Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field

Yangyang Lyu

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arXiv · 1909.01031

Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field

Yangyang Lyu

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Abstract

In this paper, we consider the continuous parabolic Anderson model (PAM) driven by a time-independent log-correlated Gaussian field (LGF). We obtain an asymptotic result of $$\mathbb{E}\exp\Bigg\{\frac{1}{2}\sum\limits_{ j,k=1}^N\int_0^t\int_0^t\gamma(B_j(s)-B_k(r))drds\Bigg\}\qquad(N\rightarrow \infty)$$ which is composed of the independent Brownian motions $\{B_j(s)\}$ and the function $\gamma$ approximating to a logarithmic potential at $0$, such as the covariances of massive free field and Bessel field. Based on the asymptotic result, we get the precise high moment asymptotics for Feynman-Kac formula of the PAM with LGF.

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Yangyang Lyu. 2019-09-03. Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field. https://arxiv.org/abs/1909.01031

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