04817nam 22006613 450 991095632590332120231110223827.097814704680951470468093(MiAaPQ)EBC6822188(Au-PeEL)EBL6822188(CKB)20058040600041(RPAM)22488281(OCoLC)1284944664(EXLCZ)992005804060004120211209d2021 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierRegularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations1st ed.Providence :American Mathematical Society,2021.©2021.1 online resource (114 pages)Memoirs of the American Mathematical Society ;v.273Print version: Chen, Le Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations Providence : American Mathematical Society,c2021 9781470450007 Includes bibliographical references.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 -- Bibliography -- Back Cover."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 The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model.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.Memoirs of the American Mathematical Society Heat equationStochastic partial differential equationsNonlinear difference equationsMalliavin calculusProbability theory and stochastic processes -- Stochastic analysis -- Stochastic partial differential equationsmscProbability theory and stochastic processes -- Stochastic processes -- Random fieldsmscPartial differential equations -- Miscellaneous topics -- Partial differential equations with randomness, stochastic partial differential equationsmscHeat equation.Stochastic partial differential equations.Nonlinear difference equations.Malliavin calculus.Probability theory and stochastic processes -- Stochastic analysis -- Stochastic partial differential equations.Probability theory and stochastic processes -- Stochastic processes -- Random fields.Partial differential equations -- Miscellaneous topics -- Partial differential equations with randomness, stochastic partial differential equations.519.2/260H1560G6035R60mscChen Le1800978Hu Yaozhong1751924Nualart David55692MiAaPQMiAaPQMiAaPQBOOK9910956325903321Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations4345990UNINA