M.Sc Student | Ben-Ari Iddo |
---|---|

Subject | Absolute Continuity/Singularity and Relative Entropy Properties for Diffusions |

Department | Department of Mathematics |

Supervisor | Professor Ross Pinsky |

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.