LEADER 04817nam 22006613 450 001 9910956325903321 005 20231110223827.0 010 $a9781470468095 010 $a1470468093 035 $a(MiAaPQ)EBC6822188 035 $a(Au-PeEL)EBL6822188 035 $a(CKB)20058040600041 035 $a(RPAM)22488281 035 $a(OCoLC)1284944664 035 $a(EXLCZ)9920058040600041 100 $a20211209d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aRegularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations 205 $a1st ed. 210 1$aProvidence :$cAmerican Mathematical Society,$d2021. 210 4$dİ2021. 215 $a1 online resource (114 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vv.273 311 08$aPrint version: Chen, Le Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations Providence : American Mathematical Society,c2021 9781470450007 320 $aIncludes bibliographical references. 327 $aCover -- 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. 330 $a"In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove u(t, x) D for measure-valued initial data, we need a localized version of Malliavin calculus. Furthermore, we prove that the (joint) density is strictly positive in the interior of the support of the law, where we allow both measure-valued initial data and unbounded diffusion coefficient. The criteria introduced by Bally and Pardoux are no longer applicable for the parabolic Anderson model. We have extended their criteria to a localized version. Our general framework includes the parabolic Anderson model as a special case"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society 606 $aHeat equation 606 $aStochastic partial differential equations 606 $aNonlinear difference equations 606 $aMalliavin calculus 606 $aProbability theory and stochastic processes -- Stochastic analysis -- Stochastic partial differential equations$2msc 606 $aProbability theory and stochastic processes -- Stochastic processes -- Random fields$2msc 606 $aPartial differential equations -- Miscellaneous topics -- Partial differential equations with randomness, stochastic partial differential equations$2msc 615 0$aHeat equation. 615 0$aStochastic partial differential equations. 615 0$aNonlinear difference equations. 615 0$aMalliavin calculus. 615 7$aProbability theory and stochastic processes -- Stochastic analysis -- Stochastic partial differential equations. 615 7$aProbability theory and stochastic processes -- Stochastic processes -- Random fields. 615 7$aPartial differential equations -- Miscellaneous topics -- Partial differential equations with randomness, stochastic partial differential equations. 676 $a519.2/2 686 $a60H15$a60G60$a35R60$2msc 700 $aChen$b Le$01800978 701 $aHu$b Yaozhong$01751924 701 $aNualart$b David$055692 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910956325903321 996 $aRegularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations$94345990 997 $aUNINA