| Abstract: |
| \textbf {Szymon Peszat (Institute of Mathematics, Jagiellonian University and Institute of Mathematics, Polish Academy of Sciences), On some smoothening effects of transition semigroups}
An important problem in the theory of Markov processes is the uniqueness of an invariant measure and its ergodicity. This holds if the transition semigroup exhibits some smoothening effects: for example if it is strong Feller and irreducible or asymptotic strong Feller. In the talk I will derive the so-called Bismut-Elworthy-Li formula for diffusions. This formula guarantees the strong Feller property. Later, I recall the classical result of Hawkes for L\`evy semigroups. It relates the strong Feller property with absolute continuity of the process. The last part of the talk will be devoted to the absolute continuity of L\`evy processes.
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