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.