$$

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

$$\begin{aligned} \int _{0}^{\infty } \int _{ \mathbb {R}^d} f(s,y ){\dot{B}} (s,y) \, \textrm{d}y \, \textrm{d}s:= \lim _{n \rightarrow \infty } {\dot{B}} (f_n). \end{aligned}$$

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

$$\begin{aligned} \int _{0}^T \int _{ \mathbb {R}^d} f(s,y ){\dot{B}} (s,y) \, \textrm{d}y \, \textrm{d}s:= \int _{0}^{\infty } \int _{ \mathbb {R}^d} \mathbb {1}_{([0,T]\times \mathbb {R}^d)}(s,y) f(s,y ){\dot{B}} (s,y) \, \textrm{d}y \, \textrm{d}s. \end{aligned}$$

By construction, for $f \in L^2(\mathbb {R}_+\times \mathbb {R}^d)$, the random variable $\int _{0}^{\infty } \int _{ \mathbb {R}^d} f(s,y ){\dot{B}} (s,y) \, \textrm{d}y \, \textrm{d}s$ is Gaussian distributed with mean zero and variance $\Vert f\Vert _{L^2(\mathbb {R}_+\times \mathbb {R}^d)}$ and for $f,g \in L^2(\mathbb {R}_+\times \mathbb {R}^d)$, the covariance $\textbf{E}[\dot{B}(f) \dot{B}(g)]$ is also given by (1.7). 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, a smooth, non-negative, spherically symmetric and compactly supported function $\phi :\mathbb {R}^d \mapsto \mathbb {R}$ such that $\int _{\mathbb {R}^d} \phi (x)\textrm{d}x = 1$. We define the Gaussian field $\left( {H}_T(\omega ) \right) _{\omega \in \Omega }$ as the Itô integral

$$\begin{aligned} {H}_T(\omega ):= \int _{0}^T \int _{ \mathbb {R}^d}\phi (\omega _s -y){\dot{B}} (s,y) \, \textrm{d}y \, \textrm{d}s, \qquad \omega \in \Omega . \end{aligned}$$

(1.8)

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

$$\begin{aligned} \textbf{E}\left[ {H}_T(\omega ) {H}_T(\omega ') \right] {=} \int _{0}^T \int _{ \mathbb {R}^d} \phi \left( \omega _s - y\right) \phi \left( \omega '_s - y\right) \, \textrm{d}y \, \textrm{d}s {=} \int _{0}^T (\phi \star \phi ) \left( \omega _s - \omega '_s\right) \textrm{d}s, \end{aligned}$$

(1.9)

where $(f\star g)(x) = \int _{ \mathbb {R}^d}f(x-y)g(y)\textrm{d}y$. Note that, for any $\omega \in \Omega$, $\textrm{Var}[H_T(\omega )]= T (\phi \star \phi )(0)$ – 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 $\Omega$

$$\begin{aligned} \mu _{\gamma ,T}(\textrm{d}\omega ):= & {} \exp \bigg (\gamma {H}_T(\omega ) - \frac{\gamma ^2}{2} \textrm{Var}(H_T(\omega )) \bigg )\mathbb {P}_0(\textrm{d}\omega )\nonumber \\= & {} \exp \bigg ( \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T(\phi \star \phi )(0) \bigg ) \mathbb {P}_0(\textrm{d}\omega ), \end{aligned}$$

(1.10)

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

$$\begin{aligned} \widehat{\mu }_{\gamma ,T}(A):=\frac{\mu _{\gamma ,T}(A)}{\mu _{\gamma ,T}(\Omega )}, \qquad A\subset \Omega \end{aligned}$$

(1.11)

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,

$$\begin{aligned} \mu _{\gamma ,T}(\Omega ) = \int _{\Omega } \exp \bigg ( \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T(\phi \star \phi )(0) \bigg ) \mathbb {P}_0(\textrm{d}\omega ) \end{aligned}$$

is adapted to the filtration $({\mathcal {F}}_T)_T$ and is a martingale. Our main results will hold in the uniform integrability phase $\gamma \in (0,\gamma _c)$ (also called weak disorder) of the martingale $(\mu _{\gamma ,T}(\Omega ))_T$ determined by the critical value $\gamma _c \in (0,\infty )$ for $d\ge 3$ defined in (3.3). We are now ready to state our main results.

2 Main results

2.1 Existence, support and characterization

For any fixed realization of $\dot{B}$ and $a>0$, we define

$$\begin{aligned} {\mathcal {T}}_a:=\bigg \{\omega \in \Omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T (\phi \star \phi )(0)}= a \bigg \} \end{aligned}$$

to be the (random) set of a-thick paths $\omega \in \Omega$. Here is our first main result:

Theorem 2.1

Fix $d\ge 3$ and $\gamma \in (0, \gamma _c)$. Then the following hold:

• (Existence) There exists a non-trivial measure $\mu _{\gamma }$ on $\Omega$ such that $\textbf{P}$-a.s., $\mu _{\gamma , T}$ converges weakly to $\mu _{\gamma }$ as $T \rightarrow \infty$. Moreover, for any fixed Borel set $A\subset \Omega$ with $\mathbb {P}_0(A)>0$, $\mu _\gamma (A)>0$ $\textbf{P}$-a.s. Similarly, there exists a probability measure $\widehat{\mu }_\gamma$ on $\Omega$ such that $\textbf{P}$-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.

• (Identification of thick points and singularity) Let $\mu _\gamma$ be the limiting measure from above. Then,

  $$\begin{aligned} \mu _\gamma \bigg \{\omega \in \Omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T (\phi \star \phi )(0)} \ne \gamma \bigg \}=0 \qquad \textbf{P}\text{-a.s. } \end{aligned}$$

  (2.1)

  That is, almost surely, any path $\omega \in \Omega$ sampled according to $\mu _\gamma$ is $\gamma$-thick. Consequently, for $\textbf{P}$-a.e. realization of $\dot{B}$, the normalized probability measure ${\widehat{\mu }}_\gamma =\mu _\gamma /\mu _\gamma (\Omega )$ (also defined above) is singular w.r.t. the Wiener measure $\mathbb {P}_0$. In particular, if $\phi$ and $\phi ^\prime$ are two mollifiers, then the limiting measures $\mu _{\gamma ,\phi }$ and $\mu _{\gamma ,\phi ^\prime }$ are singular unless $(\phi \star \phi )(0)=(\phi ^\prime \star \phi ^\prime )(0)$.

• (Characterization) Let $P=\textbf{P}\circ \dot{B}^{-1}$ be the law of the white noise, and for any measure $\nu$ on $\Omega$, which may depend on $\dot{B}$, we write

  $$\begin{aligned} \mathbb {Q}_\nu (\textrm{d}\dot{B}\textrm{d}\omega ):=\nu (\textrm{d}\omega ,\dot{B})P(\textrm{d}\dot{B}). \end{aligned}$$

  (2.2)

  Then the (unnormalized) GMC measure $\mu _\gamma = \mu _{\gamma ,\phi }$ 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

  $$\begin{aligned} {{\dot{B}} }_\phi (f)={\dot{B}} (f)+\gamma \int _{\mathbb {R}_+\times \mathbb {R}^d}f(s,y)\phi (\omega _s-y)\textrm{d}s \textrm{d}y, \qquad f \in {\mathcal {S}}(\mathbb {R}_+\times \mathbb {R}^d) \end{aligned}$$

  under $P\otimes \mathbb {P}_0$.³ In other words, $\mu _{\gamma ,\phi }$ is the unique measure satisfying

  $$\begin{aligned} \mathbb {E}^P\bigg [\int _\Omega \mu _{\gamma ,\phi }(\textrm{d}\omega ) F(\dot{B}, \omega )\bigg ]= \mathbb {E}^{P\otimes \mathbb {P}_0}\big [F(\dot{B}_\phi , \omega )\big ]. \end{aligned}$$

  (2.3)

  for any bounded measurable function $F: \Omega \times {\mathcal {E}} \mapsto \mathbb {R}$ (cf. Remark 1 below).

Remark 1

We comment on the third part of Theorem 2.1 which determines to what extent the limiting measure $\mu _\gamma$ depends on the choice of the mollification $\phi$. Let us first note that, in the infinite-dimensional setup, the measure $\mu _{\gamma ,\phi }$ does depend on the mollification $\phi$ in a strict sense, and this dependence is to be expected – indeed, it will be shown in Proposition 3.12 that for two mollifiers $\phi$ and $\phi ^\prime$, 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 $\phi =\phi ^\prime$. However, this dependence on the approximation procedure is rather small – as shown by the third statement in Theorem 2.1, the only way to perturb linearly the distribution $\dot{B}$ with the test function

$$\begin{aligned} (s,y)\mapsto \phi (\omega _s-y) \end{aligned}$$

is by using the measure $\mu _{\gamma ,\phi }$. In particular,

$$\begin{aligned} \begin{aligned}&\mu _{\gamma ,\phi }, \text{ is } \text{ the } \text{ unique } \text{ measure } \text{ such } \text{ that } \text{ the } \text{ distribution } \text{ of }\, H_T\, \text{ under }\, \mathbb {Q}_{\mu _{\gamma ,\phi }} \\&\qquad \text{ is } \text{ the } \text{ same } \text{ as } \text{ the } \text{ distribution } \text{ of }\, H_T+T(\phi \star \phi )(0)\, \text{ under }\, P\otimes \mathbb {P}_0. \end{aligned} \end{aligned}$$

Note that this characterization of $\mu _\gamma$ is reminiscent of Shamov’s definition of a subcritical GMC [33]. 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 $v:X\mapsto \mathbb {R}$ such that the law of $H+v$ is absolutely continuous w.r.t. that of H, then $\mu _{H+v}(dx)=\textrm{e} ^{v(x)}\mu _{H}(dx)$. The construction of such measure goes via approximating H by a sequence of fields $(H_n)$ such that $H_n\rightarrow H$ 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 $\omega \ne \omega '\in \Omega$,

$$\begin{aligned} \lim _{T\rightarrow \infty }\textbf{E}[H_T(\omega )H_T(\omega ')]=\int _0^\infty (\phi \star \phi )(\omega _s-\omega '_s)\textrm{d}s<\infty \qquad \mathbb {P}_0^{\otimes 2}\text {-a.s.} \quad \square \end{aligned}$$

2.2 Volume decay and Hölder exponents

Recall that by (2.1), $\mu _\gamma$ is supported only $\gamma$-thick paths $\omega \in \Omega$. Our next goal is to determine the asymptotic decay rate of small balls around each such $\gamma$-thick path, that is to determine the behavior of

$$\begin{aligned} \log {\widehat{\mu }}_\gamma (\omega \in \Omega :\Vert \omega \Vert <\varepsilon )\qquad \text{ as }\,\, \varepsilon \downarrow 0. \end{aligned}$$

(2.4)

and compare this decay rate with that of $\log \mathbb {P}_0(\omega \in \Omega :\Vert \omega \Vert <\varepsilon )$ w.r.t. the Wiener measure $\mathbb {P}_0$.

Note that $\Omega =C([0,\infty ); \mathbb {R}^d)$ is not a Banach space.⁴ To define a suitable norm, we will consider a subset $\Omega _0\subset \Omega$ with a weighted norm $\Vert \cdot \Vert _w$ such that $\mathbb {P}_0(\Omega _0)=1$ and that $(\Omega _0, \Vert \cdot \Vert _w)$ is a Banach space. For this purpose, we consider a class of maps $w:(0,\infty )\mapsto (0,\infty ]$ satisfying

$$\begin{aligned} \left( \inf _{0<t<\infty }w(t)\right) \wedge \left( \inf _{0<t<\infty }\frac{w(t)}{t}\right) >0. \end{aligned}$$

(2.5)

For a fixed w as above, we set

$$\begin{aligned} \Omega _0=\Omega _0(w):=\left\{ \omega \in \Omega : \lim _{t\rightarrow \infty }\frac{|\omega _t|}{w(t)}=0\right\} . \end{aligned}$$

(2.6)

Since $\mathbb {P}_0\left( \lim _{t\rightarrow \infty }\frac{|\omega _t|}{t}=0\right) =1$, condition (2.5) assures that $\mathbb {P}_0(\Omega _0)=1$. Moreover, under the norm $\Vert \cdot \Vert _w$, defined as

$$\begin{aligned} \Vert \omega \Vert _w:=\sup _{t>0}\frac{|\omega _t|}{w(t)}\qquad (\text {finite for all } \omega \in \Omega _0\text { by } (2.5)), \end{aligned}$$

(2.7)

$\Omega _0$ is a separable Banach space. From now on, we consider an arbitrary w satisfying (2.5).

The next theorem consists of two parts. In the first one, we will determine the decay exponents (uniform upper and pointwise lower estimates) for (2.4) valid in the whole weak disorder region $\gamma \in (0,\gamma _c)$. In the second one, we will show that, as $\gamma \rightarrow 0$, the upper and lower estimates coincide.

Theorem 2.2

(Volume decay) Fix $d\ge 3$.

• Given $\gamma \in (0,\gamma _c)$ and w satisfying (2.5), there exists $r_0>0$ such that for all $r\in (0,r_0)$, there are explicit constants $0<C_1\le C_2<\infty$ (defined in (4.13) and (4.24)) fulfilling

  $$\begin{aligned} -C_2\le & {} \liminf _{\varepsilon \rightarrow 0}\varepsilon ^2\log \widehat{\mu }_\gamma (\Vert \omega \Vert _w<r\varepsilon ) \nonumber \\\le & {} \limsup _{\varepsilon \rightarrow 0}\varepsilon ^2\sup _{\eta \in \Omega _0}\log \widehat{\mu }_\gamma (\Vert \omega -\eta \Vert _w<r \varepsilon )\le -C_1. \end{aligned}$$

  (2.8)

• Let $p_0>1$ and $q_0>0$. Then if $\gamma$ is small enough, the constants $C_1,C_2$ can be chosen as

  $$\begin{aligned} \begin{aligned} C_1(\gamma ,d,r)&:=\frac{p_0-1}{p_0 }\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\bigg )-\frac{1}{2p_0}\frac{\gamma ^2}{2}(\phi \star \phi )(0),\\ C_2(\gamma ,d,r)&:=\bigg (\frac{q_0+1}{q_0}\bigg )\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\bigg )+\frac{\gamma ^2}{2}(\phi \star \phi )(0), \end{aligned} \end{aligned}$$

  (2.9)

  where $j_{\frac{d-2}{2}}$ is the smallest positive root of the Bessel function $J_{\frac{d-2}{2}}$. $r>0$,

  $$\begin{aligned} \lim _{\gamma \rightarrow 0}C_1(\gamma ,d,r)=\lim _{\gamma \rightarrow 0}C_2(\gamma ,d,r)= & {} \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t \nonumber \\= & {} \lim _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mathbb {P}_0(\Vert \omega \Vert _w<r\varepsilon ). \end{aligned}$$

  (2.10)

  In other words, both exponents $C_1$ and $C_2$ converge as $\gamma \rightarrow 0$ to the volume decay exponent for the Wiener measure.

Remark 2

We note that the upper bound in (2.8) is uniform over shifted balls of a given radius, while the lower bound holds pointwise. In fact, we do not expect a uniform lower bound over $\eta \in \Omega _0$ in Theorem 2.2. This is reflected already in the behavior of Wiener measure $\mathbb {P}_0$ (corresponding to the case $\gamma =0$): for every function in the Cameron-Martin space, defined as the Hilbert space

$$\begin{aligned} H^1:=\big \{\eta \in \Omega _0: \dot{\eta }\in L^2([0,\infty );\mathbb {R}^d)\big \} \end{aligned}$$

with inner product induced by the norm $\Vert \eta \Vert _{H^1}:=\Vert \dot{\eta }\Vert _{L^2([0,\infty );\mathbb {R}^d)}$, we have (see (4.15) in Lemma 4.2)

$$\begin{aligned} \mathbb {P}_0(\Vert \omega -\eta \Vert<r)\ge \mathbb {P}_0(\Vert \omega \Vert <r)\textrm{e} ^{-\frac{\Vert \eta \Vert ^2_{H^1}}{2}}. \end{aligned}$$

That is, we obtain a pointwise lower bound w.r.t. $\mathbb {P}_0$.

Remark 3

We also remark that instead of considering weighted norms on $\Omega _0$, it is also possible to obtain analogous estimates as in Theorem 2.2 on $\Omega$ by considering the sets $\{\omega \in \Omega : \sup _{0\le t\le r}|\omega _t|<\varepsilon \}$ (i.e., by considering balls in the uniform metric in [0, r]) as $\varepsilon \rightarrow 0$.

2.3 Moments of $\mu _\gamma (\Omega )$ in weak disorder

Finally we turn to the positive and negative moments of the total mass $\mu _\gamma (\Omega )$. By the martingale convergence theorem, we always have $\textbf{E}[\mu _\gamma (\Omega )]=1$ for all $\gamma \in (0,\gamma _c)$. Moreover, we also have

Theorem 2.3

(Positive and negative moment) Fix $d\ge 3$ and $\gamma \in (0,\gamma _c)$. Then there is some $p>1$ and $q>0$ such that

$$\begin{aligned}&\mu _\gamma (\Omega )\in L^p(\textbf{P}), \qquad \text{ and } \end{aligned}$$

(2.11)

$$\begin{aligned}&\mu _\gamma (\Omega )\in L^{-q}(\textbf{P}) . \end{aligned}$$

(2.12)

Moreover, if $\gamma \in (0,\gamma _c)$ is chosen so that the martingale $(\mu _{\gamma ,T}(\Omega ))_T$ is bounded in $L^2(\textbf{P})$, then

$$\begin{aligned} \mu _\gamma (\Omega )\in L^{-q}(\textbf{P}) \qquad \forall q\in (0,\infty ). \end{aligned}$$

(2.13)

Remark 4

We can show (2.11) and (2.12) exactly in the same manner if $\dot{B}$ is replaced by a random environment with finite exponential moments. In [11], 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 $L^2(\textbf{P})$ bounded) and for all $q\in (-\infty ,0)$, $(\mu _{\gamma ,T}(\Omega ))_T$ is $L^q(\textbf{P})$ bounded. It remains an open problem to extend (2.13) in the full weak disorder regime $\gamma \in (0,\gamma _c)$. For negative moments of the total mass (or more generally, the tail distribution of the inverse of the partition function) of finite dimensional log-correlated GMC we refer to [16, 29].

2.4 Ideas of the proofs and comparison with literature

As a guidance to the reader, we will now outline the key technical constituents of the proofs of the above results. In this description, we will also underline similarities and differences from existing techniques in the literature.

