Ph.D Thesis

Ph.D StudentBen-Ari Iddo
SubjectTopics in Diffusion Processes
DepartmentDepartment of Mathematics
Supervisor PROF. Ross Pinsky


The thesis consists of two works. Here is a brief description of each work.

Let L be an elliptic second order partial differential operator, defined on a bounded domain in Rd, d>1, with either the Dirichlet boundary condition or an oblique-derivative boundary condition. We study the asymptotic shift for the principal eigenvalue of  L under  obstacles. An obstacle  corresponds  to a finite number of compactly supported potential wells. The main result of this work expresses the shift for the principal eigenvalue as the obstacles shrink to points in terms of the Newtonian capacity of their support and the principal eigenfunctions for the unperturbed operator and its formal adjoint. This generalizes known results on similar perturbations for selfadjoint operators. 

Consider a diffusion process on a bounded domain, absorbed on the boundary. Given a probability measure  p on the domain, the induced diffusion with random jumps from the boundary process is the process which coincides with the absorbed diffusion until hitting the boundary, at which time it starts afresh according to the distribution p. This procedure is repeated every time the process hits the boundary. We prove that the distribution of the process converges exponentially fast to the invariant measure and express the exponent in terms of the spectral gap for a corresponding differential operator.