Asymptotic Problems for Stochastic Processes and PDE's

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9803522 Freidlin Perturbations of the Hamiltonian systems will be studied in this project. The long- time behavior of such perturbed systems, even if the perturbations are purely deterministic, should be described by a stochastic process on a graph related to the Hamilton function. A class of asymptotic problems for PDEs, such as the small viscosity asymptotics for the Navier-Stokes equations in the plane or small diffusion asymptotics for reaction - diffusion in an incompressible fluid, are closely related to random perturbations of the Hamiltonian systems. The probabilistic approach is useful in analyzing these problems. Long time behavior of dynamical systems perturbed by a stochastic noise is studied in this project. This type of problem arises in many applications, for example, the long time behavior of cosmic objects or the motion of a fluid with a small viscosity. The perturbations, even if they are small, become essential for the long-time behavior of the system. The asymptotic approach, which the investigator is developing in the project, is the most promising for analyzing this type of problem.