A discussion of the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE, and moves on to probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions. The author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators, as well as Martingale problems and the Malliavin calculus. While serving as a textbook for a graduate course on diffusion theory with applications to PDE, this will also be a valuable reference to researchers in probability who are interested in PDE, as well as for analysts interested in probabilistic methods.
Diffusions and Elliptic Operators
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First Sentence:
The partial differential equations we will consider have solutions that can be represented as functionals of certain stochastic processes. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
subsequential limit point, martingale problem, inward pointing unit normal vector, weak uniqueness, pathwise solution, nondivergence form, oblique derivative problem, regular diffusion, regular conditional probability, pathwise uniqueness, positive definite square root, local martingale, support theorem, drift coefficients, speed measure, continuous martingale, quadratic variation, limit argument, natural scale, smooth functionals Key Phrases - Capitalized Phrases (CAPs): (learn more)
Moser's Harnack, Suppose Assumption New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
The partial differential equations we will consider have solutions that can be represented as functionals of certain stochastic processes. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
subsequential limit point, martingale problem, inward pointing unit normal vector, weak uniqueness, pathwise solution, nondivergence form, oblique derivative problem, regular diffusion, regular conditional probability, pathwise uniqueness, positive definite square root, local martingale, support theorem, drift coefficients, speed measure, continuous martingale, quadratic variation, limit argument, natural scale, smooth functionals Key Phrases - Capitalized Phrases (CAPs): (learn more)
Moser's Harnack, Suppose Assumption New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 26
books:
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16
books
cite this book:
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- Probabilistic Techniques in Analysis (Probability and Its Applications) by Richard F. Bass in Front Matter, Back Matter (1), and Back Matter (2)
- The Malliavin Calculus and Related Topics (Probability and Its Applications) by David Nualart in Front Matter, and Back Matter
- Elliptic Partial Differential Equations of Second Order by David Gilbarg in Back Matter
- Diffusion Processes and their Sample Paths (Classics in Mathematics) by Kiyosi Itô in Back Matter
- Continuous Martingales and Brownian Motion (Grundlehren der mathematischen Wissenschaften) by Daniel Revuz in Back Matter
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- Foundations of Modern Probability by Olav Kallenberg in Front Matter, and Back Matter
- Limit Theorems for Randomly Stopped Stochastic Processes (Probability and Its Applications) by Dmitrii S. Silvestrov in Front Matter
- Theory of Random Sets (Probability and Its Applications) by Ilya Molchanov in Front Matter
- Eigenvalues, Inequalities, and Ergodic Theory (Probability and Its Applications) by Mu-Fa Chen in Front Matter
- Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach (Probability and Its Applications) by Nils Berglund in Front Matter
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