Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations.

Format:

Book

Author/Creator:

Chen, Le.

Contributor:

Hu, Yaozhong.

Nualart, David.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.273

Language:

English

Subjects (All):

Heat equation.

Stochastic partial differential equations.

Nonlinear difference equations.

Malliavin calculus.

Physical Description:

1 online resource (114 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2021.

Summary:

"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) D 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 The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model.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"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Acknowledgements

Chapter 2. Preliminaries and Notation

2.1. Fundamental Solutions

2.2. Some Moment Bounds and Related Functions

2.3. Malliavin Calculus

Chapter 3. Nonnegative Moments: Proof of Theorem 1.5

Chapter 4. Proof of Lemma 1.6

Chapter 5. Malliavin Derivatives of the Solution

Chapter 6. Existence and Smoothness of Density at a Single Point

6.1. A Sufficient Condition

6.2. Proof of Theorem 1.1

Chapter 7. Smoothness of Joint Density at Multiple Points

7.1. Proof of Theorem 1.2

7.2. Proof of Theorem 1.3

Chapter 8. Strict Positivity of Density

8.1. Two Criteria for Strict Positivity of Densities

8.2. Proof of Theorem 1.4

8.3. Technical Propositions

Appendix A. Appendix

A.1. Some Miscellaneous Results

A.2. A General Framework from Hu et al

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Includes bibliographical references.

Other Format:

Print version: Chen, Le Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations

ISBN:

9781470468095

1470468093

OCLC:

1284944664