Harnack Inequalities for Stochastic Partial Differential Equations [electronic resource] /
Wang, Feng-Yu.author.
In this book the author presents a self-contained account of Harnack inequalities and applications for the semigroup of solutions to stochastic partial and delayed differential equations. Since the semigroup refers to Fokker-Planck equations on infinite-dimensional spaces, the Harnack inequalities the author investigates are dimension-free. This is an essentially different point from the above mentioned classical Harnack inequalities. Moreover, the main tool in the study is a new coupling method (called coupling by change of measures) rather than the usual maximum principle in the current literature.
Description based upon print version of record.
""Preface""; ""Contents""; ""Chapter 1 A General Theory of Dimension-Free Harnack Inequalities""; ""1.1 Coupling by Change of Measure and Applications""; ""1.1.1 Harnack Inequalities and Bismut Derivative Formulas""; ""1.1.2 Shift Harnack Inequalities and Integration by PartsFormulas""; ""1.2 Derivative Formulas Using the Malliavin Calculus""; ""1.2.1 Bismut Formulas""; ""1.2.2 Integration by Parts Formulas""; ""1.3 Harnack Inequalities and Gradient Inequalities""; ""1.3.1 Gradientâ€?Entropy and Harnack Inequalities""; ""1.3.2 From Gradientâ€?Gradient to Harnack Inequalities""
""1.3.3 L2 Gradient and Harnack Inequalities""""1.4 Applications of Harnack and Shift Harnack Inequalities""; ""1.4.1 Applications of the Harnack Inequality""; ""1.4.2 Applications of the Shift Harnack Inequality""; ""Chapter 2 Nonlinear Monotone Stochastic Partial Differential Equations""; ""2.1 Solutions of Monotone Stochastic Equations""; ""2.2 Harnack Inequalities for 1""; ""2.3 Harnack Inequalities for (0,1)""; ""2.4 Applications to Specific Models ""; ""2.4.1 Stochastic Generalized Porous Media Equations""; ""2.4.2 Stochastic p-Laplacian Equations""
""2.4.3 Stochastic Generalized Fast-Diffusion Equations""""Chapter 3 Semilinear Stochastic Partial Differential Equations""; ""3.1 Mild Solutions and Finite-Dimensional Approximations""; ""3.2 Additive Noise""; ""3.2.1 Harnack Inequalities and Bismut Formula""; ""3.2.2 Shift Harnack Inequalities and Integration by PartsFormula""; ""3.3 Multiplicative Noise: The Log-Harnack Inequality""; ""3.3.1 The Main Result""; ""3.3.2 Application to White-Noise-Driven SPDEs""; ""3.4 Multiplicative Noise: Harnack Inequality with Power""; ""3.4.1 Construction of the Coupling""
""3.4.2 Proof of Theorem 3.4.1""""3.5 Multiplicative Noise: Bismut Formula""; ""Chapter 4 Stochastic Functional (Partial) Differential Equations""; ""4.1 Solutions and Finite-Dimensional Approximations""; ""4.1.1 Stochastic Functional Differential Equations""; ""4.1.2 Semilinear Stochastic Functional Partial Differential Equations""; ""4.2 Elliptic Stochastic Functional Partial Differential Equations with Additive Noise""; ""4.2.1 Harnack Inequalities and Bismut Formula""; ""4.2.2 Shift Harnack Inequalities and Integration by PartsFormulas""; ""4.2.3 Extensions to Semilinear SDPDEs""
""4.3 Elliptic Stochastic Functional Partial Differential Equations with Multiplicative Noise""""4.3.1 Log-Harnack Inequality""; ""4.3.1.1 Proofs of (i)""; ""4.3.1.2 Proof of (ii)""; ""4.3.1.3 Proof of (iii)""; ""4.3.2 Harnack Inequality with Power""; ""4.3.3 Bismut Formulas for Semilinear SDPDEs""; ""4.4 Stochastic Functional Hamiltonian System""; ""4.4.1 Main Result and Consequences""; ""4.4.2 Proof of Theorem 4.4.1""; ""4.4.3 Proofs of Corollary 4.4.3 and Theorem 4.4.5 ""; ""Glossary""; ""References""; ""Index""
Includes bibliographical references and index.
In this book the author presents a self-contained account of Harnack inequalities and applications for the semigroup of solutions to stochastic partial and delayed differential equations. Since the semigroup refers to Fokker-Planck equations on infinite-dimensional spaces, the Harnack inequalities the author investigates are dimension-free. This is an essentially different point from the above mentioned classical Harnack inequalities. Moreover, the main tool in the study is a new coupling method (called coupling by change of measures) rather than the usual maximum principle in the current literature.
Partial differential equations.
Probabilities.
Mathematical analysis.
Analysis (Mathematics).
Partial Differential Equations.
Probability Theory and Stochastic Processes.
Analysis.