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

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 ofwhich is composed of the independent Brownian motions {B j (s)} and the function γ 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.

Localization of a one-dimensional simple random walk among power-law renewal obstacles

“…Since the obstacles are drawn according to a renewal process, our work provides an exemple of localization in a correlated disordered environment. Influence of spatial correlations has been investigated in other contexts such as localization of directed polymers with space-time noise [28], Brownian motion in correlated Poisson potential [29,30,36] as well as Anderson localization, both by mathematicians [25,26,31] and physicists [2,13,43]. In our model however, the extreme value statistics are more relevant than the correlation structure itself.…”

Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises

“…This property is related the moment bounds of the solution. When (1.1) is parabolic Anderson model, namely, when L is a heat operator or fractional heat operator, then the sharp (both lower and upper) moment bounds are known, see [2,6,5,7,8,17,19,25], and we also refer to [15] and references therein. However, when L is wave operators (namely the hyperbolic Anderson model) or when L is (temporal) fractional differential operators, the situation is different and as far as we know here are the progress achieved.…”