Regularity of the density for the stochastic heat equation

TL;DR

This paper investigates the smoothness of the probability density for solutions of a stochastic heat equation with multiplicative white noise, establishing the existence of negative moments for linearized solutions using Malliavin calculus.

Contribution

It demonstrates the regularity of the density for the stochastic heat equation by proving solutions have negative moments of all orders, a novel result in this context.

Findings

Density of the solution is smooth.

Solutions have negative moments of all orders.

Method reduces the problem to linear equations.

Abstract

We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.