| Abstract: |
| Right Markov processes, whose state space consists of absolutely continuous measures (resp.~probabilities) w.r.t.~a fixed measure $\lambda$ on a Polish space $M$, are the central objects in this talk.These stand in one-to-one correspondence with quasi-regular Dirichlet forms. Our first result states the quasi-regularity for a broad class of Dirichlet forms, whose diffusion part is of extrinsic derivative type and which have a non-local (killing and jumping) part. A natural way to construct closed forms of this class is to take as reference measure the push-forward of a Gaussian on $L^2(M,\lambda)$ under $f\mapsto f^2\lambda$, resp.~$f\mapsto (f^2/\lambda(f^2))\lambda$. Hence, we obtain Gaussian-type Sobolev spaces on absolutely continuous measures and associated Ornstein-Uhlenbeck processes. In case $M=\mathbb R^d$ the entropy functional is identified as a member of such a Sobolev space and we construct its stochastic extrinsic derivative flow driven by the Ornstein-Uhlenbeck process. |
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