Step 1 (Weak convergence): For the existence part of 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. This approach goes via employing a second moment method to prove convergence in the $L^2$ region, then using Girsanov’s formula to determine thick points and showing that every Liouville point $x\in D$ is $\gamma$-thick, and finally covering the entire uniform integrability phase by removing points which are thicker than this prescribed value. In the current setup, we approach the uniform integrability phase directly by using martingale properties. The starting point is that, for any Borel set $A\subset \Omega$, $(\mu _{\gamma ,T}(A))_{T\ge 0}$ is a martingale w.r.t. the white noise filtration (see Lemma 3.1 and Lemma 3.2). This guarantees convergence of $\mu _{\gamma ,T}(A)$ $\textbf{P}$-a.s. and in $L^1(\textbf{P})$ to some random variable $\mu _\gamma (A)$ in the uniform integrability phase (the same statement holds for the normalized version ${\widehat{\mu }}_{\gamma ,T}(A)\rightarrow \mu _\gamma (A)/\mu _\gamma (\Omega )$). However, it is not clear that the limiting random variable defines a probability measure, so we cannot derive weak convergence just from this fact. Thus, we first prove that the sequence $({\widehat{\mu }}_{\gamma ,T})_{T\ge 0}$ is tight, in order to obtain a weakly convergent subsequence. For this purpose, first it is necessary to find a countable collection ${\mathcal {X}}$ of Borel sets in $\Omega$ where the almost sure convergence ${\widehat{\mu }}_{\gamma ,T}(A)\rightarrow \mu _\gamma (A)/\mu _\gamma (\Omega )$ takes places simultaneously for all $A\in {\mathcal {X}}$. Using this construction and a continuity property of the limiting random variables $\mu _\gamma (\cdot )$, we then show that $\textbf{P}$-a.s., the family $\{{\widehat{\mu }}_{\gamma ,T}\}_T$ is uniformly tight, and moreover, the weak limit of ${\widehat{\mu }}_{\gamma ,T}$ is uniquely determined. We refer to Proposition 3.8 for details. It might be worth pointing out that this approach is robust in the sense that it does not rely on a $L^2$ computation, and it does not need Gaussianity of the environment. These arguments constitute Sects. 3.1, 3.2, 3.3. See also Remark 5 below for a comparison with existing literature.

Step 2 (Rooted random measures): In the second step for the proof of Theorem 2.1, we work with the measures

$$\begin{aligned} \begin{aligned}&\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}\omega \textrm{d}\dot{B})= \exp \bigg [\gamma \int _0^T\int _{\mathbb {R}^d} \phi (\omega _s-y) \dot{B}(s,y) \textrm{d}s \textrm{d}y - \frac{\gamma ^2 T(\phi \star \phi )(0)}{2}\bigg ] \\&\mathbb {P}_0(\textrm{d}\omega ) P(\textrm{d}\dot{B}), \mathbb {Q}_{\mu _\gamma }(\textrm{d}\omega \textrm{d}\dot{B})=\mu _\gamma (\textrm{d}\omega , \dot{B}) \mathbb {P}_0(\textrm{d}\omega ). \end{aligned} \end{aligned}$$

Using Theorem 2.1, we show that $\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}\omega \textrm{d}\dot{B})$ converges weakly towards $\mathbb {Q}_{\mu _\gamma }(\textrm{d}\omega \textrm{d}\dot{B})$, and apply Girsanov’s theorem to determine, under the conditional measure $\mathbb {Q}_{\mu _\gamma }(\textrm{d}\omega |\dot{B})$, the mean and the variance the Gaussian process $\{\dot{B}(f)\}_{f\in {\mathcal {S}}(\mathbb {R}_+\times \mathbb {R}^d)}$. Combining these two facts, it can be shown that the law of $\dot{B}$ under $\mathbb {Q}_{\mu _\gamma }$ is the same as the law of $\dot{B}(f)+ \int _{\mathbb {R}_+\times \mathbb {R}^d} f(s,y) \phi (\omega _s- y) \textrm{d}s \textrm{d}y$ under $\mathbb {P}_0\otimes P$. Combined with a law of large numbers of the martingale

$$\begin{aligned} \lim _{T\rightarrow \infty } \frac{H_T}{T} = 0 \qquad \text{ almost } \text{ surely } \text{ w.r.t. }\, \textbf{P}, \end{aligned}$$

it can then be shown that

$$\begin{aligned} \lim _{T\rightarrow \infty } \frac{H_T}{T} = \gamma (\phi \star \phi )(0)>0, \qquad \text{ almost } \text{ surely } \text{ w.r.t. }\, \mathbb {Q}_{\mu _\gamma }. \end{aligned}$$

These arguments constitute the second and third parts of the proof of Theorem 2.1 in Sect. 3.4.

Step 3 (Volume estimates): In this step we will show decay rate of ${\widehat{\mu }}_\gamma (\Vert \omega - \eta \Vert _w < \varepsilon )$ of microscopic balls in the weighted norm $\Vert \cdot \Vert _w$ for any $\gamma \in (0,\gamma _c)$. Note that because of uniform integrability, the total mass $\log (\mu _\gamma (\Omega ))>-\infty$ a.s., so it suffices to estimate the unnormalized volume $\mu _\gamma (\Vert \omega - \eta \Vert _w < \varepsilon )$. For the upper bound, we approximate this quantity by $\log \mu _{\gamma , S+ \varepsilon ^{-2}}(\Vert \omega - \eta \Vert _{w} < \varepsilon )$ for any fixed $\varepsilon >0$ and large $S\gg 1$, and use Markov property and Hölder’s inequality to split the latter term into three contributions, namely, the uniform small ball probabilities $\sup _\eta \log \mathbb {P}_0[\Vert \omega - \eta \Vert _{w,\varepsilon } < r \varepsilon ]$ (under a truncated weighted norm $\Vert \cdot \Vert _{w,\varepsilon }$), the uniform space-time translations

$$\begin{aligned} \mathbb {E}_0\big [\sup _{S>0} \, \big (\mu _{\gamma ,S}(\Omega )^p \circ \theta _{\varepsilon ^{-2}, \omega _{\varepsilon ^{-2}}}\big )\big ]\qquad \text{ for } \text{ some } \text{ fixed }\, p>1\, \text{ and } \text{ any }\, \gamma \in (0,\gamma _c) \end{aligned}$$

(2.14)

and then the contribution (up to time $\varepsilon ^{-2}$) from $\mu _{\varphi (\gamma ), \varepsilon ^{-2}}(\Omega )$ for a disorder $\varphi (\gamma )$ depending on the exponents coming from Hölder’s inequality. The first term $\sup _\eta \log \mathbb {P}_0[\Vert \omega - \eta \Vert _{w,\varepsilon } < r \varepsilon ]$ is handled by log-concavity of Gaussian measures and an asymptotic small ball probability in the weighted norm; this provides a rate involving Bessel functions. To handle the second term, we crucially use the space-time stationarity of the white noise $\dot{B}$ – this fact, together with Doob’s inequality and the $L^p(\textbf{P})$ moment (2.11) for any $\gamma \in (0,\gamma _c)$, dictate that the ergodic theorem for stationary processes is applicable, implying that the LHS of (2.14) is dominated by $C(\dot{B}) \varepsilon ^{-2}$ almost surely w.r.t. $\textbf{P}$, thereby removing the influence of this term on a logarithmic scale.

For the third term, the asymptotic behavior of $\varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )$, for any $\gamma >0$, is provided by a tractable variational formula over the translation-invariant compactification $\widetilde{{\mathcal {X}}}$ of probability measures introduced in [26] – briefly, this space consists of elements $\xi =\{{\widetilde{\alpha }}_i\}_i$ of orbits ${\widetilde{\alpha }}_i$ of sub-probability measures $\alpha _i$ with $\sum _i \alpha _i(\mathbb {R}^d)\le 1$, see Sect. 4.2. For our purposes, it might be worthwhile explaining the structure of this formula. First, we can decompose $\log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )$ in terms of a martingale and an additive (and shift-invariant) energy functional of the probability distributions ${\widehat{\mu }}_{\gamma ,\varepsilon ^{-2}}[\omega _{\varepsilon ^{-2}}\in \cdot ]$, but embedded into the aforementioned compactification $\widetilde{{\mathcal {X}}}$, and these distributions also define a dynamics defined by transition probabilities. This dynamics was introduced in [1] as a fixed point approach related to the cavity method from spin glasses for discrete time random walks on $\mathbb {Z}^d$ over the aforementioned compactified space (this used the countability of $\mathbb {Z}^d$ by employing a different, but equivalent, metric to the one in [26]). In the present context, we will follow the approach developed in [6] which directly used the precise metric structure from [26] in the continuum showing that the above Markovian transition kernel is continuous w.r.t. the metric on $\widetilde{{\mathcal {X}}}$, implying existence of invariant measures (over probability measures ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$ on $\widetilde{{\mathcal {X}}}$). The key aspect here hinges upon the notion of total disintegration of mass as well as a decoupling phenomenon of two independent GMC chains at large distances, both traits being compatible with the topology of $\widetilde{{\mathcal {X}}}$. Then the aforementioned representation of $\varepsilon ^{2} \log \mu _{\gamma , \varepsilon ^{-2}}(\Omega )$ and the continuity of the above map provide, for any temperature $\gamma >0$, a tractable quenched variational formula $\varepsilon ^{2} \log \mu _{\gamma , \varepsilon ^{-2}}(\Omega )= - \sup _{\vartheta } \int _{\widetilde{{\mathcal {X}}}} \Phi _{\gamma }(\xi ) \vartheta (\textrm{d}\xi )$ almost surely w.r.t. $\textbf{P}$. Here

$$\begin{aligned} \Phi _{\gamma }(\xi )= \frac{\gamma ^2}{2} \sum _i \int _{\mathbb {R}^{2d}} (\phi \star \phi )(x-y) \alpha _i(\textrm{d}x) \alpha _i(\textrm{d}y) \qquad \xi =({\widetilde{\alpha }}_i)_i \in \widetilde{{\mathcal {X}}}. \end{aligned}$$

The supremum in the variational formula is taken (and attained) over the invariant measures of the dynamic in ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$. The maximizer(s) depend on the temperature $\gamma$ and undergoes a qualitative change as $\gamma$ varies, see Sect. 4.2 for details. Combining these estimates will provide the required upper bound in Theorem 2.2. Note that since the $L^p$ bound in (2.11) holds for some (possibly implicit) $p>1$, the subsequent application of the Hölder’s inequality is valid for this (pre-determined) $p>1$. In this vein, the structure of the formula (in terms of $\gamma$ and p) is instrumental in providing a non-trivial uniform upper bound $\exp (-C_1/\varepsilon ^2)$ with an explicit $C_1>0$, for any $\gamma \in (0,\gamma _c)$ and $r>0$ sufficiently small. We refer to Proposition 4.1 for details.

We also remark that an orthogonal approach for handling asymptotic behavior of the logarithmic partition function goes via exploiting its sub-additivity [9, 12], which, unlike the present approach, does not immediately yield an explicit limiting object.

Now, for the proof of the lower bound in Theorem 2.2 we will apply reverse Hölder’s inequality to split the probabilities also into three factors similar to the one above. The applicative of this inequality then requires a negative moment from (2.12), and then the space-time translation invariance of the white noise allows the use of the ergodic theorem again. However, now a concentration bound coming from stochastic calculus allows us to replace the quenched probabilities $\varepsilon ^{2} \log \mu _{\gamma , \varepsilon ^{-2}}(\Vert \omega \Vert _{w,\varepsilon }< r\varepsilon )$ by its annealed counterpart $\varepsilon ^{2} \textbf{E}[\log \mu _{\gamma , \varepsilon ^{-2}}(\Vert \omega \Vert _{w,\varepsilon }< r\varepsilon )]$, and subsequently an application of Jensen’s inequality removes any contribution coming from this term. This leads to a lower bound $\exp (-C_2/\varepsilon ^2)$ for any $\gamma \in (0,\gamma _c)$ with $C_2 <\infty$, see Proposition 4.3. Similar to $C_1$, the lower bound constant $C_2$ is also explicit, but unlike the upper bound, $C_2$ carries only two contributions coming from the normalization constant and from the exit time estimate for small ball probability under the Wiener measure. As remarked earlier, the upper and lower bounds $C_1$ and $C_2$ differ for a fixed $\gamma \in (0,\gamma _c)$, but as $\gamma \rightarrow 0$, these exponents coincide with an explicit constant involving the Bessel function (the scaling exponent of the Wiener measure $\mathbb {P}_0$), see Corollary 4.4.

Finally, let us mention that it is shown in [5] that, for any $\gamma >0$ and $d\ge 1$, the finite-volume measure ${\widehat{\mu }}_{\gamma ,T}$ decays exponentially on small balls (in the uniform metric on paths in [0, T]). Even though the infinite-volume limit of ${\widehat{\mu }}_{\gamma ,T}$ exists for $\gamma \in (0,\gamma _c)$, it is not clear how a similar property for $\varepsilon ^2 \log {\widehat{\mu }}_\gamma (\Vert \omega \Vert < \varepsilon )$ can be deduced from the statements there directly.

Step 4 (Moments for $\gamma \in (0,\gamma _c)$): It remains to show the estimates in Theorem 2.3. For this purpose, we will adapt an argument which was developed recently in [17] to show that (2.11) and (2.12) hold for discrete polymers in a bounded environment in weak disorder. This argument involved exploiting properties of a non-negative martingale $(M_n,{\mathcal {F}}_n)_n$ satisfying three assumptions: (i) First, for any $k,\ell \in \mathbb {N}$, $\textbf{E}[f(M_{k+\ell }/M_k)| {\mathcal {F}}_k]\le \textbf{E}[f(M_\ell )]$ for any convex function f (as mentioned there, this condition is originally known to be satisfied by well-known branching processes, e.g. Galton-Watson processes); (ii) If $\textbf{P}[\lim _n M_n>0]>0$ and condition (i) holds, then $\textbf{E}[M_\infty ^\star ]< \infty$, where $M_n^\star =\sup _{0\le k \le n} M_k$ is the running maximum; (iii) If $\textbf{P}[\lim _n M_n>0]>0$ and condition (i) holds, and moreover if for some $K<\infty$, we have $M_{n+1}/M_n \le K$ for all n, then $\sup _n \Vert M_n\Vert _p < \infty$ for some $p>1$. These three properties are satisfied by the partition function of the discrete directed polymer in a bounded environment. For the continuous directed polymer (in an unbounded environment)⁵ we will modify the first step and show that the martingale $(\mu _{\gamma ,T}(\Omega ))_T$, satisfies, for a suitable passage time $\tau =\tau _u:= \inf \{t: \mu _{\gamma ,t}(\Omega )=u\}$ and for any $\gamma >0$, $\textbf{E}\big [f\big (\mu _{\gamma ,T}(\Omega )/\mu _{\gamma ,\tau }(\Omega )\big ); \tau \le T\big ] \le \textbf{P}[\tau \le T] \textbf{E}[f(\mu _{\gamma ,T}(\Omega ))]$ (see Lemma 5.2). This in turn will imply that for $\gamma \in (0,\gamma _c)$, we have $\textbf{E}[M_\infty ] < \infty$, where $M_\infty =\lim _{T\rightarrow \infty } \, \sup _{0\le S \le T} \mu _{\gamma ,S}(\Omega )$ (see Lemma 5.1). These two modifications will subsequently suffice to show (2.11) for some $p>1$, without requiring the third assumption above.

Remark 5

To compare our Step 1 (sketched above) with existing literature, let us mention that, for discrete (in time and in space $\mathbb {Z}^d$) directed polymers, Comets and Yoshida [12, Sec. 4] observed that the normalized polymer measure $\mu _n$ at time n defines a time-inhomogeneous Markov chain with transition probabilities

$$\begin{aligned}{} & {} \mu _n(\omega _{i+1}=y| \omega _i=x) \nonumber \\{} & {} = {\left\{ \begin{array}{ll} \frac{\textrm{e} ^{\beta \omega (i+1,y)}}{W_{n-i} \circ \theta _{i,x}} (W_{n-i-1} \circ \theta _{i+1,y}) P[\omega _1=y | \omega _0=x], &{} 0\le i<n,\\ P[ \omega _1=y | \omega _0=x],&{} i\ge n. \end{array}\right. } \end{aligned}$$

(2.15)

Here, P is the law of a simple random walk on $\mathbb {Z}^d$, and $W_n$ the partition function that also converges a.s to $W_\infty >0$. Thus, by using the existence of the limit $\lim _{n\rightarrow \infty }\mu _n(A)$ for $A\in \sigma \left( \bigcup _n {\mathcal {F}}_n\right)$, it can be easily shown that there is a limiting measure $\mu$ on $\sigma$-algebra generated by finite paths, and this limiting measure is a Markov chain with

$$\begin{aligned} \mu (\omega _{i+1}=y| \omega _i=x)=\frac{\textrm{e} ^{\beta \omega (i+1,y)}}{W_{\infty } \circ \theta _{i,x}} (W_{\infty } \circ \theta _{i+1,y}) P[\omega _1=y | \omega _0=x]. \end{aligned}$$

Note that, $\mu _n$ defined in (2.15) is time-inhomogeneous and also inconsistent, as the transition probabilities depend on the time horizon n. Unless $\beta =0$, there exists no Markov chain on the set of paths with infinite length such that, for any $n\ge 1$, the marginal on the set of paths of length n is $\mu _n$, see the discussion in Comets [8, p.15, Sec. 2.1.2]. We note that in the current setup, we are working from the beginning on the Borel $\sigma$-algebra of $\Omega = C(([0,\infty );\mathbb {R}^d)$ of paths of infinite length and the topology of uniform convergence on compact sets. This is different to the one generated by $\bigcup _T {\mathcal {F}}_T$, where ${\mathcal {F}}_T$ is the Borel $\sigma$-algebra generated by continuous paths up to time T under the supremum norm on [0, T]. Also, even if we were to adapt the Markovian approach from discrete polymers in the current continuum framework (and chose to work with $\sigma$-algebras on finite-time horizons), it would still require showing tightness in the continuum set up (which is not needed in the discrete setup, as mentioned). Therefore, in the current framework, we followed an alternate (and perhaps a direct) route of showing weak convergence of the measures $\mu _{\gamma ,T}$ in the space $\Omega = C(([0,\infty );\mathbb {R}^d)$, as described above in Step 1. $\square$

Organization of the article: Sects. 3.1, 3.2, 3.3, 3.4 constitute the proof of Theorem 2.1; here Sect. 3.1, 3.2,  3.3 are devoted to the existence of the measure, while Sect. 3.4 will provide the identification of thick points and characterization of the measure. Proofs of Theorem 2.2 and Theorem 2.3 can be found Sect. 4 and Sect. 5, respectively.

3 Proof of theorem 2.1

3.1 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

  $$\begin{aligned} \langle H(\omega )\rangle _T = \textbf{E}[{H}_T(\omega )^2] = \int _{0}^{T} (\phi \star \phi )(0)\textrm{d}s = T(\phi \star \phi )(0). \end{aligned}$$

• (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 $\gamma >0$,

  $$\begin{aligned} \big \{\mu _{\gamma ,T}(A)\big \}_{T>0}= \bigg \{ \int _A \exp \left\{ \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T (\phi \star \phi )(0) \right\} \textrm{d}\mathbb {P}_0(\omega ) \bigg \}_{T>0} \end{aligned}$$

  is a martingale.

Proof

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

Lemma 3.2

Let $X \in L^1(\textbf{P} \otimes \mathbb {P}_0)$ and ${\mathcal {G}}\subset \mathcal {F}$ be a $\sigma$-algebra. Then

$$\begin{aligned} \textbf{E}\bigg [ \, \int _{\Omega } X(\omega , \cdot ) \textrm{d}\mathbb {P}_0(\omega ) \, \bigg | \, {\mathcal {G}} \, \bigg ] = \int _{\Omega } \textbf{E}\big [ X(\omega , \cdot ) \, | \, {\mathcal {G}} \big ] \textrm{d}\mathbb {P}_0(\omega ). \end{aligned}$$

(3.1)

Proof

The proof is standard and a direct consequence of Fubini’s theorem for conditional expectation and is therefore skipped. $\square$

3.2 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 $\gamma '<\gamma$.

