Stochastic heat equation with rough dependence in space

TL;DR

This paper investigates the nonlinear stochastic heat equation driven by spatially rough Gaussian noise, establishing existence, uniqueness, and moment bounds, with explicit formulas in the linear case.

Contribution

It proves existence and uniqueness for the nonlinear equation with rough spatial dependence and derives explicit formulas and bounds for moments in the linear case.

Findings

Existence and uniqueness of solutions under Lipschitz conditions.

Wiener chaos expansion and Feynman-Kac formula for the linear case.

Sharp asymptotic bounds for moments of the solution.

Abstract

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.