Diffusions, Superdiffusions and Partial Differential Equations (Colloquium Publications)

Diffusions, Superdiffusions and Partial Differential Equations (Colloquium Publications)

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Description

Interactions between the theory of partial differential equations of elliptic and parabolic types and the theory of stochastic processes are beneficial for both probability theory and analysis. At the beginning, mostly analytic results were used by probabilists. More recently, analysts (and physicists) took inspiration from the probabilistic approach. Of course, the development of analysis in general and of the theory of partial differential equations in particular, was motivated to a great extent by problems in physics. A difference between physics and probability is that the latter provides not only an intuition, but also rigorous mathematical tools for proving theorems. The subject of this book is connections between linear and semilinear differential equations and the corresponding Markov processes called diffusions and superdiffusions.Most of the book is devoted to a systematic presentation (in a more general setting, with simplified proofs) of the results obtained since 1988 in a series of papers of Dynkin and Dynkin and Kuznetsov. Many results obtained originally by using superdiffusions are extended in the book to more general equations by applying a combination of diffusions with purely analytic methods. Almost all chapters involve a mixture of probability and analysis. Similar to the other books by Dynkin, ""Markov Processes"" (Springer-Verlag), ""Controlled Markov Processes"" (Springer-Verlag), and ""An Introduction to Branching Measure-Valued Processes"" (American Mathematical Society), this book can become a classical account of the presented topics.

Contents

Introduction Parabolic equations and branching exit Markov systems: Linear parabolic equations and diffusions Branching exit Markov systems Superprocesses Semilinear parabolic equations and superdiffusions Elliptic equations and diffusions: Linear elliptic equations and diffusions Positive harmonic functions Moderate solutions of $Lu=\psi(u)$ Stochastic boundary values of solutions Rough trace Fine trace Martin capacity and classes $\mathcal{N} 1$ and $\mathcal{N} 0$ Null sets and polar sets Survey of related results Basic facts of Markov processes and Martingales Facts on elliptic differential equations Epilogue Bibliography Subject index Notation index.

Product Details

  • ISBN13: 9780821831748
  • Format: Hardback
  • ID: 9780821831748
  • ISBN10: 0821831747

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