Proof

Fix $\gamma >0$ such that $(\mu _{\gamma ,T}(\Omega ))_{T\ge 0}$ is uniformly integrable. Let $\dot{B},\dot{B}'$ be independent copies of the noise, and let $\gamma '<\gamma$, so that $\gamma '=c\gamma$ for some $0<c<1$. To avoid ambiguities, we write $\mu _{\gamma ,T}(\Omega )=\mu _{\gamma ,T}(\Omega ,\dot{B})$. Note that

$$\begin{aligned} \mu _{\gamma ',T}(\Omega ,\dot{B})=\mu _{c\gamma ,T}(\Omega ,\dot{B}) =\textbf{E}\left[ \mu _{\gamma ,T}(\Omega ,c\dot{B}+\sqrt{1-c^2}\dot{B}')\vert \dot{B}\right] . \end{aligned}$$

(3.2)

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

$$\begin{aligned} \lim _{x\rightarrow \infty }\frac{f(x)}{x}=\infty , \qquad \text{ and }\qquad \sup _{T}\textbf{E}[f(\mu _{\gamma ,T}(\Omega ,\dot{B}))]<\infty . \end{aligned}$$

By (3.2) and Jensen’s inequality,

$$\begin{aligned} \textbf{E}[f(\mu _{\gamma ',T}(\Omega ,\dot{B}))]&=\textbf{E}\left[ f\left( \textbf{E}\left[ \mu _{\gamma ,T}(\Omega ,c\dot{B}+\sqrt{1-c^2}\dot{B}')\vert \dot{B}\right] \right) \right] \\&\le \textbf{E}\left[ f\left( \mu _{\gamma ,T}(\Omega ,c\dot{B}+\sqrt{1-c^2}\dot{B}')\right) \right] \\&=\textbf{E}[f(\mu _{\gamma ,T}(\Omega ,\dot{B}))]. \end{aligned}$$

It follows that $\sup _{T}\textbf{E}[f(\mu _{\gamma ,T}(\Omega ,\dot{B}))]<\infty$, which in turn implies the uniform integrability of $(\mu _{\gamma ',T}(\Omega ,\dot{B}))_{T\ge 0}$. $\square$

The last lemma allows us to define the critical parameter

$$\begin{aligned} \gamma _c=\gamma _c(d)= \sup \bigg \{\gamma : \text{ the } \text{ martingale } \mu _{\gamma ,T}(\Omega ) \text{ is } \text{ uniformly } \text{ integrable }\bigg \}. \end{aligned}$$

(3.3)

In particular, if $\gamma <\gamma _c$, then $(\mu _{\gamma ,T}(\Omega ))_{T\ge 0}$ is uniformly integrable, while for $\gamma >\gamma _c$, $(\mu _{\gamma ,T}(\Omega ))_{T\ge 0}$ is not uniformly integrable. Since we will be dealing also with the normalized probability measure $\widehat{\mu }_{\gamma , T}$ from (1.10), 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 $\lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\Omega )$ is positive $\textbf{P}$-a.s:

Lemma 3.4

If $(\mu _{\gamma ,T}(\Omega ))_{T \ge 0}$ is uniformly integrable, then $\lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\Omega )>0$ $\textbf{P}$-a.s.

Proof

First we note that $\textbf{E}[\mu _{\gamma ,T}(\Omega )]=1$, so that $\textbf{P}(\lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\Omega )>0)>0$. Moreover, the event

$$\begin{aligned} A_\gamma :=\{\mu _{\gamma }(\Omega )>0\} \end{aligned}$$

(3.4)

has probability 0 or 1 in virtue of Kolmogorov’s 0-1 law. $\square$

The non-triviality of $\gamma _c$, i.e., that $\gamma _c>0$ for $d\ge 3$ is implied by the existence of the so-called $L^2$-phase:

Lemma 3.5

For $d\ge 3$ and $\gamma >0$ small enough, the martingale $(\mu _{\gamma ,T}(\Omega ))_{T\ge 0}$ is bounded in $L^2(\textbf{P})$. In particular, $\gamma _c>0$.

Proof

Let $\omega '$ 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,

$$\begin{aligned} \textbf{E}\left[ \mu _{\gamma ,T}(\Omega )^2\right]&=\textbf{E}\left[ \int _{\Omega ^2}\textrm{e} ^{\gamma (H_T(\omega )+H_T(\omega '))-\gamma ^2 T(\phi \star \phi )(0)}\textrm{d}\mathbb {P}_0^{\otimes 2}(\omega ,\omega ')\right] \\&=\int _{\Omega ^2}\textrm{d}\mathbb {P}_0^{\otimes 2}(\omega ,\omega ')\textbf{E}\left[ \textrm{e} ^{\gamma (H_T(\omega )+H_T(\omega '))-\gamma ^2 T(\phi \star \phi )(0)}\right] . \end{aligned}$$

Recall that for $\omega ,\omega '\in \Omega$, $H_T(\omega )+H_T(\omega ')$ is a Gaussian random variable with mean 0 and variance

$$\begin{aligned} 2T(\phi \star \phi )(0)+2\int _0^T(\phi \star \phi )(\omega _s-\omega '_s)\textrm{d}s, \end{aligned}$$

and so

$$\begin{aligned} \textbf{E}\left[ \textrm{e} ^{\gamma (H_T(\omega )+H_T(\omega '))-\gamma ^2 T(\phi \star \phi )(0)}\right] =\textrm{e} ^{\gamma ^2\int _{0}^T(\phi \star \phi ) (\omega _s-\omega '_s)\textrm{d}s}. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \textbf{E}\left[ \mu _{\gamma ,T}(\Omega )^2\right]&=\int _{\Omega ^2}\textrm{d}\mathbb {P}_0^{\otimes 2}(\omega ,\omega ')\textrm{e} ^{\gamma ^2\int _{0}^T(\phi \star \phi )(\omega _s-\omega '_s)\textrm{d}s}\\&=\int _{\Omega }\textrm{d}\mathbb {P}_0(\omega )\textrm{e} ^{\gamma ^2\int _0^T(\phi \star \phi )(\sqrt{2}\omega _s)\textrm{d}s} \\&\le \int _{\Omega }\textrm{d}\mathbb {P}_0(\omega )\textrm{e} ^{\gamma ^2\int _0^\infty (\phi \star \phi )(\sqrt{2}\omega _s)\textrm{d}s}. \end{aligned} \end{aligned}$$

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

$$\begin{aligned} I(\phi ):= \sup _{x\in \mathbb {R}^d}\int _{\Omega }\textrm{d}\mathbb {P}_x(\omega ) \int _0^\infty (\phi \star \phi )(\sqrt{2}\omega _s)\textrm{d}s<\infty , \end{aligned}$$

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

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

By Khas’minski’s lemma [20] we deduce that

$$\begin{aligned} \sup _{T}\textbf{E}\left[ \mu _{\gamma ,T}(\Omega )^2\right] \le \sup _{x\in \mathbb {R}^d}\int _{\Omega }\textrm{d}\mathbb {P}_x(\omega )\textrm{e} ^{\gamma ^2\int _0^\infty \phi \star \phi (\sqrt{2}\omega _s)\textrm{d}s}<\infty . \end{aligned}$$

$\square$

Remark 6

By an application of Kahane’s inequality [19], in [25] it is proved additionally that $\gamma _c<\infty$, and

$$\begin{aligned} (\mu _{\gamma ,T}(\Omega ))_{T\ge 0} \text { is uniformly integrable}\Longleftrightarrow \lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\Omega )>0 \qquad \textbf{P}\text {-a.s.,} \end{aligned}$$

strengthening Lemma 3.4 to an “if and only if" statement. These results are not required in the sequel. We also refer to [32] where the weak and strong disorder for the partition function as well as concentration inequalities were obtained for the continuous directed polymer in a Gaussian random environment.

3.3 Weak convergence of the approximating measures ${\widehat{\mu }}_{\gamma ,T}$

The proof of the first part 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 $(\mu _{\gamma , T}(A))_{T\ge 0}$ 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 3.6

Fix a Borel set $A\subset \Omega$. For $\gamma < \gamma _c$, the sequence $(\mu _{\gamma , T}(A))_{T\ge 0}$ converges in $L^1(\textbf{P})$ and almost surely to a random variable $\mu _{\gamma }(A)$.

Proof

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

$$\begin{aligned} \mu _{\gamma , T}(A) = \int _A \exp \left\{ \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T (\phi \star \phi ) (0) \right\} \textrm{d}\mathbb {P}_0(\omega ) \end{aligned}$$

(3.5)

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 $L^1(\textbf{P})$ and almost surely to a random variable. We denote the limit by $\mu _{\gamma }(A)$.

$\square$

Remark 7

By Lemma 3.4, we know that $\mu _\gamma (\Omega )>0$ $\textbf{P}$-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

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

Lemma 3.7

Let $\mu _\gamma$ be defined as in Lemma 3.6. 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 $n \ge 1$, then $\textbf{P}$-a.s. it is true that

$$\begin{aligned} \mu _{\gamma }\left( \bigcup _{n}A_n \right) = \lim _{n \rightarrow \infty } \mu _\gamma (A_n). \end{aligned}$$

(3.6)

An analogous statement holds for decreasing events.

Proof

First note that the limit on the right hand side in (3.6) 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

$$\begin{aligned} \mu _\gamma \left( \bigcup _{n}A_n \right) \ge \lim _{n \rightarrow \infty }\mu _\gamma (A_n). \end{aligned}$$

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

$$\begin{aligned} \textbf{E}\left[ \mu _\gamma \left( \bigcup _{n}A_n \right) \right]&= \lim _{T \rightarrow \infty } \textbf{E}\left[ \mu _{\gamma , T}\left( \bigcup _{n}A_n \right) \right] \\&= \lim _{T \rightarrow \infty } \textbf{E}\left[ \int _{\bigcup _{n}A_n} \exp \left\{ \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T (\phi \star \phi ) (0) \right\} \textrm{d}\mathbb {P}_0(\omega ) \right] \\&= \lim _{T \rightarrow \infty } \int _{\bigcup _{n}A_n} \textbf{E}\left[ \exp \left\{ \gamma {H}_T(\omega ) - \frac{\gamma ^2}{2}T (\phi \star \phi ) (0) \right\} \right] \textrm{d}\mathbb {P}_0(\omega ) \\&= \lim _{T \rightarrow \infty } \mathbb {P}_0\left( \bigcup _{n}A_n \right) = \mathbb {P}_0\left( \bigcup _{n}A_n \right) = \lim _{n \rightarrow \infty } \mathbb {P}_0(A_n). \end{aligned}$$

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

$$\begin{aligned} \textbf{E}\left[ \lim _{n \rightarrow \infty } \mu _\gamma (A_n) \right]&= \lim _{n \rightarrow \infty }\textbf{E}\left[ \mu _\gamma (A_n)\right] = \lim _{n \rightarrow \infty } \textbf{E}\left[ \lim _{T \rightarrow \infty } \mu _{\gamma , T}(A_n) \right] \\&\lim _{n \rightarrow \infty } \lim _{T \rightarrow \infty } \textbf{E}\left[ \mu _{\gamma , T}(A_n) \right] = \lim _{n \rightarrow \infty } \mathbb {P}_0(A_n). \end{aligned}$$

This concludes the proof. $\square$

Remark 8

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 3.7 remains true.

The key argument for the proof of Theorem 2.1 is provided by

Proposition 3.8

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

Proof

We will carry out the proof in few steps.

Step 1: In the first step, we will show that, there is a collection ${\mathcal {X}}$ of countably many Borel measurable subsets of $\Omega$ such that

$$\begin{aligned} \widehat{\mu }_{\gamma , T}(A) \overset{T\rightarrow \infty }{\longrightarrow } \nicefrac {\mu _{\gamma }(A)}{\mu (\Omega )} \quad \textbf{P}\text{-almost } \text{ surely } \text{ for } \text{ all }\, A \in {\mathcal {X}}\, \text{ simultaneously. } \end{aligned}$$

(3.7)

Indeed, we define the sets

$$\begin{aligned} \begin{aligned}&{\mathcal {A}}_o := \{ (x_1, y_1)\times \cdots \times (x_d,y_d) \, : \, x_i,y_i \in \mathbb {Q}\}, \\&{\mathcal {A}}_c := \{ [x_1, y_1]\times \cdots \times [x_d,y_d] \, : \, x_i,y_i \in \mathbb {Q}\}, \\&{\mathcal {A}}_s := \{ [x_1, y_1)\times \cdots \times [x_d,y_d) \, : \, x_i,y_i \in \mathbb {Q}\}, \\ \end{aligned} \end{aligned}$$

(3.8)

and

$$\begin{aligned} {\mathcal {X}}_o^1 := \bigg \{ \big \{\omega : \omega (t_1) \in A_1, \ldots , \omega (t_n) \in A_n \big \} :n \in \mathbb {N}, t_1< \ldots < t_n \in \mathbb {Q}^+, A_1, \ldots , A_n \in {\mathcal {A}}_o \bigg \}. \end{aligned}$$

(3.9)

Analogously, we define ${\mathcal {X}}_c^1$ (by replacing ${\mathcal {A}}_o$ by ${\mathcal {A}}_c$ in the above definition) and ${\mathcal {X}}_s^1$ (by replacing ${\mathcal {A}}_o$ by ${\mathcal {A}}_s$), and set

$$\begin{aligned} {\mathcal {X}}_1 := {\mathcal {X}}_o^1 \cup {\mathcal {X}}_c^1 \cup {\mathcal {X}}_s^1. \end{aligned}$$

Furthermore, for any $\delta >0$, let

(3.10)

and set

$$\begin{aligned} {\mathcal {X}}_2:= & {} \Omega \cup \Bigg \{ \bigg \{\omega : \sup _{0\le t\le T} |\omega (t)|>a\bigg \} : T, a \in \mathbb {Q}^+ \Bigg \} \nonumber \\{} & {} \cup \bigg \{ \big \{\omega : m^T(\omega , \delta ) > \varepsilon \big \} : T, \delta , \varepsilon \in \mathbb {Q}^+ \bigg \},\quad \text{ and }\\ {\mathcal {X}}:= & {} {\mathcal {X}}_1 \cup {\mathcal {X}}_2.\nonumber \end{aligned}$$

(3.11)

Thus, ${\mathcal {X}}$ contains countably many Borel measurable subsets of $\Omega$. By Lemma 3.6 (actually, by Remark 7), for any Borel subset A of $\Omega$,

$$\begin{aligned} \widehat{\mu }_{\gamma , T}(A) \overset{T\rightarrow \infty }{\longrightarrow } \nicefrac {\mu _{\gamma }(A)}{\mu _\gamma (\Omega )} \end{aligned}$$

$\textbf{P}$-almost surely. But since ${\mathcal {X}}$ is countable, the above almost sure convergence happens for all $A \in {\mathcal {X}}$ simultaneously, proving (3.7).

Step 2: We will now show that, $\textbf{P}$ almost surely, for any $\ell \in \mathbb {Q}_+$ and $\eta \in \mathbb {Q}_+$, there is $a\in \mathbb {Q}_+$ such that

$$\begin{aligned} {\widehat{\mu }}_{\gamma ,T}\bigg ( \bigg \{\omega \, : \, \sup _{0\le t\le \ell } |\omega (t)|>a \bigg \} \bigg ) < \eta \qquad \forall T>0. \end{aligned}$$

(3.12)

Note that for a fixed $\ell \in \mathbb {Q}^+$, the events

$$\begin{aligned} \mathbb {Q}^+ \ni a \mapsto \left\{ \omega : \sup _{0\le t\le \ell } |\omega (t)|>a \right\} \end{aligned}$$

are decreasing. Hence, by Lemma 3.7 (and the subsequent remark), we deduce that

$$\begin{aligned} \lim _{a \rightarrow \infty , a \in \mathbb {Q}^+} \frac{\mu _{\gamma }\left( \omega : \sup _{0\le t\le \ell } |\omega (t)|>a \right) }{\mu _\gamma (\Omega )} =0 \qquad \textbf{P} \text {-a.s.} \end{aligned}$$

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

$$\begin{aligned} \frac{\mu _{\gamma }\left( \{\omega : \sup _{0\le t\le \ell } |\omega (t)|>a \} \right) }{\mu _\gamma (\Omega )}\le \frac{\eta }{2}. \end{aligned}$$

(3.13)

Now by (3.7) in Step 1, we have

$$\begin{aligned} \lim _{T \rightarrow \infty } \widehat{\mu }_{\gamma , T}\big (\big \{ \omega : \sup _{0\le t\le \ell } |\omega (t)|>a \big \}\big ) = \frac{ \mu _{\gamma }\left( \left\{ \omega : \sup _{0\le t\le \ell } |\omega (t)|>a \right\} \right) }{\mu _\gamma (\Omega )} \qquad \textbf{P}\text {-a.s.,} \end{aligned}$$

for any $\ell \in \mathbb {Q}_+$ and corresponding $a\in \mathbb {Q}_+$ simultaneously. Therefore, by (3.13), $\textbf{P}$-a.s., for any $\ell \in \mathbb {Q}_+$ and $\eta \in \mathbb {Q}_+$, there is $a\in \mathbb {Q}_+$ such that

$$\begin{aligned} \widehat{\mu }_{\gamma , T}\bigg ( \bigg \{ \omega : \sup _{0\le t\le \ell } |\omega (t)|>a \bigg \} \bigg ) \le \eta , \end{aligned}$$

for some $T>0$ large enough. Hence, we can choose $a\in \mathbb {Q}_+$ large enough which guarantees that the above inequality holds for all $T >0$. This shows (3.12).

Step 3: Recall the definition of $m^T(\omega ,\delta )$ from (3.10). We will now show that, $\textbf{P}$ almost surely, for any $\ell , \eta , \varepsilon \in \mathbb {Q}_+$, there is $\delta \in \mathbb {Q}_+$ such that

$$\begin{aligned} {\widehat{\mu }}_{\gamma ,T}\left( \left\{ \omega \, : \, m^\ell (\omega , \delta )> \varepsilon \right\} \right) < \eta \qquad \forall \, T>0. \end{aligned}$$

(3.14)

Indeed, note that for fixed $\ell , \eta , \varepsilon \in \mathbb {Q}^+$, the events

$$\begin{aligned} \mathbb {Q}^+ \ni \delta \mapsto \left\{ \omega : \sup _{0\le s, t\le \ell , |t-s|<\delta }|\omega (t)-\omega (s)|>\varepsilon \right\} \end{aligned}$$

are decreasing as $\delta \searrow 0$. Again by Lemma 3.7, it holds that

$$\begin{aligned} \lim _{\delta \rightarrow 0, \delta \in \mathbb {Q}^+} \frac{\mu _\gamma \left( \left\{ \omega : m^\ell (\omega , \delta )>\varepsilon \right\} \right) }{\mu _\gamma (\Omega )} = 0 \quad \textbf{P}\text {-a.s.} \end{aligned}$$

Thus, again using (3.7) from Step 1, $\textbf{P}$-a.s., for any $\ell , \eta , \varepsilon \in \mathbb {Q}_+$, there exists a $\delta \in \mathbb {Q}^+$ such that

$$\begin{aligned} \lim _{T \rightarrow \infty } \widehat{\mu }_{\gamma , T}\left( \left\{ \omega : m^\ell (\omega , \delta )>\varepsilon \right\} \right) = \frac{\mu _\gamma \left( \left\{ \omega : m^\ell (\omega , \delta )>\varepsilon \right\} \right) }{\mu _\gamma (\Omega )} \le \frac{\eta }{2} \end{aligned}$$

Hence, $\textbf{P}$-a.s.,

$$\begin{aligned} \widehat{\mu }_{\gamma , T}\left( \left\{ \omega : m^\ell (\omega , \delta )>\varepsilon \right\} \right) \le \eta , \end{aligned}$$

if $T>0$ is sufficiently large. If we now choose $\delta$ sufficiently small, we can assure that the inequality holds for all $T >0$, proving (3.14).

