Invariant measures of Lévy-type operators and their associated Markov processes. (English) Zbl 07854751
Summary: A distributional equation as a criterion for invariant measures of Markov processes associated to Lévy-type operators is established. This is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on solutions to Lévy-driven stochastic differential equations. The results are accompanied by various illustrative examples.
MSC:
| 60G10 | Stationary stochastic processes |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60J35 | Transition functions, generators and resolvents |
| 45B05 | Fredholm integral equations |
| 45D05 | Volterra integral equations |
| 60G51 | Processes with independent increments; Lévy processes |
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60H20 | Stochastic integral equations |
Keywords:
Feller processes; invariant distributions; Lévy-type operators; Markov processes; stochastic differential equations; Volterra-Fredholm integral equationsReferences:
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