Moments and Lyapunov exponents for the parabolic Anderson model

TL;DR

This paper derives contour integral formulas for moments and computes Lyapunov exponents for the (1+1)-dimensional parabolic Anderson model with space-time white noise, revealing detailed growth behaviors of solutions.

Contribution

It provides explicit contour integral formulas for moments and Lyapunov exponents, including for all moments in a simplified jump model, advancing understanding of the model's probabilistic structure.

Findings

Derived contour integral formulas for the second and all moments.

Computed Lyapunov exponents for moments of all orders.

Established explicit formulas for the model with jumps only to the right.

Abstract

We study the parabolic Anderson model in $(1+1)$ dimensions with nearest neighbor jumps and space-time white noise (discrete space/continuous time). We prove a contour integral formula for the second moment and compute the second moment Lyapunov exponent. For the model with only jumps to the right, we prove a contour integral formula for all moments and compute moment Lyapunov exponents of all orders.