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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.

Markov Processes and Differential Equations: Asymptotic Problems

1 Stochastic Processes Defined by ODE's.- 2 Small Parameter in Higher Derivatives: Levinson's Case.- 3 The Large Deviation Case.- 4 Averaging Principle for Stochastic Processes and for Partial Differential Equations.- 5 Averaging Principle: Continuation.- 6 Remarks and Generalizations.- 7 Diffusion Processes and PDE's in Narrow Branching Tubes.- 8 Wave Fronts in Reaction-Diffusion Equations.- 9 Wave Fronts in Slowly Changing Media.- 10 Large Scale Approximation for Reaction-Diffusion Equations.- 11 Homogenization in PDE's and in Stochastic Processes.- References.