A nonlinear stochastic heat equation: H\"older continuity and smoothness of the density of the solution

TL;DR

This paper develops a Feynman-Kac formula for multidimensional stochastic heat equations driven by semimartingales, enabling analysis of solution smoothness and H"older continuity under nonhomogeneous Gaussian noise.

Contribution

It introduces a novel Feynman-Kac formula for complex stochastic heat equations and uses it to analyze the regularity and density properties of solutions.

Findings

Explicit Malliavin derivatives of solutions obtained

Density of the solution's law shown to be smooth

Solutions demonstrated to have H"older continuity

Abstract

In this paper, we establish a version of the Feynman-Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman-Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogenous Gaussian noise: First, it is obtained an explicit expression for the Malliavin derivatives of the solutions. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the H\"older continuity of the solutions.