Mathematical Sciences: Asymptotic Problems for Nonlinear PDE's and Limit Theorems for Random Procesess and Fields

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The principal investigators will study a number of asymptotic problems for linear and nonlinear partial differential equations (PDE's) and systems of PDE's using their connections with Markov processes. Such topics as motion of wave fronts, impulses and other structures in semilinear parabolic systems, coupled elliptic and parabolic equations with small diffusivity will be considered. From a probabilistic point of view the results will be of the averaging principle type or of central limit theorem and large deviation type for the corresponding random processes. The investigators also will study random perturbations of evolutionary PDE's. The investigators plan to develop a theory similar to the theory of random perturbations of finite dimensional systems. Many kinds of limit theorems for the random fields defined by the perturbed PDE's will be considered. The principal investigators will study a number of asymptotic problems for linear and nonlinear partial differential equations (PDE's) and systems of PDE's using their connections with Markov processes. This work brings together two fields of mathematical research, partial differential equations and probability theory.