|M.Sc Student||Iddo Ben-Ari|
|Subject||Absolute Continuity/Singularity and Relative Entropy|
Properties for Diffusions
|Department||Department of Mathematics||Supervisor||Full Professor Pinsky Ross|
Let and be the generators of non-explosive diffusion processes on a domain. and induce sets of measures and , respectively, on , the space of continuous functions from to . A simple probabilistic criterion for and , not involving stochastic integrals, is given. We prove that for a certain class of generators on called Fuchsian, either or and in the latter case the relative entropy of with respect to is bounded in . In the one-dimensional setup, and we assume both diffusion processes are transient to the positive direction of the real axis. We show that if satisfies , then iff ; otherwise . A number of examples are given, amongst the following: Let with and , , be defined on . Then iff . For , an explicit event , , satisfying and is provided. In the final part of the work, a simple model of random diffusion is introduced and a sufficient criterion for absolute continuity/singularity of the associated annealed measure with respect to the one-dimensional Wiener measure is proved.