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 $$H_T(\omega )$$ H T ( ω ) is a random field defined w.r.t. space-time white noise $$\dot{B}$$ B ˙ and integrated w.r.t. Brownian paths in $$d\ge 3$$ d ≥ 3 , we consider the renormalized exponential $$\mu _{\gamma ,T}$$ μ γ , T , weighted w.r.t. the Wiener measure $$\mathbb {P}_0(\textrm{d}\omega )$$ P 0 ( d ω ) . We construct the almost sure limit $$\mu _\gamma = \lim _{T\rightarrow \infty } \mu _{\gamma ,T}$$ μ γ = lim T → ∞ μ γ , T in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that $$\begin{aligned} \mu _\gamma \Big \{\omega : \lim _{T\rightarrow \infty } \frac{H_T(\omega )}{T(\phi \star \phi )(0)} \ne \gamma \Big \}=0 \qquad \text{ almost } \text{ surely, } \end{aligned}$$ μ γ { ω : lim T → ∞ H T ( ω ) T ( ϕ ⋆ ϕ ) ( 0 ) ≠ γ } = 0 almost surely, meaning that $$\mu _\gamma $$ μ γ is supported almost surely only on $$\gamma $$ γ -thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit $$\mu _\gamma $$ μ γ w.r.t. the mollification scheme $$\phi $$ ϕ in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of $$\dot{B}$$ B ˙ under the random rooted measure $$\mathbb Q_{\mu _\gamma }(\textrm{d}\dot{B}\textrm{d}\omega )= \mu _\gamma (\textrm{d}\omega ,\dot{B})P(\textrm{d}\dot{B})$$ Q μ γ ( d B ˙ d ω

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