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, we consider a random field defined with respect to space-time white noise integrated w.r.t. Brownian paths in $d\geq 3$ and construct the infinite volume limit of the normalized exponential of this field, weighted w.r.t. the Wiener measure, in the entire weak disorder (subcritical) regime. Moreover, we characterize this infinite volume measure, which we call the {\it subcritical GMC on the Wiener space}, w.r.t. the mollification scheme in the sense of Shamov \cite{S14} and determine its support by identifying its {\it thick points}. This in turn implies that almost surely, the subcritical GMC on the Wiener space is singular w.r.t. the Wiener measure. We also prove, in the subcritical regime, existence of negative and positive ($L^p$ for $p>1$) moments of the total mass of the limiting GMC, and deduce its H\"older exponents (small ball probabilities) explicitly. While the uniform H\"older exponent (the upper bound) and the pointwise scaling exponent (the lower bound) differ for a fixed disorder, we show that, as the disorder goes to zero, the two exponents agree, coinciding with the scaling exponent of the Wiener measure.