Step 4: We will now conclude that

$$\begin{aligned} \textbf{P}\text{-a.s., } \text{ the } \text{ family }\quad \big \{\widehat{\mu }_{\gamma , T}\big \}_{T\ge 0} \,\,\,\,\text{ is } \text{ uniformly } \text{ tight }. \end{aligned}$$

(3.15)

Fix $\eta \in \mathbb {Q}_+$ and also $\ell , m \in \mathbb {N}$. Then by (3.12) and (3.14), $\textbf{P}$-a.s., there is $a_\ell , \delta _{m,\ell }\in \mathbb {Q}_+$ such that for all $T>0$,

$$\begin{aligned} \begin{aligned}&{\widehat{\mu }}_{\gamma ,T}(A_\ell ^c) \le \frac{\eta }{2^{\ell +1}}, \qquad {\widehat{\mu }}_{\gamma ,T}(B_{m,\ell }^c) \le \frac{\eta }{2^{\ell +1+m}}, \qquad \text{ where }\\&A_\ell :=\big \{\omega : \sup _{0\le t \le \ell } |\omega (t)| {\le } a_\ell \big \}, \qquad B_{m,\ell } =\big \{\omega : m^\ell (\omega ,\delta _{m,\ell }) {\le }\frac{1}{m} \big \}. \end{aligned} \end{aligned}$$

(3.16)

Note that for each $\ell ,m \in \mathbb {N}$, $A_\ell$ and $B_{m,\ell }$ are closed. Therefore,

$$\begin{aligned} K:= \big (\cap _{\ell \in \mathbb {N}} A_\ell \big ) \bigcap \big ( \cap _{m,\ell \in \mathbb {N}} B_{m,\ell }\big ) \subset \Omega \qquad \text{ is } \text{ also } \text{ closed, } \end{aligned}$$

(3.17)

By (3.12), the family of functions K is uniformly bounded and by (3.14), K is equicontinuous, and hence, by Arzela-Ascoli theorem, K is relatively compact. Together with (3.17), we conclude that $K\subset \Omega$ is compact. Finally, by (3.16), $\textbf{P}$-a.s., for any $\eta \in \mathbb {Q}_+$ and all $T>0$, ${\widehat{\mu }}_{\gamma ,T}(K) \ge 1- \sum _{\ell \in \mathbb {N}} \eta /2^{\ell +1} - \sum _{\ell ,m\in \mathbb {N}} \eta /2^{\ell +1+m}=1- \eta$. This proves (3.15).

Step 5: We will now conclude the proof of the proposition. By tightness from Step 4, for each subsequence of $(\widehat{\mu }_{\gamma , T})_{T\ge 0}$ 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 that is, this limit is 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

$$\begin{aligned} \widehat{\mu }_{\gamma }(A)= \nicefrac {\mu _\gamma (A)}{\mu _\gamma (\Omega )}. \end{aligned}$$

More precisely, it suffices to show that this is true for all sets $A \in {\mathcal {X}}_s^1$ (recall (3.9)). 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 3.7,

$$\begin{aligned} \frac{\mu _\gamma (A)}{\mu _\gamma (\Omega )} = \sup _{A_1', \ldots , A_n'} \frac{\mu _\gamma (\{ \omega : \omega (t_1)\in A_1', \ldots , \omega (t_n) \in A_n' \} )}{\mu _\gamma (\Omega )} \quad \textbf{P}\text {-a.s.}, \end{aligned}$$

(3.18)

where the supremum is over $A_1', \ldots , A_n' \in {\mathcal {A}}_c$ (recall (3.8)) such that $A_i' \subset A_i$ for all $1 \le i \le n$. Similarly, we have

$$\begin{aligned} \frac{\mu _\gamma (A)}{\mu _\gamma (\Omega )} = \inf _{A_1', \ldots , A_n'} \frac{\mu _\gamma (\{ \omega : \omega (t_1)\in A_1', \ldots , \omega (t_n) \in A_n' \} )}{\mu _\gamma (\Omega )} \qquad \textbf{P}\text {-a.s.,} \end{aligned}$$

(3.19)

where the infimum is over all $A_1', \ldots , A_n' \in {\mathcal {A}}_o$ such that $\bar{A_i} \subset A_i'$ for all $1\le i\le n$. By Lemma 3.7 and Portmanteau’s Theorem, for any $A \in {\mathcal {X}}_s^1$ and any $A_1', \ldots , A_n' \in {\mathcal {A}}_o$ with $\bar{A_i}\subset A_i'$ for all $1\le i\le n$, $\textbf{P}$-a.s., we have

$$\begin{aligned}&\widehat{\mu }_{\gamma } ( \{\omega : \omega (t_1)\in A_1', \ldots , \omega (t_n)\in A_n' \} ) \\&\quad \le \liminf _{T \rightarrow \infty } \frac{\mu _{\gamma , T}( \{\omega : \omega (t_1)\in A_1', \ldots , \omega (t_n)\in A_n' \} )}{\mu _{\gamma , T}(\Omega )} \\&\quad =\lim _{T \rightarrow \infty } \frac{\mu _{\gamma , T}( \{\omega : \omega (t_1)\in A_1', \ldots , \omega (t_n)\in A_n' \} )}{\mu _{\gamma , T}(\Omega )} \\&\quad = \frac{\mu _\gamma ( \{\omega : \omega (t_1)\in A_1', \ldots , \omega (t_n)\in A_n' \} )}{\mu _\gamma (\Omega )}. \end{aligned}$$

Since $A_1', \ldots , A_n'$ are arbitrary, by taking the infimum and using (3.19), we can deduce that $\textbf{P}$-a.s., $\widehat{\mu }_{\gamma }(A)\le \nicefrac {\mu _\gamma (A)}{\mu _\gamma (\Omega )}$. Proceeding analogously with sets $A_1', \ldots , A_n' \in {\mathcal {A}}_c$ such that $A_i' \subset A_i$ for all $1 \le i \le n$ and using (3.18), we obtain $\textbf{P}$-a.s., $\widehat{\mu }_\gamma (A)\ge \nicefrac {\mu _\gamma (A)}{\mu _\gamma (\Omega )}$. Therefore, $\widehat{\mu }_{\gamma }(A)= \nicefrac {\mu _\gamma (A)}{\mu _\gamma (\Omega )}$ $\textbf{P}$-a.s. as desired. The proof of the weak convergence for the unnormalized measures can be performed by the same methods, by noting that $\mu _\gamma (\Omega )>0$ $\textbf{P}$-a.s. $\square$

We can now complete the

Proof of existence Theorem 2.1

By Lemma 3.6, for each Borel set $A\subset \Omega$, the sequence $(\widehat{\mu }_{\gamma , T}(A))$ converges $\textbf{P}$-a.s. to $\nicefrac {\mu _\gamma (A)}{\mu _\gamma (\Omega )}$ as $T \rightarrow \infty$. Hence, by Proposition 3.8, $\widehat{\mu }_{\gamma , T}$ converges weakly $\textbf{P}$-a.s. to the probability measure $\widehat{\mu }_{\gamma }$. This completes the existence part of the proof of Theorem 2.1. $\square$

3.4 Thick points, support and characterization

We turn to the proofs of the second and the third parts of Theorem 2.1. We start with the third part, which will be shown in

Proposition 3.9

(Characterization of $\mu _\gamma )$ Fix $d\ge 3$, $\gamma \in (0,\gamma _c)$ and a mollifier $\phi$, Then the (unnormalized) GMC measure $\mu _\gamma = \mu _{\gamma ,\phi }$ is the unique measure such that the law of ${\dot{B}}$ under $\mathbb {Q}_{\mu _{\gamma ,\phi }}$ (recall the notation (2.2)) is the same as the law of the Schwartz distribution

$$\begin{aligned} {{\dot{B}} }_\phi (f)={\dot{B}} (f)+\gamma \int _{\mathbb {R}_+\times \mathbb {R}^d}f(s,y)\phi (\omega _s-y)\textrm{d}s \textrm{d}y, \qquad f \in {\mathcal {S}}(\mathbb {R}_+\times \mathbb {R}^d) \end{aligned}$$

under $P\otimes \mathbb {P}_0$. In other words, $\mu _{\gamma ,\phi }$ is the unique measure satisfying

$$\begin{aligned} \mathbb {E}^P\bigg [\int _\Omega \mu _{\gamma ,\phi }(\textrm{d}\omega ) F(\dot{B}, \omega )\bigg ]= \mathbb {E}^{P\otimes \mathbb {P}_0}\big [F(\dot{B}_\phi , \omega )\big ]. \end{aligned}$$

(3.20)

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

Proof

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

$$\begin{aligned} \mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}{\dot{B}}, \textrm{d}\omega ) = \exp \left\{ \gamma {H}_T(\omega , {\dot{B}} ) - \frac{\gamma ^2}{2}T(\phi \star \phi )(0) \right\} \mathbb {P}_0(\textrm{d}\omega ) P(\textrm{d}{\dot{B}} ). \end{aligned}$$

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

$$\begin{aligned} \mathbb {Q}_{\mu _{\gamma }}(\textrm{d}{\dot{B}}, \textrm{d}\omega ) = \mu _\gamma (\textrm{d}\omega , {\dot{B}} ) P(\textrm{d}{\dot{B}} ). \end{aligned}$$

Before continuing, we show that

Lemma 3.10

$\mathbb {Q}_{\mu _{\gamma ,T}} \rightarrow \mathbb {Q}_{\mu _{\gamma }}$ weakly as $T \rightarrow \infty$.

Proof

We first prove first that $(\mathbb {Q}_{\mu _{\gamma ,T}})_{T\ge 0}$ is tight. It is enough to verify that the marginals

$$\begin{aligned} \begin{aligned}&\mathbb {Q}_{\mu _{\gamma ,T}}^{1}(\textrm{d}\omega ):=\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}\omega \times {\mathcal {S}}')=\mathbb {P}_0(\textrm{d}\omega )\quad \text{ and } \\&\mathbb {Q}_{\mu _{\gamma ,T}}^2(\textrm{d}\dot{B}):=\mathbb {Q}_{\mu _{\gamma ,T}}(\Omega \times \textrm{d}\dot{B})=\mu _{\gamma ,T}(\Omega ) P(\textrm{d}\dot{B}) \quad \text{ are } \text{ tight }. \end{aligned} \end{aligned}$$

Clearly $\mathbb {Q}_{\mu _{\gamma ,T}}^{1}$ is tight since it is a single probability measure. On the other hand, since ${\mathcal {S}}'$ is $\sigma$-compact (by the Banach-Alaoglu theorem) and $\mathbb {Q}_{\mu _{\gamma ,T}}(\Omega \times {\mathcal {S}}')$ is uniformly bounded in T by the uniform integrability of $(\mu _{\gamma ,T}(\Omega ))_{T\ge 0}$, then $\mathbb {Q}_{\mu _{\gamma ,T}}^{2}$ is also tight. Therefore, $\mathbb {Q}_{\mu _{\gamma ,T}}$ converges weakly to some measure $\mathbb {Q}$ as $T\rightarrow \infty$. We claim that $\mathbb {Q}=\mathbb {Q}_{\mu _{\gamma }}$. Indeed, for Borel subsets $A_1\subset \Omega , A_2\subset {\mathcal {S}}'$, it holds that $(\mu _{\gamma ,T}(A_1))_{T\ge 0}$ is uniformly integrable and $\mu _{\gamma ,T}(A_1)\rightarrow \mu _{\gamma }(A_1)$ P-a.s. and in $L^1$. Therefore,

$$\begin{aligned} \big |\mathbb {Q}_{\mu _{\gamma ,T}}(A_1\times A_2)-\mathbb {Q}_{\mu _{\gamma }}(A_1\times A_2)\big | \le \int _{A_2} \big |\mu _{\gamma ,T}(A_1,\dot{B})-\mu _{\gamma }(A_1,\dot{B})\big |P(\textrm{d}\dot{B})\rightarrow 0 \end{aligned}$$

as $T\rightarrow \infty$. Thus, for any Borel sets $A_1\subset \Omega$ and $A_2\subset {\mathcal {S}}'$, $\lim _{T\rightarrow \infty }\mathbb {Q}_{\mu _{\gamma ,T}}(A_1\times A_2)=\mathbb {Q}_{\mu _{\gamma }}(A_1\times A_2)$. By Portmanteau’s theorem, we deduce that $\mathbb {Q}=\mathbb {Q}_{\mu _{\gamma }}$. $\square$

We now continue with the proof of Proposition 3.9. By the previous lemma, if $n \in \mathbb {N}, f_1, \ldots , f_n \in {\mathcal {S}}(\mathbb {R}_{+}\times \mathbb {R}^d)$ and $g: \Omega \mapsto \mathbb {R}$ 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 $\Omega \times {\mathcal {S}}'$, then

$$\begin{aligned} \begin{aligned}&\lim _{T \rightarrow \infty } E \left[ {\dot{B}} (f_1)\cdot \ldots \cdot {\dot{B}} (f_n) \int _{\Omega } g(\omega )\exp \left\{ \gamma {H}_T(\omega , {\dot{B}} ) - \frac{\gamma ^2}{2}T(\phi \star \phi )(0) \right\} \mathbb {P}_0(\textrm{d}\omega ) \right] \\&\quad = E \left[ {\dot{B}} (f_1)\cdot \ldots \cdot {\dot{B}} (f_n) \int _{\Omega } g(\omega ) \mu _{\gamma }(\textrm{d}\omega , {\dot{B}} ) \right] . \end{aligned} \end{aligned}$$

(3.21)

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

$$\begin{aligned} \mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}{\dot{B}} \, | \, \omega ) =\exp \left\{ \gamma {H}_T(\omega , {\dot{B}} ) - \frac{\gamma ^2}{2}T(\phi \star \phi )(0) \right\} P(\textrm{d}{\dot{B}} ). \end{aligned}$$

(3.22)

By the Cameron-Martin-Girsanov Theorem, we know that under the measure $\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}{\dot{B}} \, | \, \omega )$, $({\dot{B}} (f))_f$ is a Gaussian process with the same covariance structure as in (1.7) and mean given by

$$\begin{aligned} \int {\dot{B}} (f) \textrm{d}\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}{\dot{B}} \, | \, \omega )= & {} \gamma {{\,\textrm{Cov}\,}}\left( {\dot{B}} (f), {H}_T(\omega ) \right) \\= & {} \gamma \int _{\mathbb {R}_{+}\times \mathbb {R}^d}{\mathbb {1}}_{[0,T]}(s)f(s,y)\phi (\omega _s - y)\textrm{d}s \textrm{d}y. \end{aligned}$$

Hence, the expression in (3.21) is equal to

$$\begin{aligned}&\lim _{T \rightarrow \infty } \int _{\Omega }g(\omega ) \int {\dot{B}} (f_1)\cdot \ldots \cdot {\dot{B}} (f_n) \textrm{d}\mathbb {Q}_{\mu _{\gamma ,T}}(\textrm{d}{\dot{B}} , \omega ) \mathbb {P}_0(\textrm{d}\omega ) \\&\quad =\lim _{T \rightarrow \infty } \int _{\Omega }g(\omega ) \int \prod _{i=1}^{n} \left( {\dot{B}} (f_i) + \gamma {{\,\textrm{Cov}\,}}\left( {\dot{B}} (f_i), {H}_T(\omega ) \right) \right) P(\textrm{d}{\dot{B}} ) \mathbb {P}_0(\textrm{d}\omega )\\&\quad =\lim _{T \rightarrow \infty } \int _{\Omega }g(\omega ) \int \prod _{i=1}^{n} \left( {\dot{B}} (f_i) + \gamma \int _{\mathbb {R}_{+}\times \mathbb {R}^d}{\mathbb {1}}_{[0,T]}(s)f_i(s,y)\phi (\omega _s - y)\textrm{d}s \textrm{d}y \right) P(\textrm{d}{\dot{B}} ) \mathbb {P}_0(\textrm{d}\omega ). \end{aligned}$$

Writing $\mathbb {Q}_{\mathbb {P}_0}(\textrm{d}{\dot{B}}, \textrm{d}\omega ) {=} \mathbb {P}_0(\textrm{d}\omega ) P(\textrm{d}{\dot{B}} )$, and using (3.21), we deduce that

$$\begin{aligned}&\int g(\omega ) \prod _{i=1}^{n} \left( {\dot{B}} (f_i) + \gamma \int _{\mathbb {R}_{+}\times \mathbb {R}^d}f_i(s,y)\phi (\omega _s - y)\textrm{d}s \textrm{d}y \right) \mathbb {Q}_{\mathbb {P}_0}(\textrm{d}{\dot{B}} , \textrm{d}\omega ) \\&\quad = \int g(\omega ) \prod _{i=1}^n{\dot{B}} (f_i) \mathbb {Q}_{\mu _\gamma }(\textrm{d}{\dot{B}} , \textrm{d}\omega ). \end{aligned}$$

Since $g, f_1, \ldots , f_n$ are arbitrary, we conclude that the law of ${\dot{B}}$ under $\mathbb {Q}_{\mu _\gamma }$ is the same as the law of $\widetilde{{\dot{B}} }$ defined by $\widetilde{{\dot{B}} }(f) {:=} {\dot{B}} (f) + \gamma \int _{R_{+}\times \mathbb {R}^d} f(s,y) \phi (\omega _s - y) \textrm{d}s \textrm{d}y$ under $\mathbb {Q}_{\mathbb {P}_0}$. The uniqueness is immediate from this construction. This completes the proof of Proposition 3.9. $\square$

Finally, the second part of Theorem 2.1 will be shown in

Proposition 3.11

Fix $d\ge 3$ and $\gamma \in (0,\gamma _c)$ and let $\mu _\gamma$ be the infinite volume measure. Then,

$$\begin{aligned} \mu _\gamma \bigg \{\omega \in \Omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T (\phi \star \phi )(0)} \ne \gamma \bigg \}=0 \qquad \textbf{P}\text{-a.s. } \end{aligned}$$

Proof

First recall that, for every $\omega \in \Omega$, the stochastic integral $H_T(\omega )= \int _0^T \int _{\mathbb {R}^d} \phi (\omega _s-y) \dot{B}(s,y) \textrm{d}s \textrm{d}y$ is a continuous martingale with quadratic variation

$$\begin{aligned} \langle H \rangle _T= T (\phi \star \phi )(0) \rightarrow \infty \quad \text{ as }\, T\rightarrow \infty . \end{aligned}$$

It follows that, for every $\omega \in \Omega$,⁶

$$\begin{aligned} \lim _{T \rightarrow \infty } \frac{{H}_T(\omega )}{T} = 0\quad \textbf{P}\text {-a.s}. \end{aligned}$$

