Markov Processes and Differential Equations

The book explores the application of probabilistic methods to asymptotic problems in second-order linear and non-linear partial differential equations (PDEs). It examines four key areas: the Dirichlet problem with small parameters, the averaging principle for stochastic processes and PDEs, homogenization, and wave front propagation in semilinear equations. By linking these topics to random perturbations and pattern formation, the text presents new findings in probability theory, supported by examples for clarity. A background in probability and PDEs is required for readers.