Parabolic stochastic PDEs on bounded domains with rough initial conditions: moment and correlation bounds

TL;DR

This paper investigates nonlinear parabolic stochastic PDEs on bounded domains with rough initial data, establishing existence, uniqueness, moment bounds, and correlation estimates, and demonstrating intermittency under certain conditions.

Contribution

It provides new existence and moment bounds for solutions with measure-valued initial data and analyzes correlation functions, extending results to less regular initial conditions.

Findings

Established existence and uniqueness of solutions.

Derived explicit bounds for correlation functions.

Proved solution intermittency for high noise levels.

Abstract

We consider nonlinear parabolic stochastic PDEs on a bounded Lipschitz domain driven by a Gaussian noise that is white in time and colored in space, with Dirichlet or Neumann boundary condition. We establish existence, uniqueness and moment bounds of the random field solution under measure-valued initial data $\nu$. We also study the two-point correlation function of the solution and obtain explicit upper and lower bounds. For $C^{1, \alpha}$-domains with Dirichlet condition, the initial data $\nu$ is not required to be a finite measure and the moment bounds can be improved under the weaker condition that the leading eigenfunction of the differential operator is integrable with respect to $|\nu|$. As an application, we show that the solution is fully intermittent for sufficiently high level $\lambda$ of noise under the Dirichlet condition, and for all $\lambda > 0$ under the Neumann condition.