Lyapunov exponents for the one-dimensional parabolic Anderson model with drift

TL;DR

This paper derives formulas for Lyapunov exponents in a one-dimensional parabolic Anderson model with drift, showing convergence properties and intermittency, and analyzes the random walk speed under different potential conditions.

Contribution

It provides explicit representations for quenched and annealed Lyapunov exponents for all p, and establishes their convergence and intermittency properties in the model.

Findings

Annealed Lyapunov exponents converge to the quenched exponent as p approaches zero.

The solution u exhibits p-intermittency for large p.

A phase transition in the random walk speed occurs depending on the potential's negativity.

Abstract

We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for {\it all} $p \in (0, \infty).$ These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p \downarrow 0.$ Furthermore, we show that $u$ is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of $u$ under the corresponding Gibbs measure. In this context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears.