Regularity of the density for a stochastic heat equation

TL;DR

This paper investigates the smoothness of the probability density function of solutions to a nonlinear stochastic heat equation driven by Levy noise and white noise, using Malliavin calculus techniques.

Contribution

It demonstrates that the solution's law admits a smooth density for all positive times and spatial points, extending previous results to Levy-driven equations.

Findings

The solution's density exists and is smooth for all t > 0.

Malliavin calculus is effectively applied to Levy-driven stochastic PDEs.

The law of the solution is absolutely continuous with respect to Lebesgue measure.

Abstract

We study the smoothness of the density of the solution to the nonlinear heat equation u_t=Lu(t,x)+\sigma(u(t,x))W on a torus with a periodic boundary condition, where L is the generator of a Levy process on the torus, and W is white noise. We use Malliavin calculus techniques to show that the law of the solution has a density with respect to the Lebesgue measure for all t >0 and x in R.