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

Chen, Le/ Hu, Yaozhong/ Nualart, David

ISBN 10: 1470450003 / ISBN 13: 9781470450007

Published by Amer Mathematical Society, 2021

About this Item

277 pages. In Stock.

Seller Inventory # __1470450003

Bibliographic Details

• Title: Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations
• Publisher: Amer Mathematical Society
• Publication Date: 2021
• Language: English
• ISBN 10: 1470450003
• ISBN 13: 9781470450007
• Binding: Paperback
• Condition: Brand New
• Weight: 0.21 kilograms

About this title

Synopsis

"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 ofall 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 The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model.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"--