Dirichlet form approach to diffusions with discontinuous scale
A talk in the Bielefeld Stochastic Afternoon series by
Liping Li
| Abstract: | It is well known that a regular diffusion on an interval $I$ without killing inside is uniquely determined by a canonical scale function $s$ and a canonical speed measure $m$. Note that $s$ is a strictly increasing and continuous function and $m$ is a fully supported Radon measure on $I$. In this talk we will associate a general triple $(I,s,m)$, where $s$ is only assumed to be increasing and $m$ is not necessarily fully supported, to certain Markov processes by way of Dirichlet forms. A straightforward generalization of the Dirichlet form associated with regular diffusion will be first put forward, and we will find out its corresponding continuous Markov process $\dot X$, for which the strong Markov property fails whenever $s$ is not continuous. Then by operating regular representations on Dirichlet form and Ray-Knight compactification on $\dot X$ respectively, the same unique desirable symmetric Hunt process associated to $(I,s,m)$ is eventually obtained. This Hunt process is homeomorphic to a quasi-diffusion, which is known as a celebrated generalization of regular diffusion. This talk is based on recent papers arXiv: 2211.12369, arXiv:2303.07574, arXiv:2303.07571, and arXiv:2303.07567. Within the CRC this talk is associated to the project(s): A5 |