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arXiv · 2410.10115

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

D. Farazakis·G. Karali·A. Stavrianidi
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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.

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D. Farazakis, G. Karali, A. Stavrianidi. 2024-10-14. Malliavin Calculus for the stochastic heat equation and results on the density. https://arxiv.org/abs/2410.10115

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