Markov Processes and Stochastic Extrinsic Derivative Flows on the Space of Absolutely Continuous Measures

TL;DR

This paper develops criteria for quasi-regular Dirichlet forms on measure spaces and constructs Markov processes and stochastic flows on these spaces, linking Dirichlet forms with SDE solutions involving extrinsic derivatives.

Contribution

It introduces a new criterion for quasi-regularity of Dirichlet forms on measure spaces and constructs associated Markov processes and stochastic flows using extrinsic derivatives.

Findings

Established a quasi-regularity criterion based on $(L^1+L^ty)$-norm bounds.

Constructed Markov processes with diffusion, jump, and killing components on measure spaces.

Provided martingale solutions to SDEs driven by extrinsic derivatives of entropy functionals.

Abstract

Let $E$ be the class of finite (resp. probability) measures absolutely continuous with respect to a $\sigma$-finite Radon measure on a Polish space. We present a criterion on the quasi-regularity of Dirichlet forms on $E$ in terms of upper bound conditions given by the uniform $(L^1+L^\infty)$-norm of the extrinsic derivative. As applications, we construct a class of general type Markov processes on $E$ via quasi-regular Dirichlet forms containing the diffusion, jump and killing terms. Moreover, stochastic extrinsic derivative flows on $E$ are studied by using quasi-regular Dirichlet forms, which in particular provide martingale solutions to SDEs on these two spaces, with drifts given by the extrinsic derivative of entropy functionals.