New results on pathwise uniqueness for the heat equation with colored noise

TL;DR

This paper establishes new conditions for pathwise uniqueness of solutions to the stochastic heat equation with colored noise, extending previous results and confirming a conjecture in the field.

Contribution

It improves the sufficient conditions for strong uniqueness in the stochastic heat equation with colored noise, generalizing prior work and confirming a key conjecture.

Findings

Pathwise uniqueness is proven under new, less restrictive conditions.

Confirms a conjecture regarding strong uniqueness for colored noise in the heat equation.

Extends techniques from white noise to colored noise in higher dimensions.

Abstract

We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in q dimensional space ($q \geq 1$) with a singular correlation kernel. The noise coefficient is H\"older continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the H\"older coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper.