Localization of generators of Markov processes via the Dynkin operator
Abstract
In this article, we consider a Feller process whose resolvent has the strong Feller property. Using the Dynkin operator of this process, we define local realizations of the generator on an open subset U of its state space. In particular, this allows us to describe the generator of the process killed upon leaving a regular, open subset U as a realization of a maximal operator subject to Dirichlet boundary conditions. Besides the definition via the Dynkin operator, we also consider other possible definitions of such a local realization: a definition in terms of the unkilled semigroup, a definition based on approximation with elements of the generator and a probabilistic definition using martingales. Under mild additional assumptions, we prove that all these definitions are equivalent.
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BibTeX
@misc{kunze2026localization,
title = {{Localization of generators of Markov processes via the Dynkin operator}},
author = {Markus Kunze and Jonathan Mui},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38761},
eprint = {2610.38761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38761},
}
RIS
TY - UNPB
TI - Localization of generators of Markov processes via the Dynkin operator
AU - Markus Kunze
AU - Jonathan Mui
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38761
UR - https://arxiv.org/abs/2610.38761
AB - In this article, we consider a Feller process whose resolvent has the strong Feller property. Using the Dynkin operator of this process, we define local realizations of the generator on an open subset U of its state space. In particular, this allows us to describe the generator of the process killed upon leaving a regular, open subset U as a realization of a maximal operator subject to Dirichlet boundary conditions. Besides the definition via the Dynkin operator, we also consider other possible definitions of such a local realization: a definition in terms of the unkilled semigroup, a definition based on approximation with elements of the generator and a probabilistic definition using martingales. Under mild additional assumptions, we prove that all these definitions are equivalent.
ER -