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OverviewIn 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 The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model.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. Full Product DetailsAuthor: Le Chen , Yaozhong Hu , David NualartPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.210kg ISBN: 9781470450007ISBN 10: 1470450003 Publication Date: 30 March 2022 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsReviewsAuthor InformationLe Chen, Emory University, Atlanta, GA. Yaozhong Hu, University of Alberta at Edmonton, Canada. David Nualart, University of Kansas, Lawrence, KS. Tab Content 6Author Website:Countries AvailableAll regions |