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Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equation
About this Title
Le Chen, Yaozhong Hu and David Nualart
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 273, Number 1340
ISBNs: 978-1-4704-5000-7 (print); 978-1-4704-6809-5 (online)
DOI: https://doi.org/10.1090/memo/1340
Published electronically: November 3, 2021
Keywords: Stochastic heat equation,
space-time white noise,
Malliavin calculus,
negative moments,
regularity of density,
strict positivity of density,
measure-valued initial data,
parabolic Anderson model
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries and Notation
- 3. Nonnegative Moments: Proof of Theorem
- 4. Proof of Lemma
- 5. Malliavin Derivatives of the Solution
- 6. Existence and Smoothness of Density at a Single Point
- 7. Smoothness of Joint Density at Multiple Points
- 8. Strict Positivity of Density
- A. Appendix
Abstract
In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove $u(t,x)\in \mathbb {D}^\infty$ for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case.- Vlad Bally and Etienne Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998), no. 1, 27–64. MR 1644120, DOI 10.1023/A:1008686922032
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