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

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 E exp {1/2 Sigma(N)(j,k=1) integral(t)(0) integral(t )(0)gamma(B-j(s) - B-k(r))drds} (N -> infinity) 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. (C) 2019 Elsevier B.V. All rights reserved.