Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs

• Published in: Journal of evolution equations Vol. 21; no. 1; pp. 601 - 623
• Main Authors: Lee, Haesung, Trutnau, Gerald
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2021; Springer Nature B.V
• Subjects: Analysis; Differential equations; Diffusion coefficient; Dirichlet problem; Divergence; Invariants; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Regularity; Uniqueness; Strong Feller property; 60J60; Elliptic and parabolic regularity; 35J15; Itô-SDE; Krylov type estimate; Primary 47D07; Invariant measure; 35B65; Uniqueness in law; Moment inequalities; Secondary 60H20; 60J35
• ISSN: 1424-3199; 1424-3202
• DOI: 10.1007/s00028-020-00593-y

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 ( R d , m ) , s ∈ [ 1 , ∞ ] , 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.