Strong Feller semigroups and Markov processes: a counterexample
Springer Science and Business Media LLC - Trang 1-16 - 2023
Tóm tắt
The aim of this note is to show, by providing an elementary way to construct counterexamples, that the strong Feller and the joint (space-time) continuity for a semigroup of Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counterexample is the Brownian semigroup on
$${\mathbb {R}}$$
restricted to
$${\mathbb {R}}\setminus \{0\}$$
, for which it is shown that there is no associated càdlàg Markov process. Using the same idea and results from potential theory we then prove that the analogous result with càdlàg Markov process replaced by right Markov process also holds, even if one allows to change the Polish topology to another Polish topology with the same Borel
$$\sigma $$
-algebra.
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