Ph.D Thesis | |

Ph.D Student | Ben-Ari Iddo |
---|---|

Subject | Topics in Diffusion Processes |

Department | Department 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 R^{d},
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.