Abstract
We show existence of an infinitesimally invariant measure m for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in \(L^s({\mathbb {R}}^d,m)\), \(s\in [1,\infty ]\), including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of m, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.
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This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(NRF-2017R1D1A1B03035632)
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Lee, H., Trutnau, G. Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs. J. Evol. Equ. 21, 601–623 (2021). https://doi.org/10.1007/s00028-020-00593-y
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DOI: https://doi.org/10.1007/s00028-020-00593-y
Keywords
- Elliptic and parabolic regularity
- Strong Feller property
- Invariant measure
- Krylov type estimate
- Moment inequalities
- Uniqueness in law
- Itô-SDE