LARGE DEVIATION PRINCIPLE FOR THE DIFFUSION-TRANSMUTATION PROCESSES AND DIRICHLET PROBLEM FOR PDE SYSTEMS WITH SMALL-PARAMETER

The diffusion-transmutation processes are considered as the diffusivit ies are of order epsilon, epsilon --> 0 and the transmutation intensit ies are of order epsilon(-1). We prove a large deviation principle for the position joint with the type occupation times as epsilon --> 0 an d study the exit problem for this process. We consider the Levinson ca se where a trajectory of the average drift field exits from a domain i n finite time in a regular way and the large deviation case where the average drift field on the boundary points inward at the domain. The e xit place and the type distribution at the exit time are determined as epsilon --> 0; this gives the limit of the Dirichlet problems for the corresponding PDE systems with a parameter epsilon --> 0.