Quenched asymptotics for Brownian motion in generalized Gaussian potential

Abstract

In this paper, we study the long-term asymptotics for the quenched moment Ex exp t 0

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

“…Because the kernel k(x, y) may be non-stationary, we need to find a function satisfying the condition (H) instead of the k(x, y) in the moment representation (7). By (3)…”

“…Becasue the Riesz potential | · | −α in (9) satisfies the Dalang's condition, t 0 t 0 k(B(s), B(r))dsdr is exponentially integrable. Hence, (5) and (7) are well defined. Based on it, we obtain the following main result about the precise high moment asymptotics for the Feynman-Kac formula (5) of the PAM with LGF.…”

The Parabolic Anderson Model

“…Canonical choices of non-negative clouds are W D C½ K for some compact set K R d (say, a centred ball) containing the origin and for some C 2 .0; 1, or W some non-negative continuous function with compact support, or W.x/ D Cjxj q for some C 2 .0; 1/ and q 2 .0; 1/. Recently there was also some efforts to study the PAM under much less regularity assumptions [Che14], where the potential V is not even a function, but only a measure, and see Example 1.21 for the uncorrelated case. However, one must be careful, as, for W.x/ D Cjxj q with q Ä d, the potential V is infinite almost everywhere (i.e., V Á 1), almost surely [ x i / i , one can also pick the random field .x i / i to be a Gibbsian point field, i.e., a point field that, unlike a Poisson process, has some nontrivial correlation between the particles.…”

“…Þ Remark 5.14 (Generalised Gaussian potential) A Gaussian potential V on R d with much less regularity was considered in [Che14]. Indeed, following a question raised in [CarMol95], it was assumed that the covariance function B has a behaviour of the form B.x/ 2 jxj ˛f or x !…”

“…Indeed, following a question raised in [CarMol95], it was assumed that the covariance function B has a behaviour of the form B.x/ 2 jxj ˛f or x ! The main result of [Che14] is that, almost surely, as t ! Away from the origin, B is assumed to be continuous and bounded.…”