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 \cite{K85}. 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 $d\geq 3$, we consider the renormalized exponential, weighted w.r.t. the Wiener measure $\mathbb P_0$. We construct the almost sure limit $\mu_\gamma$ in the {\it entire weak disorder (subcritical)} regime and call it {\it subcritical GMC} on the Wiener space. We show that $$ \mu_\gamma\Big\{\omega: \lim_{T\to\infty} \frac{H_T(\omega)}{T(\phi\star\phi)(0)} \ne \gamma\Big\}=0 \qquad \mbox{almost surely,} $$ meaning, $\mu_\gamma$ is supported only on $\gamma$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $\mu_\gamma$ w.r.t. the mollification scheme $\phi$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{\mu_\gamma}(d\dot B d\omega)= \mu_\gamma(d\omega,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $\gamma$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehat\mu_\gamma(\|\omega\| < r) \leq \limsup_{r\to 0}\sup_\eta r^2 \log \widehat\mu_\gamma(\|\omega-\eta \| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) H\"older exponents which are {\it 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.