Malliavin Calculus for the stochastic heat equation and results on the density

TL;DR

This paper applies Malliavin calculus to the one-dimensional stochastic heat equation with unbounded nonlinear coefficients, establishing the existence of a density for the solution using a localization approach instead of comparison principles.

Contribution

It introduces a new localization-based method to prove the existence of a density for the solution of the stochastic heat equation with unbounded coefficients.

Findings

Established regularity of the solution using a piecewise approximation.

Provided a new proof of density existence without comparison principles.

Handled unbounded coefficients effectively with localization.

Abstract

We study the one-dimensional stochastic heat equation with unbounded, nonlinear,Lipschitz coefficients with Dirichlet boundary conditions. Using Malliavin calculus, we construct a piecewise approximation of the solution u and establish regularity results. This approximation enables us to provide a new proof of the existence of a density for the random variable u(t, x) at any fixed t, x. Unlike existing proofs, which rely on comparison principles ([10], [12]), our approach is based purely on a localization argument, which allows us to handle the unbounded coefficients.