Next, by Theorem 3.9, we know that the law of ${H}_T$ under the measure $\mathbb {Q}_{\mu _\gamma }(\textrm{d}{\dot{B}} ,\textrm{d}\omega )=\mu _\gamma (\textrm{d}\omega ,{\dot{B}} )P(d{\dot{B}} )$ is the same as the law of ${H}_T + \gamma T (\phi \star \phi )(0)$ under $P\otimes \mathbb {P}_0$. Combining these two facts, we have

$$\begin{aligned} \lim _{T \rightarrow \infty } \frac{{H}_T}{T} = \gamma (\phi \star \phi )(0) \quad \mathbb {Q}_{{\mu _\gamma }}\text {-a.s.}, \end{aligned}$$

so that $\mu _{\gamma }\left( \lim _{T \rightarrow \infty } \frac{{H}_T}{T} \ne \gamma (\phi \star \phi )(0) \right) =0$ $\textbf{P}$-a.s. This completes the proof of Proposition 3.11, and therefore, that of Theorem 2.1. $\square$

We will end this section with

Proposition 3.12

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

Proof

Since for any $A\subset \Omega$ Borel measurable, $(\mu _{\gamma ,T,\phi }(A))_{T\ge 0}$ and $(\mu _{\gamma ,T,\phi '}(A))_{T\ge 0}$ are uniformly integrable martingales converging respectively to $\mu _{\gamma ,\phi }(A)$ and $\mu _{\gamma ,\phi '}(A)$, we have $\textbf{P}$-a.s. the following identities:

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,\phi }(A)|{\mathcal {F}}_T]&=\mu _{\gamma ,T,\phi }(A),\\ \textbf{E}[\mu _{\gamma ,\phi '}(A)|{\mathcal {F}}_T]&=\mu _{\gamma ,T,\phi '}(A). \end{aligned}$$

Thus, if $\mu _{\gamma ,\phi }(A)=\mu _{\gamma ,\phi '}(A)$ $\textbf{P}$-a.s., then we deduce that for all $T\ge 0$, $\mu _{\gamma ,T,\phi }(A)=\mu _{\gamma ,T,\phi '}(A)$ $\textbf{P}$-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 3.8, one can deduce that $\textbf{P}\otimes \mathbb {P}_0$-a.s. for all $T\in \mathbb {Q}^+$ and assuming that $\phi \star \phi (0)=\phi '\star \phi '(0)$,

$$\begin{aligned} \textrm{e} ^{\gamma H_{T,\phi }-\frac{\gamma ^2}{2}T\phi \star \phi (0)}=\textrm{e} ^{\gamma H_{T,\phi '}-\frac{\gamma ^2}{2}T\phi '\star \phi '(0)}=\textrm{e} ^{\gamma H_{T,\phi '}-\frac{\gamma ^2}{2}T\phi \star \phi (0)}, \end{aligned}$$

concluding that $\textbf{P}\otimes \mathbb {P}_0$-a.s., $H_{T,\phi }=H_{T,\phi '}$. Note that that for $\omega \in \Omega$, the quadratic variation of $H_{T,\phi }(\omega )-H_{T,\phi '}(\omega )$ is equal to

$$\begin{aligned} 2T\phi \star \phi (0)-2T\int _{\mathbb {R}^d}\phi (y)\phi '(y)\textrm{d}y=2T\left( \int _{\mathbb {R}^d}\phi (y)^2\textrm{d}y-\int _{\mathbb {R}^d}\phi (y)\phi '(y)\textrm{d}y\right) . \end{aligned}$$

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

Remark 9

Looking at (2.1) in Theorem 2.1, one may wonder if for a fixed $\phi$, $\mu _{\gamma ,\phi }$ is the unique measure such that

$$\begin{aligned} \mu _{\gamma ,\phi }\left( \lim _{T \rightarrow \infty } \frac{{H}_T(\omega )}{T} \ne \gamma (\phi \star \phi )(0) \right) = 0 \qquad \textbf{P}\text {-a.s.} \end{aligned}$$

(3.23)

However, this is false: by perturbing $\mathbb {P}_0$ by $\textrm{e} ^{\gamma (H_T+H_1)-\frac{\gamma ^2}{2}\phi \star \phi (0)(T+1+2(T\wedge 1))}$ instead of by $\textrm{e} ^{\gamma H_T-\frac{\gamma ^2}{2}T\phi \star \phi (0)}$ leads to a measure $\tilde{\mu }_{\gamma ,\phi }$ (at least for $\gamma$ small enough so that the limiting measure also exists) such that (3.23) still holds. Indeed, one can follow the proof of Theorem 3.9 and notice that the law of $\dot{B}$ under $\mathbb {Q}_{\tilde{\mu }_{\gamma ,\phi }}$ is the same as the law of

$$\begin{aligned} {{\dot{B}} }_\phi (f)= & {} {\dot{B}} (f)+\gamma \int _{\mathbb {R}_+\times \mathbb {R}^d}f(s,y)\phi (\omega _s-y)\textrm{d}s \textrm{d}y\\{} & {} +\gamma \int _{[0,1]\times \mathbb {R}^d}f(s,y)\phi (\omega _s-y)\textrm{d}s \textrm{d}y, \qquad f \in {\mathcal {S}}^\prime (\mathbb {R}_+\times \mathbb {R}^d) \end{aligned}$$

under $P\otimes \mathbb {P}_0$. In particular, $\mu _{\gamma ,\phi }\ne \tilde{\mu }_{\gamma ,\phi }$. Nevertheless, the distribution of $(H_T)_{T\ge 1}$ under $\mathbb {Q}_{\tilde{\mu }_{\gamma ,\phi }}$ is the same as the distribution of $(H_T+\gamma (T+1)\phi \star \phi (0))_{T\ge 1}$ under under $P\otimes \mathbb {P}_0$. Thus, $\tilde{\mu }_{\gamma ,\phi }$ also satisfies (3.23).

4 Volume decay and Hölder exponents

4.1 Proof of theorem 2.2

Recall the definition of $(\Omega _0, \Vert \cdot \Vert _w)$ from Sect. 2.2. The proof of Theorem 2.2 is split in four parts. First, we will show the (uniform on $\eta \in \Omega _0$) upper bound, stated as

Proposition 4.1

Given $\gamma \in (0,\gamma _c)$ and w satisfying (2.5), there exists $r_0>0$ such that for all $r\in (0,r_0)$, there is an explicit constant $C_1\in (0,\infty )$ (defined in (4.13)) such that

$$\begin{aligned} \begin{aligned}&\limsup _{\varepsilon \rightarrow 0}\varepsilon ^2\sup _{\eta \in \Omega _0}\log \widehat{\mu }_\gamma (\Vert \omega -\eta \Vert _w<r \varepsilon )\le -C_1. \end{aligned} \end{aligned}$$

(4.1)

Proof

We will prove this result in four steps.

Step 1: Since

$$\begin{aligned} \varepsilon ^2\log \widehat{\mu }_\gamma (\Vert \omega -\eta \Vert _w<r\varepsilon )=\varepsilon ^2\log \mu _{\gamma }(\Vert \omega -\eta \Vert _w<r\varepsilon )-\varepsilon ^2\log \mu _{\gamma }(\Omega ) \qquad \forall \eta \in \Omega _0, \end{aligned}$$

(4.2)

and since $\gamma <\gamma _c$, we know that $\textbf{P}$-a.s., when letting $\varepsilon \rightarrow 0$, the second term vanishes. So we consider from now on only the asymptotic behavior of the first term $\varepsilon ^2\log \mu _{\gamma }(\Vert \omega -\eta \Vert <r\varepsilon )$. Let us set

$$\begin{aligned} \Vert f\Vert _{w,\varepsilon }:=\sup _{0<t<\varepsilon ^{-2}}\frac{|f(t)|}{w(t)}. \end{aligned}$$

(4.3)

Using that for every $\varepsilon >0$, $\mu _\gamma = \lim _{s\rightarrow \infty } \mu _{\gamma ,s+ \varepsilon ^{-2}}(\cdot )$ and the monotonicity

$$\begin{aligned} \big \{\Vert \omega -\eta \Vert _w<r\varepsilon \}\subset \{\Vert \omega -\eta \Vert _{w,\varepsilon }<r\varepsilon \big \}, \end{aligned}$$

(4.4)

we obtain

$$\begin{aligned} \begin{aligned}&\log \mu _\gamma (\Vert \omega -\eta \Vert<r \varepsilon )\\&\quad =\limsup _{S\rightarrow \infty }\log \mu _{\gamma ,S+\varepsilon ^{-2}}(\Vert \omega -\eta \Vert _w<r \varepsilon )\\&\quad \le \sup _{S\ge 0}\log \mu _{\gamma ,S+\varepsilon ^{-2}}(\Vert \omega -\eta \Vert _{w,\varepsilon }<r \varepsilon ) \\&\quad = \sup _{S\ge 0}\log \mathbb {E}_0\bigg [\textrm{e} ^{\gamma H_{S+\varepsilon ^{-2}}(\omega ) - \frac{\gamma ^2}{2} (S+ \varepsilon ^{-2}) (\phi \star \phi )(0)} \, \mathbb {1}_{\Vert \omega - \eta \Vert _{w,\varepsilon }< r \varepsilon }\bigg ] \\&\quad \le \sup _{S \ge 0} \log \mathbb {E}_0\left[ \textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega )-\frac{\gamma ^2}{2}\varepsilon ^{-2}\phi \star \phi (0)} \mu _{\gamma ,S}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\, \mathbb {1}_{\Vert \omega -\eta \Vert _{w,\varepsilon }<r \varepsilon } \right] \\&\quad \le \log \mathbb {E}_0\left[ \textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega )-\frac{\gamma ^2}{2}\varepsilon ^{-2}\phi \star \phi (0)}\sup _{S\ge 0} \big (\mu _{\gamma ,S}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big ) \mathbb {1}_{\Vert \omega -\eta \Vert _{w,\varepsilon } <r \varepsilon } \right] ,\\ \end{aligned} \end{aligned}$$

(4.5)

where in the penultimate step we used the Markov property (upon conditioning on the $\sigma$-algebra generated by the Brownian path until time $\varepsilon ^{-2}$; we remind the reader that $\theta _{t,x}$ denotes the time-space shift on the noise $\dot{B}$).

Step 2: Now, since $\gamma <\gamma _c$, by Part (i) of Theorem 2.3 and Doob’s maximal inequality (applied to the Martingale $(\mu _{\gamma ,S}(\Omega ))_{S\ge 0}$),

$$\begin{aligned} \sup _{S\ge 0}\mu _{\gamma ,S}(\Omega )\in L^{p_0}(\textbf{P})\qquad \text{ for } \text{ some } p_0>1. \end{aligned}$$

(4.6)

For any $q,\ell >1$ such that $\frac{1}{p_0}+\frac{1}{q}+\frac{1}{\ell }=1$, we apply Hölder’s inequality to (4.5), so that

$$\begin{aligned} \log \mu _\gamma \big (\Vert \omega -\eta \Vert _w<r \varepsilon \big )\le & {} \frac{1}{q}\log \mathbb {E}_0\left[ \textrm{e} ^{q\gamma H_{\varepsilon ^{-2}}(\omega )-\frac{q}{2}\gamma ^2 \varepsilon ^{-2}\phi \star \phi (0)}\right] \nonumber \\{} & {} +\frac{1}{\ell }\log \mathbb {P}_0(\Vert \omega -\eta \Vert _\varepsilon<r \varepsilon )\nonumber \\{} & {} +\frac{1}{p_0}\log \mathbb {E}_0\left[ \sup _{S\ge 0} \big (\mu _{\gamma ,S}^{p_0}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big )\right] \nonumber \\= & {} \frac{1}{q}\log \mathbb {E}_0\left[ \textrm{e} ^{q\gamma H_{\varepsilon ^{-2}}(\omega )-\frac{q^2}{2}\gamma ^2{\varepsilon ^{-2}}\phi \star \phi (0)}\right] \nonumber \\{} & {} +\frac{1}{\ell }\log \mathbb {P}_0(\Vert \omega -\eta \Vert _{w,\varepsilon } <r \varepsilon )\nonumber \\{} & {} +\frac{1}{p_0}\log \mathbb {E}_0\left[ \sup _{S\ge 0} \big (\mu _{\gamma ,S}^{p_0}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big )\right] \nonumber \\{} & {} +\frac{q-1}{2}\gamma ^2\varepsilon ^{-2}\phi \star \phi (0). \end{aligned}$$

(4.7)

The first term on the right hand side above is controlled by an explicit constant $\lambda (\cdot )$ such that for any $\gamma >0$ and $\textbf{P}$-a.s.

$$\begin{aligned} \frac{1}{q}\,\, \lim _{\varepsilon \rightarrow 0} \varepsilon ^2 \log \mathbb {E}_0\left[ \textrm{e} ^{q\gamma H_{\varepsilon ^{-2}}(\omega )-\frac{q^2}{2}\gamma ^2{\varepsilon ^{-2}}\phi \star \phi (0)}\right] = \frac{1}{q}\lambda \left( q\gamma \right) . \end{aligned}$$

(4.8)

In order to not ebb the flow of the proof, we defer the proof of the above fact to Theorem 4.8 in Sect. 4.2. We therefore bound the second term in the last display of the right hand side of (4.7) as follows. By Anderson’s inequality (see (4.14) below in Lemma 4.2), for any $\eta \in \Omega _0$,

$$\begin{aligned} \mathbb {P}_0(\Vert \omega -\eta \Vert _{w,\varepsilon }<r \varepsilon )\le \mathbb {P}_0(\Vert \omega \Vert _{w,\varepsilon } <r \varepsilon ). \end{aligned}$$

To estimate the right hand side, we will need the following estimate valid for the Wiener measure $\mathbb {P}_0$ on $\Omega _0$ (see [21, Theorem 1.4]): Let $j_{\frac{d-2}{2}}$ be the smallest positive root of the Bessel function $J_{\frac{d-2}{2}}$. If w satisfies (2.5), then

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mathbb {P}_0\left( \Vert \omega \Vert _w<\varepsilon \right) =-\frac{j^2_{\frac{d-2}{2}}}{2}\int _{0}^\infty w^{-2}(t)\textrm{d}t. \end{aligned}$$

(4.9)

Thus, applying (4.9) leads to

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\sup _{\eta \in \Omega _0}\frac{\varepsilon ^2}{\ell }\log \mathbb {P}_0(\Vert \omega -\eta \Vert _{w,\varepsilon } <r \varepsilon )\le -\frac{1}{2\ell r^2}j^2_{\frac{d-2}{2}}\int _{0}^\infty w^{-2}(t)\textrm{d}t. \end{aligned}$$

(4.10)

Step 3: Thus, we concentrate on the third term of the last display of the (r.h.s.) of (4.7). Set

$$\begin{aligned} f_{\varepsilon }:=\mathbb {E}_0\Big [\sup _{S\ge 0} \big (\mu _{\gamma ,S}^{p_0}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big )\Big ]. \end{aligned}$$

(4.11)

Since $\dot{B}$ is stationary with respect to space-time shifts $(\theta _{t,x})_{t>0,x\in \mathbb {R}^d}$, then $f_{\varepsilon }$ is stationary and by (4.6), $\textbf{E}[f_{\varepsilon }]=\textbf{E}[\sup _{S\ge 0}\mu _{\gamma ,S}(\Omega )^p]< \infty$. By the ergodic theorem for stationary processes,⁷ there is a (possibly random) $C=C(\dot{B})$ such that for all $\varepsilon >0$ sufficiently small,

$$\begin{aligned} \mathbb {E}_0\left[ \sup _{S\ge 0} \big (\mu _{\gamma ,S}^{p_0}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big )\right] \le C\varepsilon ^{-2} \qquad \textbf{P}\,\text {-a.s.}, \end{aligned}$$

and consequently,

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0} \varepsilon ^2 \log \mathbb {E}_0\left[ \sup _{S\ge 0}\big (\mu _{\gamma ,S}^{p_0}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\big )\right] \le 0\qquad \textbf{P}\text {-a.s.} \end{aligned}$$

(4.12)

Combining (4.7–4.12) and optimizing over $q,\ell$, we obtain

$$\begin{aligned} \begin{aligned}&\limsup _{\varepsilon \rightarrow 0}\varepsilon ^2\log \widehat{\mu }_\gamma (\Vert \omega \Vert _w<r \varepsilon )\\&\quad \le -\sup _{\begin{array}{c} q,\ell >1:\\ \frac{1}{q}+\frac{1}{\ell }=\frac{p_0-1}{p_0} \end{array}} \left[ \frac{1}{2\ell r^2}j^2_{\frac{d-2}{2}}\int _{0}^\infty w^{-2}(t)\textrm{d}t-\frac{q-1}{2}\gamma ^2\phi \star \phi (0)-\frac{1}{q}\lambda \left( q\gamma \right) \right] \\ {}&=:-C_1(d,\gamma ,g,r), \end{aligned} \end{aligned}$$

(4.13)

and observe that $C_1>0$ if we choose r small enough. This completes the proof of the proposition.

Step 4: In Step 2 above, we have used the following general result for Gaussian measures:

Lemma 4.2

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

• (i) For any symmetric convex subset $A\subset E$ and $x\in E$,

  $$\begin{aligned} \mu (A+ x) \le \mu (A). \end{aligned}$$

  (4.14)

• (ii) If $H_\mu$ is the Cameron-Martin space of $\mu$,⁸ then for any $r>0$, and $\eta \in H_\mu$,

  $$\begin{aligned} \textrm{e} ^{-\frac{1}{2}\Vert \eta \Vert _\mu ^2} \mu (\omega \in E: \Vert \omega \Vert \le r) \le \mu (\omega \in E:\Vert \omega - \eta \Vert \le r) \le \mu (\omega \in E: \Vert \omega \Vert \le r). \end{aligned}$$

  (4.15)

Proof

(4.14) is Anderson’s inequality, which follows from log-concavity of Gaussian measures (see [4, Thm. 2.8.10], or [22, Thm. 2.13]). The upper bound of (4.15) follows from (4.14), while the lower bound follows from the Cameron-Martin formula

$$\begin{aligned} \mu (A-\eta )= \int _A \exp \big (-\frac{1}{2} \Vert \eta \Vert _\mu ^2 + \langle \omega , \eta \rangle _\mu \big ) \mu (\textrm{d}\omega ), \qquad A\subset E, \eta \in H_\mu , \end{aligned}$$

as well as Hölder’s inequality and the symmetry of $\langle \omega , \eta \rangle _\mu$ on $\{\omega \in E: \Vert \omega \Vert \le r\}$, see [22, Thm. 3.1]. $\square$

We now turn to the lower bound in Theorem 2.2, stated as

Proposition 4.3

Given $\gamma \in (0,\gamma _c)$ and w satisfying (2.5), there exists $r_0>0$ such that for all $r\in (0,r_0)$, there is an explicit constant $C_2 \in (0,\infty )$ (defined in (4.24)) such that

$$\begin{aligned} \begin{aligned}&\liminf _{\varepsilon \rightarrow 0}\varepsilon ^2\log \widehat{\mu }_\gamma (\Vert \omega \Vert _w<r\varepsilon ) \ge - C_2. \end{aligned} \end{aligned}$$

(4.16)

Proof

This will also be shown in four steps.

