Path properties of the solution to the stochastic heat equation with L\'evy noise

TL;DR

This paper investigates the path regularity of solutions to the stochastic heat equation driven by Lévy noise, identifying conditions under which solutions have continuous or unbounded paths based on the Lévy noise's Blumenthal-Getoor index.

Contribution

It establishes the existence of càdlàg modifications in fractional Sobolev spaces and determines critical Blumenthal-Getoor index values for path regularity in space and time.

Findings

Solutions have càdlàg modifications in fractional Sobolev spaces of index less than -d/2.

Lévy noise with smaller Blumenthal-Getoor index yields continuous sample paths.

Lévy noise with larger index results in unbounded sample paths on open sets.

Abstract

We consider sample path properties of the solution to the stochastic heat equation, in $\mathbb{R}^d$ or bounded domains of $\mathbb{R}^d$, driven by a L\'evy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a c\`adl\`ag modification in fractional Sobolev spaces of index less than $-\frac d 2$. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal-Getoor index of the L\'evy noise such that noises with a smaller index entail continuous sample paths, while L\'evy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative L\'evy noises, and to light- as well as heavy-tailed jumps.