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

• Autore: Chen Le
• Pubblicazione: Providence : , : American Mathematical Society, , 2021; ©2021
• Edizione: 1st ed.
• Descrizione fisica: 1 online resource (114 pages)
• Disciplina: 519.2/2
• Soggetto topico: Heat 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
• Classificazione: 60H1560G6035R60
• Altri autori: HuYaozhong NualartDavid
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: 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.
• Sommario/riassunto: "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"--
• Titolo autorizzato: Regularity and Strict Positivity of Densities for the Nonlinear Stochastic Heat Equations
• ISBN: 9781470468095; 1470468093
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910956325903321
• Lo trovi qui: Univ. Federico II
• Serie: Memoirs of the American Mathematical Society