Step 1: Recall the norm $\Vert \cdot \Vert _{w,\varepsilon }$ from (4.3). We will first prove a lower bound on $\liminf _{\varepsilon \rightarrow 0}\varepsilon ^2\log \widehat{\mu }_\gamma (\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon )$. As remarked below (4.2), for $\gamma \in (0,\gamma _c)$, it is sufficient to handle

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\varepsilon ^2\log {\mu }_\gamma (\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon ). \end{aligned}$$

For a fixed $\gamma <\gamma _c$, we apply part (ii) of Theorem 2.3 to find some $q_0>0$ such that for all $0<q\le q_0$,

$$\begin{aligned} \textbf{E}[\mu _\gamma (\Omega )^{-q}]<\infty . \end{aligned}$$

(4.17)

Then

$$\begin{aligned} \begin{aligned}&\mu _{\gamma }(\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon )\\&\quad =\liminf _{S\rightarrow \infty } \mu _{\gamma ,S+\varepsilon ^{-2}}(\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon )\\&\quad =\liminf _{S\rightarrow \infty }\mathbb {E}_0\left[ \left( \textrm{e} ^{\gamma H_{\varepsilon ^{-2}}-\frac{\gamma ^2}{2}\varepsilon ^{-2}\phi \star \phi (0)} \mathbb {1}_{\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon }\right) ~\left( \mu _{\gamma ,S}(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\right) \right] \\&\quad \ge \mathbb {E}_0\bigg [\bigg (\textrm{e} ^{\gamma H_{\varepsilon ^{-2}}-\frac{\gamma ^2}{2}\varepsilon ^{-2}\phi \star \phi (0)}\mathbb {1}_{\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon }\bigg )~\Big (\mu _{\gamma }(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\Big ) \bigg ]\\&\quad \ge \mathbb {E}_0\bigg [\textrm{e} ^{\frac{q\gamma }{q+1} H_{\varepsilon ^{-2}} -\frac{\gamma ^2q}{2(q+1)}\varepsilon ^{-2}\phi \star \phi (0)} \mathbb {1}_{\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon } \bigg ]^{1+\frac{1}{q}}\mathbb {E}_0\bigg [\mu _{\gamma }(\Omega )^{-q}\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}} \bigg ]^{-\frac{1}{q}}. \end{aligned} \end{aligned}$$

(4.18)

In the second equality above we used the Markov property (again by conditioning on the $\sigma$-algebra generated by the Brownian path until time $\varepsilon ^{-2}$), in the subsequent lower bound we used Fatou’s lemma and in the last lower bound we invoked reverse Hölder’s inequality. To justify this step, recall that Hölder’s inequality implies that if $\theta >1$ and $f(\omega ),g(\omega )$ are measurable functions satisfying $\Vert fg\Vert _1 < \infty$ and $\Vert g\Vert _{-\frac{1}{\theta -1}} <\infty$, then

$$\begin{aligned} \Vert fg \Vert _1 \ge \Vert f\Vert _{\frac{1}{\theta }}\Vert g\Vert _{-\frac{1}{\theta -1}}. \end{aligned}$$

To deduce the last lower bound in (4.18), we apply the above inequality for

$$\begin{aligned} f(\omega )= & {} \textrm{e} ^{\gamma H_{\varepsilon ^{-2}}-\frac{\gamma ^2}{2}\varepsilon ^{-2}\phi \star \phi (0)} \mathbb {1}_{\Vert \omega \Vert _{w,\varepsilon } <r\varepsilon }, \\ g(\omega )= & {} \mu _{\gamma }(\Omega )\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}}\qquad \text{ and }\,\,\, \theta =\frac{q+1}{q}>1, \end{aligned}$$

Thus,

$$\begin{aligned}{} & {} \varepsilon ^{2}\log \mu _{\gamma }(\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon )\nonumber \\{} & {} \quad \ge \varepsilon ^2 \left( \frac{q+1}{q}\right) \log \mathbb {E}_0 \left[ \textrm{e} ^{\frac{q\gamma }{q+1} H_{\varepsilon ^{-2}} -\frac{q\gamma ^2}{2(q+1)}\varepsilon ^{-2}\phi \star \phi (0)} \mathbb {1}_{\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon } \right] \nonumber \\{} & {} \qquad -\frac{1}{q}\varepsilon ^2\log \mathbb {E}_0\left[ \mu _{\gamma }(\Omega )^{-q} \circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}} \right] \nonumber \\{} & {} \quad =-\frac{\gamma ^2}{2(q+1)}\phi \star \phi (0)+\varepsilon ^2 \left( \frac{q+1}{q}\right) \log \mu _{\frac{q\gamma }{q+1}, {\varepsilon ^{-2}}}(\Vert \omega \Vert _{w,\varepsilon } <r\varepsilon )\nonumber \\{} & {} \qquad -\frac{1}{q}\varepsilon ^2\log \mathbb {E}_0\left[ \mu _{\gamma }(\Omega )^{-q} \circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}} \right] . \end{aligned}$$

(4.19)

Step 2: In this step, we will show that for all $\gamma >0$,

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(\Vert \omega \Vert _{w,\varepsilon } <r \varepsilon )\ge -\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t+\frac{\gamma ^2}{2}(\phi \star \phi )(0)\bigg ) \end{aligned}$$

(4.20)

This in turn will imply that

$$\begin{aligned}{} & {} \liminf _{\varepsilon \rightarrow 0}\varepsilon ^2 \left( \frac{q+1}{q}\right) \log \mu _{\frac{q\gamma }{q+1},{\varepsilon ^{-2}}}(\Vert \omega \Vert _{w,\varepsilon } <r\varepsilon )\nonumber \\{} & {} \quad \ge -\left( \frac{q+1}{q}\right) \bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t+\frac{\gamma ^2}{2} \frac{q^2}{(q+1)^2} (\phi \star \phi )(0)\bigg ) \end{aligned}$$

(4.21)

To show (4.20), we first note that, for any event $A_\varepsilon \subset \Omega$ with $\mathbb {P}_0(A_\varepsilon )>0$ it holds $\textbf{P}$-almost surely that

$$\begin{aligned} {\liminf _{\varepsilon \downarrow 0}} \varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )= {\liminf _{\varepsilon \downarrow 0}} \varepsilon ^2 \textbf{E}\big [\log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )\big ]. \end{aligned}$$

(4.22)

We refer to Lemma 4.9 for a proof of this result. Therefore,

$$\begin{aligned} \varepsilon ^2 \textbf{E}\big [\log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )\big ]= & {} \varepsilon ^2 \textbf{E}\bigg [ \log \mathbb {E}_0\Big [ \textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega ) - \frac{\gamma ^2}{2} \varepsilon ^{-2}(\phi \star \phi )(0)} \,\, \mathbb {1}_{A_\varepsilon }\Big ]\bigg ] \\= & {} \varepsilon ^2 \textbf{E}\bigg [ \log \mathbb {E}_0\Big [ \mathbb {E}_0\Big (\textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega ) - \frac{\gamma ^2}{2} \varepsilon ^{-2}(\phi \star \phi )(0)} \,\, \mathbb {1}_{A_\varepsilon }\Big | A_\varepsilon \Big )\Big ]\bigg ] \\= & {} \varepsilon ^2 \textbf{E}\bigg [ \log \mathbb {E}_0\bigg [ \mathbb {1}_{A_\varepsilon }\,\, \mathbb {E}_0\Big (\textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega ) - \frac{\gamma ^2}{2} \varepsilon ^{-2}(\phi \star \phi )(0)} \Big | A_\varepsilon \Big )\bigg ] \\= & {} \varepsilon ^2 \textbf{E}\bigg [ \log \mathbb {P}_0(A_\varepsilon ) + \log \mathbb {E}_0\bigg (\textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega ) - \frac{\gamma ^2}{2} \varepsilon ^{-2}(\phi \star \phi )(0)} \Big | A_\varepsilon \Big )\bigg ] \\= & {} \varepsilon ^2 \log \mathbb {P}_0(A_\varepsilon ) - \frac{\gamma ^2}{2} (\phi \star \phi )(0)+ \varepsilon ^2 \textbf{E}\bigg [\log \mathbb {E}_0\Big (\textrm{e} ^{\gamma H_{\varepsilon ^{-2}}(\omega ) } \Big | A_\varepsilon \Big )\bigg ] \\\ge & {} \varepsilon ^2 \log \mathbb {P}_0(A_\varepsilon ) - \frac{\gamma ^2}{2} (\phi \star \phi )(0) + \varepsilon ^2 \mathbb {E}_0\big [\gamma \textbf{E}(H_{\varepsilon ^{-2}}(\omega )) \big | A_\varepsilon ]\big ] \\= & {} \varepsilon ^2 \log \mathbb {P}_0(A_\varepsilon ) - \frac{\gamma ^2}{2} (\phi \star \phi )(0). \end{aligned}$$

We note that in the lower bound above we applied Jensen’s inequality and in the following display we used that $H_{\varepsilon ^{-2}}$ is an Itô integral with $\textbf{E}[H_{\varepsilon ^{-2}}]=0$. We now choose $A_\varepsilon =\{\omega :\Vert \omega \Vert _{w,\varepsilon } < r\varepsilon \}$ and note that (4.4) and (4.9) imply in particular that for this choice, $\mathbb {P}_0(A_\varepsilon )>0$ for $\varepsilon >0$ sufficiently small. Thus these two facts, combined with the above estimate imply (4.20).

Step 3: Now using (4.17), the stationarity of the white noise, combined with the ergodic theorem and invoking (4.17) (similar to the argument for deducing (5.7)), we obtain that

$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0}\frac{1}{q}\varepsilon ^2\log \mathbb {E}_0\left[ \mu _{\gamma }(\Omega )^{-q}\circ \theta _{\varepsilon ^{-2},\omega _{\varepsilon ^{-2}}} \right] \le 0 \qquad \textbf{P}\text{-a.s. } \end{aligned}$$

(4.23)

Finally, combining (4.19)-(4.23), and optimizing over $0<q<q_0$, it holds $\textbf{P}$-a.s.

$$\begin{aligned}{} & {} \liminf _{\varepsilon \rightarrow 0}\varepsilon ^{2}\log \mu _{\gamma }(\Vert \omega \Vert _{w,\varepsilon }<r\varepsilon )\nonumber \\{} & {} \quad \ge -\inf _{0<q<q_0}\left[ \left( \frac{q+1}{q}\right) \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t+\frac{\gamma ^2}{2}\phi \star \phi (0)\right] \nonumber \\{} & {} \quad = -\left( \frac{q_0+1}{q_0}\right) \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t-\frac{\gamma ^2}{2}\phi \star \phi (0) \nonumber \\{} & {} \quad =:-C_2(d,\gamma ,w,r). \end{aligned}$$

(4.24)

Clearly, $C_2<\infty$.

Step 4: Note that we have shown Proposition 4.3 for the restricted weighted norm $\Vert \cdot \Vert _{w,\varepsilon }$. To extend the argument to the weighted norm $\Vert \cdot \Vert _w$, we can apply again the reverse Hölder’s inequality to deduce, for any $p>1$,

$$\begin{aligned}&\liminf _{\varepsilon \rightarrow 0}\varepsilon ^2 \log \mu _\gamma (\Vert \omega \Vert _w<\varepsilon )\ge p\liminf _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mu _\gamma (\Vert \omega \Vert _{w,\varepsilon }<\varepsilon )\\&\qquad -(p-1)\limsup _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mu _\gamma \left( \sup _{t>\varepsilon ^{-2}}\frac{|\omega (t)|}{w(t)}<\varepsilon \right) \\&\quad \ge -p C_2-(p-1)\limsup _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mu _\gamma \left( \Omega \right) \\&\quad =-p C_2. \end{aligned}$$

As $p>1$ is arbitrary, we conclude the proof of (2.8). $\square$

To finish the proof of Theorem 2.2, we show

Corollary 4.4

Let $p_0>1$ and $q_0>0$. Then for any $\gamma$ small enough, the constants $0<C_1\le C_2<\infty$ can be chosen as

$$\begin{aligned} \begin{aligned} C_1&:=\frac{p_0-1}{p_0 }\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\bigg )-\frac{1}{2p_0}\frac{\gamma ^2}{2}(\phi \star \phi )(0),\\ C_2&:=\bigg (\frac{q_0+1}{q_0}\bigg )\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\bigg )+\frac{\gamma ^2}{2}(\phi \star \phi )(0), \end{aligned} \end{aligned}$$

(4.25)

where $j_{\frac{d-2}{2}}$ is the smallest positive root of the Bessel function $J_{\frac{d-2}{2}}$. In particular, for any $r>0$,

$$\begin{aligned} \lim _{\gamma \rightarrow 0}C_1(\gamma ,d,r)= & {} \lim _{\gamma \rightarrow 0}C_2(\gamma ,d,r)=\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t \nonumber \\= & {} \lim _{\varepsilon \rightarrow 0}\varepsilon ^2\log \mathbb {P}_0(\Vert \omega \Vert _w<r\varepsilon ). \end{aligned}$$

(4.26)

In other words, both exponents converge as $\gamma \rightarrow 0$ to the volume decay exponent for the Wiener measure.

Proof

First we check (4.25). Since the constant $C_2$ is the same from (4.24), we only show the corresponding estimate for $C_1$. We note that by Theorem 4.8, it holds that $\lambda (\gamma )\le 0$ for all $\gamma >0$. Therefore, if $\gamma$ is small enough so that $\mu _\gamma \in L^{2 p_0}(\textbf{P})$, the formula from (4.13) can be bounded by below by (choosing $\ell =\frac{p_0}{p_0-1}$ and $q=2p_0$),

$$\begin{aligned} C_1:=\frac{p_0-1}{p_0 }\bigg (\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\bigg )-\frac{1}{2p_0}\frac{\gamma ^2}{2}(\phi \star \phi )(0). \end{aligned}$$

Next, we prove (4.26). By the previous part, let $\varepsilon >0$ and $p_0>1$, $q_0>0$ such that $\frac{p_0-1}{p_0}>1-\varepsilon$ and $\frac{q_0+1}{q_0}<1+\varepsilon$. Then if $\gamma$ is small enough so that (4.25) holds and $\frac{\gamma ^2}{2}(\phi \star \phi )(0)<\varepsilon$, then

$$\begin{aligned} 0\le \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t-C_1\le \varepsilon \left( \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\right) +\varepsilon \end{aligned}$$

and

$$\begin{aligned} 0\le C_2-\frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\le \varepsilon \left( \frac{j^2_{\frac{d-2}{2}}}{2r^2}\int _{0}^\infty w^{-2}(t)\textrm{d}t\right) +\varepsilon . \end{aligned}$$

Since $\varepsilon >0$ is arbitrary, this completes the proof of Corollary 4.4. $\square$

4.2 A variational formula on a group-invariant compactification

In Step 2 of the proof of Proposition 4.1 we have used (4.8) which gives an explicit variational formula for the free energy $\lim _{T\rightarrow \infty }\frac{1}{T}\log \mu _{\gamma ,T}(\Omega )$ as well its properties for all $\gamma >0$. Description of this formula and its properties need some setting up and further notation.

We denote by ${{\mathcal {M}}}_1= {{{\mathcal {M}}}_1}(\mathbb {R}^d)$ (resp., ${{\mathcal {M}}}_{\le 1}$) 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 $\widetilde{\mu }=\{\mu \star \delta _x:\, x\in \mathbb {R}^d\}\in {\widetilde{{{\mathcal {M}}}}}_1$. Then we define

$$\begin{aligned} \widetilde{{\mathcal {X}}}=\Big \{\xi :\xi =\{\widetilde{\alpha }_i\}_{i\in I},\alpha _i\in {\mathcal {M}}_{\le 1},\sum _{i\in I}\alpha _i(\mathbb {R}^d)\le 1\Big \} \end{aligned}$$

(4.27)

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 $k\ge 2$, ${\mathcal {H}}_k$ 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 $h\in {\mathcal {H}}=\bigcup _{k\ge 2}{\mathcal {H}}_k$, the functionals

$$\begin{aligned} {\mathscr {J}} (h,\xi )=\sum _{{\widetilde{\alpha }}\in \xi }\int _{(\mathbb {R}^d)^k }h(x_1,\ldots , x_k)\alpha (\textrm{d}x_1)\cdots \alpha (\textrm{d}x_k). \end{aligned}$$

(4.28)

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

$$\begin{aligned} {\mathscr {J}}(h,\xi _n)\rightarrow {\mathscr {J}}(h,\xi )\qquad \forall \,\, h\in {\mathcal {H}}. \end{aligned}$$

This leads to the following definition of the metric $\textbf{D}$ on $\widetilde{{\mathcal {X}}}$. For any $\xi _1,\xi _2\in \widetilde{{\mathcal {X}}}$, set

$$\begin{aligned} \textbf{D}(\xi _1,\xi _2)&=\sum _{r=1}^{\infty }\frac{1}{2^r}\frac{1}{1+\Vert h_r\Vert _{\infty }} \bigg |{\mathscr {J}}(h_r,\xi _1)- {\mathscr {J}}(h_r,\xi _2)\bigg | \\&=\sum _{r=1}^{\infty }\frac{1}{2^r}\frac{1}{1+\Vert h_r\Vert _{\infty }}\bigg |\sum _{\widetilde{\alpha }\in \xi _1}\int h_r(x_1,...,x_{k_r})\prod _{i=1}^{k_r}\alpha (\textrm{d}x_i)\\&-\sum _{\widetilde{\alpha }\in \xi _2}\int h_r(x_1,...,x_{k_r})\prod _{i=1}^{k_r}\alpha (\textrm{d}x_i)\bigg |. \end{aligned}$$

The following result was proved in [26, Theorem 3.1$-$3.2].

Theorem 4.5

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

• $\textbf{D}$ is a metric on $\widetilde{{\mathcal {X}}}$ and the space ${\widetilde{{{\mathcal {M}}}}}_1(\mathbb {R}^d)$ is dense in $(\widetilde{{\mathcal {X}}},\textbf{D})$.

• 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 $\textbf{D}$.

• Let a sequence $(\xi _n)_n$ in $\widetilde{{\mathcal {X}}}$ consist of a single orbit ${\widetilde{\gamma }}_n$ and $\textbf{D}(\xi _n,\xi )\rightarrow 0$ where $\xi =({\widetilde{\alpha }}_i)_i\in \widetilde{{\mathcal {X}}}$ such that $\alpha _1(\mathbb {R}^d)\ge \alpha _2(\mathbb {R}^d) \ge \dots$. 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

    $$\begin{aligned} \begin{aligned} \alpha _{n,i}\star \delta _{a_{n,i}} \Rightarrow \alpha _i \quad \text{ with }\quad \lim _{n\rightarrow \infty } \, \inf _{i\ne j}\,\, |a_{n,i}- a_{n,j}| =\infty . \end{aligned} \end{aligned}$$

  • The sequence $\beta _n$ totally disintegrates, meaning that for any $r>0$, $\sup _{x\in \mathbb {R}^d} \beta _n\big (B_r(x)\big )\rightarrow 0$.

