Existence and regularity of infinitesimally invariant measures, transition functions and time-homogeneous Itô-SDEs
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...
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Published in | Journal of evolution equations Vol. 21; no. 1; pp. 601 - 623 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
01.03.2021
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 1424-3199 1424-3202 |
DOI: | 10.1007/s00028-020-00593-y |