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050  4 QA377 .S855 2008 
082 0  515.353 
100 1  Stroock, Daniel W 
245 10 Partial Differential Equations for Probabilists 
264  1 New York :|bCambridge University Press,|c2008 
264  4 |c©2008 
300    1 online resource (233 pages) 
336    text|btxt|2rdacontent 
337    computer|bc|2rdamedia 
338    online resource|bcr|2rdacarrier 
490 1  Cambridge Studies in Advanced Mathematics ;|vv.112 
505 0  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 
505 8  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 
505 8  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 
520    This book provides probabilists with sufficient background
       to begin applying PDEs to probability theory and 
       probability theory to PDEs 
588    Description based on publisher supplied metadata and other
       sources 
590    Electronic reproduction. Ann Arbor, Michigan : ProQuest 
       Ebook Central, 2020. Available via World Wide Web. Access 
       may be limited to ProQuest Ebook Central affiliated 
       libraries 
650  0 Differential equations, Partial.;Differential equations, 
       Parabolic.;Differential equations, Elliptic.;Probabilities
655  4 Electronic books 
776 08 |iPrint version:|aStroock, Daniel W.|tPartial Differential
       Equations for Probabilists|dNew York : Cambridge 
       University Press,c2008|z9780521886512 
830  0 Cambridge Studies in Advanced Mathematics 
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