Recall the definition of $\Phi _\gamma :\widetilde{{\mathcal {X}}}\rightarrow \mathbb {R}$ from (4.35):

$$\begin{aligned} \Phi _\gamma (\xi )=\frac{\gamma ^2}{2}\sum _{i\in I}\int _{\mathbb {R}^d\times \mathbb {R}^d}V(x_1-x_2)\prod _{j=1}^2\alpha _i(\textrm{d}x_j), \qquad \xi =(\widetilde{\alpha }_i)_{i\in I}. \end{aligned}$$

(4.29)

Because of shift-invariance of the integrand in (4.29), $\Phi _\gamma$ is well-defined on $\widetilde{{\mathcal {X}}}$. Moreover, we have

Lemma 4.6

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

Proof

For the continuity of $\Phi _\gamma$, we refer to [26, Corollary 3.3]. Recall that $V=\phi \star \phi$ and $\phi$ is rotationally symmetric. Hence, for any $\alpha \in {{\mathcal {M}}}_{\le 1}(\mathbb {R}^d)$, by Cauchy-Schwarz inequality,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2d}}&V(x_1-x_2) \, \alpha (\textrm{d}x_1)\, \alpha (\textrm{d}x_2) =\int _{\mathbb {R}^{2d}} \alpha (\textrm{d}x_1)\alpha (\textrm{d}x_2) \, \int _{\mathbb {R}^d} \textrm{d}z \, \phi (x_1-z) \, \phi (x_2-z) \\&\le \int _{\mathbb {R}^{2d}} \alpha (\textrm{d}x_1)\alpha (\textrm{d}x_2) \, \bigg [\int _{\mathbb {R}^d} \textrm{d}z \phi ^2(x_1-z)\bigg ]^{1/2} \,\, \bigg [\int _{\mathbb {R}^d} \textrm{d}z \phi ^2(x_2-z)\bigg ]^{1/2} \\&\le \alpha \big (\mathbb {R}^d\big )^2 \Vert \phi \Vert _2^2. \end{aligned} \end{aligned}$$

(4.30)

Thus, $\Phi _\gamma (\xi )\le \frac{\gamma ^2 (\phi \star \phi )(0)}{2} \sum _{i\in I} (\alpha _i(\mathbb {R}^d))^2 \le \frac{\gamma ^2 (\phi \star \phi )(0)}{2}$ since, for $\xi =({\widetilde{\alpha }}_i)_{i\in I} \in \widetilde{{\mathcal {X}}}$, we have $\sum _{i\in I} \alpha _i(\mathbb {R}^d)\le 1$. Moreover, since $V=\phi \star \phi$ is non-negative, also ${\Phi _\gamma }(\cdot )\ge 0$.

$\square$

To define the required variational formula, we also need a Markovian dynamics on the space ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$ of probability measures on the compactification ${\mathcal {X}}$. To this end, we recall the basic notations and properties of this dynamics.

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

$$\begin{aligned} {\mathscr {F}}_t(\alpha )=\int _{\mathbb {R}^d}\int _{\mathbb {R}^d}\alpha (\textrm{d}z) \mathbb {E}_z\bigg [\mathbb {1}{\{\omega _t\in \textrm{d}x\}}\, \exp \bigg \{\gamma {\mathscr {H}}_t(\omega )-\frac{\gamma ^2}{2}tV(0)\bigg \}\bigg ], \end{aligned}$$

and note that for any $a\in \mathbb {R}^d$ and $t>0$, ${\mathscr {F}}_t(\alpha _i)\overset{{\scriptscriptstyle {({d}})}}{=}{\mathscr {F}}_t(\alpha _i\star \delta _a)$. Hence, we may define ${\mathscr {F}}_t, \overline{{\mathscr {F}}}_t: \widetilde{{\mathcal {X}}}\mapsto \mathbb {R}$ as

$$\begin{aligned} \begin{aligned}&{\mathscr {F}}_t(\xi )=\sum _i {\mathscr {F}}_t(\alpha _i), \quad \overline{{\mathscr {F}}}_t(\xi )={\mathscr {F}}_t(\xi )+\textbf{E}\big [ Z_t-{\mathscr {F}}_t(\xi )\big ]\\&\quad \forall \xi =({\widetilde{\alpha }}_i)_{i\in I}\in \widetilde{{\mathcal {X}}}, \,\,\, Z_t=\mathbb {E}_0[\textrm{e} ^{\gamma H_t}]. \end{aligned} \end{aligned}$$

(4.31)

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

$$\begin{aligned} \begin{aligned}&\alpha _i^{{\scriptscriptstyle {({t}})}}(\textrm{d}x):=\frac{1}{\overline{{\mathscr {F}}}_t(\xi )}\int _{\mathbb {R}^d}\alpha _i (\textrm{d}z)\mathbb {E}_z\bigg [\mathbb {1}{\{\omega _t\in \textrm{d}x\}}\, \exp \big \{\gamma H_t(\omega )-\frac{\gamma ^2}{2}tV(0)\big \}\bigg ] ,\\&\xi ^{{\scriptscriptstyle {({t}})}}:= \big ({\widetilde{\alpha }}_i^{{\scriptscriptstyle {({t}})}}\big )_{i\in I} \in \widetilde{{\mathcal {X}}}. \end{aligned} \end{aligned}$$

(4.32)

Recall that ${\mathscr {F}}_t(\alpha _i)\overset{{\scriptscriptstyle {({d}})}}{=}{\mathscr {F}}_t(\alpha _i\star \delta _a)$ and likewise, $(\alpha _i\star \delta _a)^{{\scriptscriptstyle {({t}})}}(\textrm{d}x)\overset{(d)}{=}(\alpha _i^{{\scriptscriptstyle {({t}})}}\star \delta _a)(\textrm{d}x)$. For any $\vartheta \in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$, then (4.32) further defines a transition kernel

$$\begin{aligned} \Pi _t(\vartheta ,\textrm{d}\xi ^\prime )= \int _{\widetilde{{\mathcal {X}}}} \pi _t(\xi ,\textrm{d}\xi ^\prime ) \vartheta (\textrm{d}\xi ) \qquad \text{ where }\quad \pi _t(\xi ,\textrm{d}\xi ^{\prime })=\mathbb {P}\big [\xi ^{{\scriptscriptstyle {({t}})}}\in \textrm{d}\xi ^\prime |\xi \big ]\in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}). \end{aligned}$$

(4.33)

For any $\gamma >0$, the set of fixed points of this dynamics is non-empty, as shown by

Lemma 4.7

The set

$$\begin{aligned} \mathfrak {m}_\gamma =\big \{\vartheta \in {\mathcal {M}}_1(\widetilde{{\mathcal {X}}}):\Pi _t\,\vartheta =\vartheta \text { for all } t>0\big \} \end{aligned}$$

(4.34)

of fixed points of $\Pi _t$ is a non-empty, compact subset of ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$.

Proof

Note that ${\mathfrak m_\gamma }\ne \emptyset$, because $\delta _{\widetilde{0}}\in {\mathfrak m_\gamma }$. Moreover, using the definition of the metric $\textbf{D}$ on $\widetilde{{\mathcal {X}}}$ and by the resulting convergence criterion determined by Theorem 4.5, it was shown in [6, Theorem 3.1] that the map

$$\begin{aligned} \widetilde{{\mathcal {X}}}\ni \xi \mapsto \pi _t(\xi ,\cdot )\in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}) \qquad \text{ is } \text{ continuous. } \end{aligned}$$

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 ${\mathfrak m_\gamma }$. $\square$

The following result provides the required variational formula (4.8) and additional information about the maximizers of the continuous map ${\mathcal {E}}_{\Phi _{\gamma }}$, defined in (4.35) in the compact set $\mathfrak {m}_\gamma$.

Theorem 4.8

Fix $d\in \mathbb {N}$ and $\gamma >0$, recall the set $\mathfrak m_\gamma$ from (4.34) and the functional

$$\begin{aligned} \begin{aligned}&\Phi _{\gamma }(\xi )=\frac{\gamma ^2}{2}\sum _{{\widetilde{\alpha }}\in \xi }\int _{\mathbb {R}^d\times \mathbb {R}^d}(\phi \star \phi )(x_1-x_2)\prod _{j=1}^2\alpha (\textrm{d}x_j) \quad \forall \, \xi \in \widetilde{{\mathcal {X}}}, \quad \text{ and } \text{ we } \text{ define }\\&{{\mathscr {I}}}_{\Phi _{\gamma }}(\vartheta )= \int _{\widetilde{{\mathcal {X}}}} \Phi _{\gamma }(\xi )\,\vartheta (\textrm{d}\xi ) \qquad \vartheta \in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}). \end{aligned} \end{aligned}$$

(4.35)

Then we have the following implications:

• $\textbf{P}$-almost surely,

  $$\begin{aligned} \begin{aligned} \lim _{\varepsilon \downarrow 0}\varepsilon ^2\log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )&= \lim _{\varepsilon \downarrow 0}\varepsilon ^2 \textbf{E}\big [\log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )\big ] \\&= \lambda (\gamma ):=-\sup _{\vartheta \in {{\mathfrak m_\gamma }}}{{\mathscr {I}}}_{\Phi _{\gamma }}(\vartheta ) =- \sup _{\vartheta \in {\mathfrak m_\gamma }} \,\,\int _{\widetilde{{\mathcal {X}}}} \Phi _{\gamma }(\xi )\vartheta (\textrm{d}\xi ) \end{aligned} \end{aligned}$$

  (4.36)

  The above supremum in (4.36) is attained, and we always have $\lambda (\gamma ) \in [-\gamma ^2(\phi \star \phi )(0)/2, 0]$.

• Moreover, there exists $\gamma _1=\gamma _1(d)$ such that $\gamma _1>0$ if $d\ge 3$ 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, $\sup _{{\mathfrak m_\gamma }}{{\mathcal {E}}}_{\Phi _{\gamma }}(\cdot ) =0$ and also,

  $$\begin{aligned} \lambda (\gamma )=0 \qquad d\ge 3 \text{ and } \gamma \in (0,\gamma _c)\text{. } \end{aligned}$$

  (4.37)

  Moreover, if $\gamma >\gamma _1$, then $\sup _{{\mathfrak m_\gamma }}{{\mathscr {I}}}_{\Phi _{\gamma }}>0$.

• Finally, if $\vartheta \in {\mathfrak m_\gamma }$ is a maximizer of ${{\mathscr {I}}}_{\Phi _{\gamma }}(\cdot )$ and $\vartheta (\xi )>0$ for $\xi =({\widetilde{\alpha }}_i)_{i\in I}\in \widetilde{{\mathcal {X}}}$, then $\sum _{i\in I}\alpha _i(\mathbb {R}^d)=1$. In other words, any maximizer of (4.36) assigns positive mass only to those elements of $\widetilde{{\mathcal {X}}}$ whose total mass add up to one.

Proof

We will briefly sketch the argument and refer to [6] for details. To derive (4.36), note that $\log \mu _{\gamma ,T}(\Omega )$ can be decomposed

$$\begin{aligned} \log \mu _{\gamma ,T}(\Omega )= M_T - \int _0^T \Phi _\gamma (\widetilde{\mathbb Q}_t) \textrm{d}t \end{aligned}$$

(4.38)

in terms of a square-integrable martingale $M_T=\gamma \int _0^T \int _{\mathbb {R}^d} \mathbb {E}^{{\widehat{\mu }}_{\gamma ,t}}\big [\phi (y- \omega _t)\big ] \dot{B}(t,y) \textrm{d}y \textrm{d}t$ and an additive functional $\int _0^T \Phi _\gamma (\widetilde{\mathbb Q}_t) \textrm{d}t$, where $\mathbb {Q}_t={{\widehat{\mu }}_{\gamma ,t}[\omega _t \in \cdot ]}\in {{\mathcal {M}}}_1(\mathbb {R}^d)$ and ${\widetilde{\mathbb {Q}}}_t\in \widetilde{{\mathcal {X}}}$. As a consequence,

$$\begin{aligned} \begin{aligned}&\frac{1}{T}\textbf{E}[ \log \mu _{\gamma ,T}(\Omega )]= \frac{1}{T}\int _0^T \textbf{E}[ \Phi _\gamma ({\widetilde{\mathbb {Q}}}_t)]\textrm{d}t= \textbf{E}\big [ {\mathscr {I}}_{\Phi _{\gamma }}(\nu _T)\big ], \quad \text{ where }\\&\nu _T=\frac{1}{T}\int _0^T \delta _{{\widetilde{\mathbb {Q}}}_t} \textrm{d}t \in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}), \quad \text{ and } \text{ also } \\&\frac{1}{T}\mu _{\gamma ,T}(\Omega ) - \frac{1}{T} \int _0^T \Phi _\gamma ({\widetilde{\mathbb {Q}}}_t) \textrm{d}t\rightarrow 0,\qquad \text{ almost } \text{ surely } \text{ w.r.t. }\, \textbf{P}. \end{aligned} \end{aligned}$$

Here ${\mathscr {I}}_{\Phi _\gamma }$ is defined in (4.36). Therefore

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )= \liminf _{\varepsilon \rightarrow 0} {\mathscr {I}}_{\Phi _\gamma }(\nu _{\varepsilon ^{-2}}) \qquad \textbf{P}\text{-a.s. } \end{aligned}$$

Thus, studying the behavior of the left hand side reduces to studying the asymptotic behavior of $\nu _T$ in the space ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$. Because of the properties of the dynamics $\Pi _t$ mentioned in Lemma 4.7 and its proof, it can be shown that $d(\nu _T, m_\gamma )\rightarrow 0$ almost surely, for any $\gamma >0$ (where d is a metric which induces the weak topology on probability measures of the compactification $\widetilde{{\mathcal {X}}}$). This can be used to derive the variational formula for the (quenched) free energy $\lim _{\varepsilon \rightarrow 0}\varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(\Omega )= - \sup _{\vartheta \in \mathfrak m_\gamma } \int \Phi _\gamma (\xi ) \vartheta (\textrm{d}\xi )$, where the infimum is taken (and given the continuity of the map $\Pi _t$), attained over the compact set $\mathfrak m=\{\vartheta \in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}):\Pi _t(\vartheta ,\cdot )=\vartheta \,\forall t\ge 0\}$ of fixed points of $\Pi _t(\vartheta ,\cdot )=\int \pi _t(\xi ,\cdot )\vartheta (\textrm{d}\xi )$ for $\vartheta \in {{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}})$. We refer to the proof of [6, Theorem 3.7] for details.

Note that the second part of the first item follows from Lemma 4.6 (which implies continuity of $\widetilde{{\mathcal {X}}} \ni \xi \mapsto \Phi _\gamma (\xi )$, and therefore that of ${{\mathcal {M}}}_1(\widetilde{{\mathcal {X}}}) \ni \vartheta \rightarrow {\mathscr {I}}_{\Phi _\gamma }(\vartheta )$) and the compactness of $\mathfrak m_\gamma$ shown in Lemma 4.7. The bounds on $\sup _{{\mathfrak m_\gamma }}{{\mathscr {I}}}_{\Phi _{\gamma }}$ also follows from the bounds on $\Phi _\gamma$ shown in Lemma 4.6.

For the second item, we recall the known fact that for any $d\ge 1$, $\gamma \mapsto \lambda (\gamma )$ is non-decreasing and continuous in $(0,\infty )$. Thus, (4.37) follows since $\lambda (\gamma )>0$ clearly implies that $\lim _{T\rightarrow \infty }\mu _{\gamma ,T}(\Omega )=0$. Thus, with $\gamma _1:=\inf \{\gamma>0:\lambda (\gamma )>0\}$, it follows that $\gamma _1\ge \gamma _c$ with $\gamma _c$ being defined by uniform integrability as in Remark 6 (in fact, it is conjectured that $\gamma _1=\gamma _c$). This shows (4.37). For the third item we refer to [6, Lemma 4.4]. $\square$

Lemma 4.9

For any sequence of events $(A_\varepsilon )_{\varepsilon } \subset \Omega$ with $\mathbb {P}_0(A_\varepsilon )>0$ for all $\varepsilon >0$, it holds $\textbf{P}$-almost surely that

$$\begin{aligned} \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )= \liminf _{\varepsilon \downarrow 0} \varepsilon ^2 \textbf{E}\big [\log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )\big ]. \end{aligned}$$

(4.39)

Proof

We fix an event $A_\varepsilon$ such that $\mathbb {P}_0(A_\varepsilon )>0$. Then, for any $\varepsilon , t>0$ and $x\in \mathbb {R}^d$,

$$\begin{aligned} D_{t,x} \log \mu _{\gamma ,t}(A_\varepsilon )= \gamma \mathbb {E}^{{\widehat{\mu }}_{\gamma ,t}}\big [ \phi (\omega _t - x) \, | A_\varepsilon \big ], \end{aligned}$$

where $D_{t,x}$ denotes the Malliavin derivative. Moreover, $\textbf{P}$-a.s. it holds that

$$\begin{aligned}&\int _0^{\varepsilon ^{-2}} \textrm{d}t \int _{\mathbb {R}^d} \textrm{d}x \big | D_{t,x} \log \mu _{\gamma ,t}(A_\varepsilon )\big |^2 \\&\quad = \gamma ^2 \int _0^{\varepsilon ^{-2}} \textrm{d}t\int _{\mathbb {R}^d} \textrm{d}x \mathbb {E}^{{\widehat{\mu }}^{\otimes 2}_{\gamma ,t}} \big [ \phi (\omega ^{{\scriptscriptstyle {({1}})}}_t- x) \phi (\omega ^{{\scriptscriptstyle {({2}})}}_t - x) \big | A^{\otimes 2}_\varepsilon \big ] \\&\quad = \gamma ^2 \int _0^{\varepsilon ^{-2}} \textrm{d}t \,\, \mathbb {E}^{{\widehat{\mu }}^{\otimes 2}_{\gamma ,t}} \big [ (\phi \star \phi )\big (\omega ^{{\scriptscriptstyle {({1}})}}_t- \omega ^{{\scriptscriptstyle {({2}})}}_t \big ) \big | A^{\otimes 2}_\varepsilon \big ] \\&\quad \le \gamma ^2 \int _0^{\varepsilon ^{-2}} \textrm{d}t \,\, \mathbb {E}^{{\widehat{\mu }}^{\otimes 2}_{\gamma ,t}}\big [ (\phi \star \phi )(0)| A^{\otimes 2}_\varepsilon \big ] \\&\quad =\gamma ^2 \varepsilon ^{-2} (\phi \star \phi )(0). \end{aligned}$$

In the second line of the last display, $\omega ^{{\scriptscriptstyle {({1}})}}$ and $\omega ^{{\scriptscriptstyle {({2}})}}$ denote two independent Brownian paths, ${\widehat{\mu }}_{\gamma ,t}^{\otimes 2}$ stands for the corresponding normalized probability measure⁹ and the conditioning event $A_\varepsilon ^{\otimes 2}$ denotes two independent copies of the same event (corresponding to $\omega ^{{\scriptscriptstyle {({1}})}}$ and $\omega ^{{\scriptscriptstyle {({2}})}}$). In the second line, we have used Fubini’s theorem and symmetry of $\phi (\cdot )$, and in the third line we have estimated $(\phi \star \phi )(\cdot ) \le (\phi \star \phi )(0)$. Hence, for any $\varepsilon , \gamma >0$, we have that $\Vert D \log \mu _{\gamma ,\cdot }(A_\varepsilon )\Vert _{L^2([0,\varepsilon ^{-2}]\otimes \mathbb {R}^d)}^2 \le \gamma ^2 \varepsilon ^{-2} (\phi \star \phi )(0)$. Thus, by the Gaussian concentration inequality [34, Theorem B.8.1], for any $u>0$,

$$\begin{aligned} \textbf{P}(|\log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )-\textbf{E} \log \mu _{\gamma ,\varepsilon ^{-2}}(A_\varepsilon )|> u)\le 2\exp \bigg (-\frac{u^2}{2\gamma ^2 \varepsilon ^{-2}(\phi \star \phi )(0)}\bigg ). \end{aligned}$$

