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Author Stroock, Daniel W
Title Partial Differential Equations for Probabilists
Imprint New York : Cambridge University Press, 2008
©2008
book jacket
Descript 1 online resource (233 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Cambridge Studies in Advanced Mathematics ; v.112
Cambridge Studies in Advanced Mathematics
Note Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Chapter 1 Kolmogorov's Forward, Basic Results -- 1.1 Kolmogorov's Forward Equation -- 1.1.1. Derivation of Kolmogorov's Forward Equation -- 1.1.2. Solving Kolmogorov's Forward Equation -- 1.2 Transition Probability Functions -- 1.2.1. Lipschitz Continuity and Convergence -- 1.3 Some Important Extensions -- 1.3.1. Higher Moments -- 1.3.2. Introduction of a Potential -- 1.3.3. Lower Triangular Systems -- 1.4 Historical Notes and Some Commentary -- Chapter 2 Non-Elliptic Regularity Results -- 2.1 The Existence-Uniqueness Duality -- 2.1.1. The Basic Duality Result -- 2.2 Smoothness in the Backward Variable -- 2.2.1. Differentiating the Backward Variable -- 2.1.2. Uniqueness of Solutions to Kolmogorov's Equations -- 2.2.2. The Distribution of Derivatives -- 2.2.3. Uniqueness of Solutions to Kolmogorov's Equation -- 2.3 Square Roots -- 2.3.1. The Non-Degenerate Case -- 2.3.2. The Degenerate Case -- 2.4 Oleinik's Approach -- 2.4.1. The Weak Minimum Principle -- 2.4.2. Oleinik's Estimate -- 2.5 The Adjoint Semigroup -- 2.5.1. The Adjoint Transition Function -- 2.6 Historical Notes and Commentary -- Chapter 3 Preliminary Elliptic Regularity Results -- 3.1 Integration by Parts -- 3.1.1. Gaussian Integration by Parts -- 3.1.2. A General Formula -- 3.2 Application to the Backward Variable -- 3.2.1. A Formula of Bismut Type -- 3.2.2. Higher Order Derivatives -- 3.3 Application to the Forward Variable -- 3.3.1. A Criterion for Absolute Continuity -- 3.3.2. The Inverse of the Jacobian -- 3.3.3. The Fundamental Solution -- 3.3.4. Smoothness in the Forward Variable -- 3.3.5. Gaussian Estimates -- 3.4 Hypoellipticity -- 3.4.1. Hypoellipticity of L -- 3.4.2. Hypoellipicity of… -- 3.5 Historical Notes and Commentary -- Chapter 4 Nash Theory -- 4.1 The Upper Bound
4.1.1. Nash's Inequality -- 4.1.2. Off-Diagonal Upper Bound -- 4.2 The Lower Bound -- 4.2.1. A Poincare Inequality -- 4.2.2. Nash's Other Inequality -- 4.2.3. Proof of the Lower Bound -- 4.3 Conservative Perturbations -- 4.4 General Perturbations of Divergence Form Operators -- 4.4.1. The Upper Bound -- 4.4.2. The Lower Bound -- 4.5 Historical Notes and Commentary -- Chapter 5 Localization -- 5.1 Diffusion Processes on RN -- 5.2 Duhamel's Formula -- 5.2.1. The Basic Formula -- 5.2.2. Application of Duhamel's Formula to Regularity -- 5.2.3. Application of Duhamel's Formula to Positivity -- 5.2.4. A Refinement -- 5.3 Minimum Principles -- 5.3.1. The Weak Minimum Principle Revisited -- 5.3.2. A Mean Value Property -- 5.3.3. The Strong Minimum Principle -- 5.3.4. Nash's Continuity Theorem -- 5.3.5. The Harnack Principle of De Giorgi and Moser -- 5.4 Historical Notes and Commentary -- Chapter 6 On a Manifold -- 6.1 Diffusions on a Compact Riemannian Manifold -- 6.1.1. Splicing Measures -- 6.1.2. Existence of Solutions -- 6.1.3. Uniqueness of Solutions -- 6.2 The Transition Probability Function on M -- 6.2.1. Local Representation of… -- 6.2.2. The Transition Probability Density -- 6.3 Properties of the Transition Probability Density -- 6.3.1. Smoothness in the Forward Variable -- 6.3.2. Smoothness in all Variables -- 6.4 Nash Theory on a Manifold -- 6.4.1. The Lower Bound -- 6.4.2. The Upper Bound -- 6.5 Long Time Behavior -- 6.5.1. Doeblin's Theorem -- 6.5.2. Ergodic Property -- 6.5.3. A Poincar e Inequality -- 6.6 Historical Notes and Commentary -- Chapter 7 Subelliptic Estimates and Hörmander's Theorem -- 7.1 Elementary Facts about Sobolev Spaces -- 7.1.1. Tempered Distributions -- 7.1.2. The Sobolev Spaces -- 7.2 Pseudodifferential Operators -- 7.2.1. Symbols -- 7.2.3. Composition of Pseudodi erential Operators
7.2.4. General Pseudodifferential Operators -- 7.3 Hypoellipticity Revisited -- 7.3.1. Elliptic Estimates -- 7.3.2. Hypoellipticity -- 7.4 Hörmander's Theorem -- 7.4.1. A Preliminary Reduction -- 7.4.2. Completing Hormander's Theorem -- 7.4.3. Some Applications -- 7.5 Historical Notes and Commentary -- Notation -- Index
This book provides probabilists with sufficient background to begin applying PDEs to probability theory and probability theory to PDEs
Description based on publisher supplied metadata and other sources
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: Stroock, Daniel W. Partial Differential Equations for Probabilists New York : Cambridge University Press,c2008 9780521886512
Subject Differential equations, Partial.;Differential equations, Parabolic.;Differential equations, Elliptic.;Probabilities
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