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Markov Processes and Differential Equations

Asymptotic Problems

  • Book
  • © 1996

Overview

Authors:
  1. Mark Freidlin

    1. Department of Mathematics, University of Maryland, College Park, USA

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Part of the book series: Lectures in Mathematics. ETH Zürich (LM)

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Table of contents (11 chapters)

  1. Front Matter

    Pages I-VI
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  2. Stochastic Processes Defined by ODE’s

    • Mark Freidlin
    Pages 1-11

  3. Small Parameter in Higher Derivatives: Levinson’s Case

    • Mark Freidlin
    Pages 13-24

  4. The Large Deviation Case

    • Mark Freidlin
    Pages 25-39

  5. Averaging Principle for Stochastic Processes and for Partial Differential Equations

    • Mark Freidlin
    Pages 41-53

  6. Averaging Principle: Continuation

    • Mark Freidlin
    Pages 55-66

  7. Remarks and Generalizations

    • Mark Freidlin
    Pages 67-78

  8. Diffusion Processes and PDE’s in Narrow Branching Tubes

    • Mark Freidlin
    Pages 79-89

  9. Wave Fronts in Reaction-Diffusion Equations

    • Mark Freidlin
    Pages 91-108

  10. Wave Fronts in Slowly Changing Media

    • Mark Freidlin
    Pages 109-123

  11. Large Scale Approximation for Reaction-Diffusion Equations

    • Mark Freidlin
    Pages 125-135

  12. Homogenization in PDE’s and in Stochastic Processes

    • Mark Freidlin
    Pages 137-148

  13. Back Matter

    Pages 149-154
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Keywords

About this book

Probabilistic methods can be applied very successfully to a number of asymptotic problems for second-order linear and non-linear partial differential equations. Due to the close connection between the second order differential operators with a non-negative characteristic form on the one hand and Markov processes on the other, many problems in PDE's can be reformulated as problems for corresponding stochastic processes and vice versa. In the present book four classes of problems are considered: - the Dirichlet problem with a small parameter in higher derivatives for differential equations and systems - the averaging principle for stochastic processes and PDE's - homogenization in PDE's and in stochastic processes - wave front propagation for semilinear differential equations and systems. From the probabilistic point of view, the first two topics concern random perturbations of dynamical systems. The third topic, homog- enization, is a natural problem for stochastic processes as well as for PDE's. Wave fronts in semilinear PDE's are interesting examples of pattern formation in reaction-diffusion equations. The text presents new results in probability theory and their applica- tion to the above problems. Various examples help the reader to understand the effects. Prerequisites are knowledge in probability theory and in partial differential equations.

Authors and Affiliations

  • Department of Mathematics, University of Maryland, College Park, USA

    Mark Freidlin

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