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

TL;DR

This paper proves the existence of a density for the solution $u(t,x)$ of the 1D stochastic heat equation with unbounded, Lipschitz nonlinearities under Dirichlet boundary conditions. It introduces a localization-based approach, constructing piecewise approximations $u_n$ and establishing local $D_{1,2}$ regularity via Malliavin calculus; density follows from the strict positivity of the Malliavin derivative norm. Unlike prior works that rely on comparison principles, the method handles unbounded coefficients by localization and careful cancellation arguments in the Malliavin derivative. The results extend density existence to nonlinear SPDEs with unbounded coefficients and showcase a robust localization technique for Malliavin analysis of SPDEs with singular kernels.

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.

Theorems & Definitions (12)

• Theorem 1.1
• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• Remark 4.1
• Lemma 5.1
• proof
• Theorem 5.2
• proof