Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay
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2022
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| author | Bazaes, Rodrigo Lammers, Isabel Mukherjee, Chiranjib |
| author_facet | Bazaes, Rodrigo Lammers, Isabel Mukherjee, Chiranjib |
| contents | We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(ω)$ 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 $μ_γ$ 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, $μ_γ$ is supported only on $γ$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $μ_γ$ w.r.t. the mollification scheme $ϕ$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{μ_γ}(d\dot B dω)= μ_γ(dω,\dot B)P(d\dot B)$.
We then determine the fractal properties of the measure around $γ$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehatμ_γ(\|ω\| < r) \leq \limsup_{r\to 0}\sup_ηr^2 \log \widehatμ_γ(\|ω-η\| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) 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 $μ_γ$ in the weak disorder regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_08996 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay Bazaes, Rodrigo Lammers, Isabel Mukherjee, Chiranjib Probability Mathematical Physics We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(ω)$ 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 $μ_γ$ 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, $μ_γ$ is supported only on $γ$-{\it thick paths}, and consequently, the normalized version is singular w.r.t. the Wiener measure. We characterize uniquely the limit $μ_γ$ w.r.t. the mollification scheme $ϕ$ in the sense of Shamov \cite{S14} and the random {\it rooted} measure $\mathbb Q_{μ_γ}(d\dot B dω)= μ_γ(dω,\dot B)P(d\dot B)$. We then determine the fractal properties of the measure around $γ$-thick paths: $-C_2 \leq \liminf_{r\to 0} r^2 \log \widehatμ_γ(\|ω\| < r) \leq \limsup_{r\to 0}\sup_ηr^2 \log \widehatμ_γ(\|ω-η\| < r) \leq -C_1$ w.r.t a weighted norm $\|\cdot\|$. Here $C_1>0$ and $C_2<\infty$ are the uniform upper (resp. pointwise lower) 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 $μ_γ$ in the weak disorder regime. |
| title | Subcritical Gaussian Multiplicative Chaos in the Wiener Space: Construction, Moments and Volume Decay |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2211.08996 |