Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise

TL;DR

This paper studies the long-term behavior of solutions to a stochastic heat equation driven by correlated noise in space and time, deriving formulas for moments and growth rates.

Contribution

It establishes existence and uniqueness of solutions, provides a Feynman-Kac formula for moments, and computes Lyapunov exponents and growth indices for the model.

Findings

Existence and uniqueness of mild solutions.

Feynman-Kac formula for moments.

Explicit formulas for Lyapunov exponents and growth indices.

Abstract

We consider the linear stochastic heat equation on $\mathbb{R}^\ell$, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in (\frac 14, \frac 12]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.