טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentIddo Ben-Ari
SubjectAbsolute Continuity/Singularity and Relative Entropy
Properties for Diffusions
DepartmentDepartment of Mathematics
Supervisor Full Professor Pinsky Ross


Abstract


  Let  and  be the generators of  non-explosive diffusion processes on  a domainand  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.