Transportation inequalities under uniform metric for a stochastic heat equation driven by time-white and space-colored noise

TL;DR

This paper establishes transportation inequalities for the solution of a stochastic heat equation driven by Gaussian noise that is white in time and colored in space, using a novel moment inequality under the uniform metric.

Contribution

It introduces a new moment inequality for stochastic convolution under the uniform metric, enabling transportation inequalities for solutions to stochastic heat equations with space-time colored noise.

Findings

Proves transportation inequalities for the law of the stochastic heat equation solution

Develops a new moment inequality for stochastic convolution under the uniform metric

Provides tools applicable to equations driven by space-colored Gaussian noise

Abstract

In this paper, we prove transportation inequalities on the space of continuous paths with respect to the uniform metric, for the law of solution to a stochastic heat equation defined on $[0,T]\times [0,1]^d$. This equation is driven by the Gaussian noise, white in time and colored in space. The proof is based on a new moment inequality under the uniform metric for the stochastic convolution with respect to the time-white and space-colored noise, which is of independent interest.