Path properties of the solution to the stochastic heat equation with Lévy noise

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 Lévy 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 càdlàg 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 Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy 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 Lévy noises, and to light- as well as heavy-tailed jumps.

References in corpus (1)

• Stochastic PDEs with heavy-tailed noise

Cited by in corpus (4)

• The almost-sure asymptotic behavior of the solution to the stochastic heat equation with Lévy noise
• Stochastic wave equation with Lévy white noise
• Comparison principle for stochastic heat equations driven by $\alpha$-stable white noises
• The compact support property for solutions to stochastic heat equations with stable noise