The main goal of this project is to investigate a large class of problems in which asymptotic analysis allows for a detailed study of multi-scale phenomena. Among the problems to be studied are the asymptotic behavior of branching diffusion processes, transition between the averaging and homogenization regimes for transport by cellular flows, and asymptotics of solutions to nonlinear parabolic PDEs. Several problems concerning random perturbations of incompressible flows will also be studied. These include the transport properties of the Benard convection, large deviations for randomly perturbed incompressible flows, and mathematical analysis of Brownian motors.
The proposed research will provide a rigorous mathematical foundation for several phenomena that have been actively discussed in natural sciences. In particular, we will consider branching diffusion processes, which are central in the study of evolution of various populations such as bacteria, cancer cells, carriers of a particular gene, etc., where each member of a population may die or produce offspring independently of the rest. We plan to describe the long-time behavior of the population in different regions of space when, in addition to branching, the members of the population move in space and the branching mechanism depends on the location. Other models to be considered describe the movement of particles (e.g., molecules) due to a combination of a macroscopic motion and a small random diffusion. In particular, we'll examine mechanisms that create directed motion out of fluctuations of a random or periodic velocity field even in situations when the field itself has no preferred direction. Such mechanisms are very important in many applications and have been discussed in hundreds of physics and chemistry papers, although mostly at computational and experimental levels.