The result is now a consequence of the Borel-Cantelli lemma. Indeed, if we define $t_1:= 1$ and

$$\begin{aligned} t_{n+1}:= t_n + t_n^\delta \quad \text{ for } \text{ some } \delta \in (\frac{1}{2}, 1) \text{ to } \text{ be } \text{ chosen, } \text{ then }\quad t_n= n^{\frac{1}{1-\delta }+ o(1)} \qquad \text{ as }\, n\rightarrow \infty .\nonumber \\ \end{aligned}$$

(4.40)

By the above upper bound and Borel-Cantelli lemma, we have that $\textbf{P}$-a.s.

$$\begin{aligned} \limsup _{n\rightarrow \infty } \frac{1}{t_n} \bigg |\log \mu _{\gamma ,t_n}(A_{t_n})- \textbf{E}[\log \mu _{\gamma ,t_n}(A_{t_n})] \bigg | =0. \end{aligned}$$

(4.41)

(For notational simplicity, we write $A_{t_n}$ instead of $A_{\frac{1}{\sqrt{t_n}}}$). To complete the proof, we again apply Itô’s formula to decompose $\log \mu _{\gamma ,t}(A_t)= M_t - \frac{1}{2} \langle M_t\rangle + \frac{\gamma ^2}{2} t (\phi \star \phi )(0)$ for a continuous martingale $M_t:= \gamma \int _0^t \int _{\mathbb {R}^d} \mathbb {E}^{{\widehat{\mu }}_{\gamma ,t}}[ \phi (\omega _t - y) \mathbb {1}_{A_t}] \dot{B}(t,y) \textrm{d}y \textrm{d}t$ which satisfies

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t} \langle M_t\rangle \le \gamma ^2 (\phi \star \phi )(0). \end{aligned}$$

(4.42)

Let us fix a sequence

$$\begin{aligned} \delta _n = \frac{1}{n^{o(1)}} \quad \text{ such } \text{ that }\quad \gamma ^2 (\phi \star \phi )(0) t_n^\delta < \delta _n t_{n+1} \qquad \text{(recall } (\text {4.40})). \end{aligned}$$

(4.43)

Then

$$\begin{aligned}{} & {} \textbf{P}\bigg (\sup _{t_n \le t \le t_{n+1}} \big | \log \mu _{\gamma ,t}(A_t)- \log \mu _{\gamma ,t_n}(A_{t_n}) - \textbf{E}[\log \mu _{\gamma ,t}(A_t)] + \textbf{E}(\log \mu _{\gamma ,t_n}(A_{t_n}))> 2 \delta _n t_{n+1}\bigg ) \nonumber \\{} & {} \quad \le \textbf{P}\big (\sup _{t_n \le t \le t_{n+1}} |M_t - M_{t_n}| > \delta _n t_{n+1} \big ) \le (\delta _n t_{n+1})^{-2} \textbf{E}\big ( \langle M_{t_{n+1}} \rangle - \langle M_{t_n}\rangle \big ) \end{aligned}$$

(4.44)

by Doob’s inequality. Since

$$\begin{aligned} \textbf{E}\big ( \langle M_{t_{n+1}} \rangle - \langle M_{t_n}\rangle \big ) {\mathop {\le }\limits ^{(\text {4.42})}} \gamma ^2 (\phi \star \phi )(0) (t_{n+1} - t_n) {\mathop {=}\limits ^{(\text {4.40})}} \gamma ^2 (\phi \star \phi )(0) t_n^\delta {\mathop {<}\limits ^{(\text {4.43})}} \delta _n t_{n+1}, \end{aligned}$$

(4.45)

we have that the probability on the left hand side of (4.44) is bounded above by $\frac{1}{\delta _n t_{n+1}}$. If we now choose $\delta \in (\frac{1}{2}, 1)$ sufficiently large, then these probabilities are summable and by Borel-Cantelli lemma, combined with (4.41), we have $\textbf{P}$-a.s.,

$$\begin{aligned} \limsup _{t\rightarrow \infty } \frac{1}{t} \bigg |\log \mu _{\gamma ,t}(A_{t})- \textbf{E}[\log \mu _{\gamma ,t}(A_{t})] \bigg | =0. \end{aligned}$$

(4.46)

To conclude (4.39), we use that for any sequence $X_t$ and $Y_t$ satisfying $\limsup _{t\rightarrow \infty } |X_t - Y_t|=0$, we have $\liminf _{t\rightarrow \infty } X_t= \liminf _{t\rightarrow \infty } (Y_t + (X_t - Y_t)) \ge \liminf _{t\rightarrow \infty } Y_t$ and by reverting the argument we also have $\liminf _{t\rightarrow \infty } Y_t \ge \liminf _{t\rightarrow \infty } X_t$. Hence, $\liminf _{t\rightarrow \infty } X_t= \liminf _{t\rightarrow \infty } Y_t$. Invoking (4.46) and the latter fact for $X_t= \frac{1}{t} \log \mu _{\gamma ,t}(A_{t})$ and $Y_t= \frac{1}{t} \textbf{E}[\log \mu _{\gamma ,t}(A_{t})]$ we have (4.39). $\square$

5 Moments: proof of theorem 2.3

The proof of Theorem 2.3 will require the following result:

Lemma 5.1

Given $T>0$ and $\gamma <\gamma _c$, let $M_T:=\sup _{0\le s\le T}\mu _{\gamma ,s}(\Omega )$ and $M_\infty :=\lim _{T\rightarrow \infty }M_T$. Then $\textbf{E}[M_\infty ]<\infty$.

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

$$\begin{aligned} \tau =\tau _u:=\inf \{T\ge 0:\mu _{\gamma ,T}(\Omega )=u\}. \end{aligned}$$

(5.1)

Lemma 5.2

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

$$\begin{aligned} \textbf{E}\bigg [f\left( \frac{\mu _{\gamma ,T}(\Omega )}{\mu _{\gamma ,\tau }(\Omega )}\right) ,\tau \le T\bigg ]\le \textbf{P}(\tau \le T)\textbf{E}[f(\mu _{\gamma ,T}(\Omega ))]. \end{aligned}$$

(5.2)

Proof

Let $(\tau _n)_n$ be a discrete approximation of $\tau$ such that $\tau _n\searrow \tau$. If (5.2) holds for $\tau _n$, for each n, then by Fatou’s lemma we can deduce (5.2) 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 $s\le T$,

$$\begin{aligned} \begin{aligned} \frac{\mu _{\gamma ,T}(\Omega )}{\mu _{\gamma ,s}(\Omega )}&=\frac{\mathbb {E}_0\Big [\textrm{e} ^{\gamma H_s(\omega )-\frac{\gamma ^2s}{2}\phi \star \phi (0)}\mu _{\gamma ,T-s}(\Omega )\circ \theta _{s,\omega _s}\Big ]}{\mathbb {E}_0\Big [\textrm{e} ^{\gamma H_s(\omega )-\frac{\gamma ^2s}{2}\phi \star \phi (0)}\Big ]} \\&=:\mathbb {E}_{0,s}\Big [\mu _{\gamma ,T-s}(\Omega )\circ \theta _{s,\omega _s}\Big ], \end{aligned} \end{aligned}$$

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

If $[0,T]\cap \{t_i\}_{n\in \mathbb {N}}=\{t_1,\ldots ,t_n\}$, then using the convexity of f and Jensen’s inequality,

$$\begin{aligned}&\textbf{E}\bigg [f\left( \frac{\mu _{\gamma ,T}(\Omega )}{\mu _{\gamma ,\tau }(\Omega )}\right) ,\tau \le T\bigg ]\\&\quad \le \textbf{E}\bigg [\mathbb {E}_{0,\tau }\bigg (f\left( \mu _{\gamma ,T-\tau }(\Omega )\circ \theta _{\tau ,\omega _\tau }\right) \bigg ),\tau \le T\bigg ]\\&\quad =\textbf{E}\bigg [\mathbb {E}_0\bigg (\frac{\textrm{e} ^{\gamma H_\tau (\omega )-\frac{\gamma ^2}{2}\tau \phi \star \phi (0)}f\left( \mu _{\gamma ,T-\tau }(\Omega )\circ \theta _{\tau ,\omega _\tau }\right) }{\mathbb {E}_0[\textrm{e} ^{\gamma H_\tau (\omega )-\frac{\gamma ^2}{2}\tau \phi \star \phi (0)}]}\bigg ),\tau \le T\bigg ]\\&\quad =\sum _{i=1}^n \textbf{E}\bigg [\mathbb {E}_0\bigg (\frac{\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}f\left( \mu _{\gamma ,T-t_i}(\Omega )\circ \theta _{t_i,\omega _{t_i}}\right) }{\mathbb {E}_0[\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}]}\bigg ),\tau =t_i\bigg ]\\&\quad =\sum _{i=1}^n \textbf{E}\bigg [\textbf{E}\bigg (\mathbb {E}_0\bigg [\frac{\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}f\left( \mu _{\gamma ,T-t_i}(\Omega )\circ \theta _{t_i,\omega _{t_i}}\right) }{\mathbb {E}_0[\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}]}\bigg ]\bigg \vert {\mathcal {F}}_{t_i}\bigg ),\tau =t_i\bigg ] \end{aligned}$$

By Lemma 3.2, the last expression is equal to

$$\begin{aligned}&\sum _{i=1}^n \textbf{E}\bigg [\mathbb {E}_0\bigg (\textbf{E}\bigg [\frac{\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}f\left( \mu _{\gamma ,T-t_i}(\Omega )\circ \theta _{t_i,\omega _{t_i}}\right) }{\mathbb {E}_0[\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}]}\bigg \vert {\mathcal {F}}_{t_i}\bigg )\bigg ],\tau =t_i\bigg ]\\&\quad =\sum _{i=1}^n \textbf{E}\bigg [\mathbb {E}_0\bigg (\frac{\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}}{\mathbb {E}_0[\textrm{e} ^{\gamma H_{t_i}(\omega )-\frac{\gamma ^2}{2}t_i\phi \star \phi (0)}]}\textbf{E}\bigg [f\left( \mu _{\gamma ,T-t_i}(\Omega )\circ \theta _{t_i,\omega _{t_i}}\right) \big \vert {\mathcal {F}}_{t_i}\bigg ]\bigg ),\tau =t_i\bigg ] \end{aligned}$$

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

$$\begin{aligned} \sum _{i=1}^n \textbf{P}\left[ \tau =t_i\right] \textbf{E}\left[ f\left( \mu _{\gamma ,T-t_i}(\Omega )\right) \right] \le \textbf{P}(\tau \le T)\textbf{E}\left[ f\left( \mu _{\gamma ,T}(\Omega )\right) \right] , \end{aligned}$$

where we used that f is convex, so that $(f(\mu _{\gamma ,T}(\Omega )))_{T\ge 0}$ is a submartingale. $\square$

Proof of Lemma 5.1

Given $\varepsilon >0$, let

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

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

$$\begin{aligned} \mathbb {1}_{[\varepsilon ,\infty )}(x)\ge f_{\varepsilon }(x)\ge \mathbb {1}(x)_{[2\varepsilon ,\infty )}-\mathbb {1}_{[0,\varepsilon ]}(x). \end{aligned}$$

(5.3)

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

$$\begin{aligned} \textbf{P}(M_T>u)\le 2\textbf{P}(\mu _{\gamma ,T}(\Omega )>u\varepsilon ). \end{aligned}$$

(5.4)

Indeed,

$$\begin{aligned} \textbf{E}[M_T]=\int _{0}^1\textbf{P}(M_T>u)\textrm{d}u+\int _{1}^{\infty }\textbf{P}(M_T>u)\textrm{d}u\le 1+ \frac{2}{\varepsilon }\textbf{E}[\mu _{\gamma ,T}(\Omega )]=1+\frac{2}{\varepsilon }. \end{aligned}$$

For a fixed $u>1$, recall the definition of $\tau =\tau _u$ from (5.1), so that $\mu _{\gamma ,\tau }(\Omega )=u$. Hence, by Lemma 5.2 and Eq. (5.3)

$$\begin{aligned} \textbf{P}(\mu _{\gamma ,T}(\Omega )>u\varepsilon )\ge & {} \textbf{P}\left( \tau \le T,\frac{\mu _{\gamma ,T}(\Omega )}{\mu _{\gamma ,\tau }(\Omega )}>\varepsilon \right) \nonumber \\\ge & {} \textbf{E}\left[ f_{\varepsilon }\left( \frac{\mu _{\gamma ,T}(\Omega )}{\mu _{\gamma ,\tau }(\Omega )}\right) ,\tau \le T\right] \nonumber \\\ge & {} \textbf{P}(\tau \le T)\textbf{E}\left[ f_{\varepsilon }(\mu _{\gamma ,T}(\Omega ))\right] \nonumber \\\ge & {} \textbf{P}(\tau \le T)\inf _{T\ge 0}\textbf{E}\left[ f_{\varepsilon }(\mu _{\gamma ,T}(\Omega ))\right] \nonumber \\\ge & {} \textbf{P}(\tau \le T)\textbf{E}\left[ \inf _{T\ge 0} f_{\varepsilon }(\mu _{\gamma ,T}(\Omega ))\right] . \end{aligned}$$

(5.5)

We use again (5.3) to deduce

$$\begin{aligned} \textbf{E}\left[ \inf _{T\ge 0} f_{\varepsilon }(\mu _{\gamma ,T}(\Omega ))\right]&\ge \textbf{E}\left[ \inf _{T\ge 0} \mathbb {1}_{\mu _{\gamma ,T}(\Omega )\ge 2\varepsilon }\right] - \textbf{E}\left[ \sup _{T\ge 0}\mathbb {1}_{\mu _{\gamma ,T}(\Omega )\le \varepsilon }\right] \\&=\textbf{P}\left( \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )\ge 2\varepsilon \right) - \textbf{P}\left( \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )\le \varepsilon \right) . \end{aligned}$$

To see the last equality, note that

$$\begin{aligned} \inf _{T\ge 0} \mathbb {1}_{\mu _{\gamma ,T}(\Omega )\ge 2\varepsilon }=1 \quad \text{ if } \text{ and } \text{ only } \text{ if }\quad \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )\ge 2\varepsilon \end{aligned}$$

and

$$\begin{aligned} \sup _{T\ge 0}\mathbb {1}_{\mu _{\gamma ,T}(\Omega )\le \varepsilon }=1\quad \text{ if } \text{ and } \text{ only } \text{ if }\quad \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )\le \varepsilon . \end{aligned}$$

Letting $\varepsilon \rightarrow 0$ in the last display and noting that $\textbf{P}(\inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )=0)=\textbf{P}(\mu _{\gamma }(\Omega )=0)=0$, we conclude that

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\textbf{E}\left[ \inf _{T\ge 0} f_{\varepsilon }(\mu _{\gamma ,T}(\Omega ))\right] \ge \textbf{P}\left( \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )\ge 0\right) -\textbf{P}\left( \inf _{T\ge 0}\mu _{\gamma ,T}(\Omega )=0\right) =1. \end{aligned}$$

Thus, for $\varepsilon >0$ small enough, and recalling the computations from (5.5), we conclude that

$$\begin{aligned} \textbf{P}(\mu _{\gamma ,T}(\Omega )>u\varepsilon )\ge \frac{1}{2}\textbf{P}(\tau \le T)=\frac{1}{2}\textbf{P}(M_T>u), \end{aligned}$$

which is (5.4). $\square$

Now we are ready to give the

Proof of Theorem 2.3

Set $\gamma <\gamma _c$. For some fixed $u>1$, we recall the stopping time $\tau =\tau _u$ defined in (5.1). For any $p>1$ and $T>0$,

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,T}(\Omega )^p]= & {} \textbf{E}[\mu _{\gamma ,T}(\Omega )^p,\tau >T]+\textbf{E}[\mu _{\gamma ,T}(\Omega )^p,\tau \le T]\nonumber \\\le & {} u^p+u^p\textbf{E}\left[ \left( \frac{\mu _{\gamma ,T(\Omega )}}{\mu _{\gamma ,\tau }(\Omega )}\right) ^p,\tau \le T\right] \nonumber \\\le & {} u^p+u^p\textbf{P}(\tau \le T)\textbf{E}\left[ \mu _{\gamma ,T}(\Omega )^p\right] , \end{aligned}$$

(5.6)

where in the last line we used Lemma 5.2. By Lemma 5.1,

$$\begin{aligned} \textbf{E}[M_\infty ]=\int _{0}^\infty \textbf{P}(M_\infty>u)\textrm{d}u=1+\int _{1}^\infty \textbf{P}(M_\infty >u)\textrm{d}u<\infty , \end{aligned}$$

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

$$\begin{aligned} \textbf{P}(\tau \le T)=\textbf{P}(M_T>u)\le \textbf{P}(M_\infty >u)\le \frac{1}{2u}, \end{aligned}$$

we deduce from (5.6) the upper bound

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,T}(\Omega )^p]\le u^p+\frac{u^{p-1}}{2}\textbf{E}[\mu _{\gamma ,T}(\Omega )^p]. \end{aligned}$$

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

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,T}(\Omega )^p]\le \frac{2u^p}{(2-u^{p-1})}. \end{aligned}$$

We turn to the proof of (2.12) and (2.13). To show (2.12), we use the stopping time $\tau =\tau _{1/u}$ for $u>1$ (recall (5.1)). Proceeding as in (5.6), noting that $x\mapsto x^{-q}$ is convex on $(0,\infty )$ and using Lemma 5.2, we have

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q}]= & {} \textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q},\tau >T]+\textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q},\tau \le T]\nonumber \\\le & {} u^q+u^q\textbf{E}\left[ \left( \frac{\mu _{\gamma ,T(\Omega )}}{\mu _{\gamma ,\tau }(\Omega )}\right) ^{-q},\tau \le T\right] \nonumber \\\le & {} u^q+u^q\textbf{P}(\tau \le T)\textbf{E}\left[ \mu _{\gamma ,T}(\Omega )^{-q}\right] . \end{aligned}$$

(5.7)

Since $\textbf{P}(\tau \le T)\le \textbf{P}(\inf _{T\ge 0} \mu _{\gamma ,T}\le u^{-1})\rightarrow 0$ as $u\rightarrow \infty$, we infer

$$\begin{aligned} \sup _{T\ge 0}\textbf{P}(\tau \le T)\le \frac{1}{4} \end{aligned}$$

for $u>1$ large enough. If $q\in (0,1)$ is chosen so that $u^q\le 2$, from (5.7) we conclude that

$$\begin{aligned} \textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q}]\le 2+\frac{\textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q}]}{2}, \end{aligned}$$

and hence

$$\begin{aligned} \sup _{T\ge 0}\textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q}]\le 4. \end{aligned}$$

This finishes the proof of (2.12). To show (2.13), we appeal to [11, 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 $L^2(\textbf{P})$), for all $q\in (0,\infty )$,

$$\begin{aligned} \sup _{T\ge 0}\textbf{E}[\mu _{\gamma ,T}(\Omega )^{-q}]<\infty . \end{aligned}$$

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 (2.13) for all negative q for $L^2$ disorder. $\square$