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 $d\ge 3$, 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 $$ μ_γ\Big\{ω: \lim_{T\to\infty} \frac{H_T(ω)}{T(ϕ\starϕ)(0)} \ne γ\Big\}=0 \qquad \mbox{almost surely,} $$ 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 $\varphi$ 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: $-C_2\le {\mathrm{liminf}}_{r\to 0}{r}^2\log {\hat{\mu }}_{\gamma }(\Vert \omega \Vert <r)\le {\mathrm{limsup}}_{r\to 0}{\sup }_{\eta }{r}^2\log {\hat{\mu }}_{\gamma }(\Vert \omega -\eta \Vert <r)\le -C_1$ 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 {\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.

Minor revision,