Stochastics Seminar

Speaker: Olav Kallenberg

Title: Local time, excursions, and regeneration

Abstract: A random process $X$ is said to be regenerative at a state $a$, if it enjoys the strong Markov property at visits to $a$. The set of times $t\geq 0$ with $X_t=a$ is typically perfect and nowhere dense and supports a singular and diffuse random measure $\xi$, called the local time of $X$ at $a$. Furthermore, the set of excursions of $X$ from $a$ is given by a Poisson process on the time scale of $\xi$. Our aim is to study the local hitting and conditioning properties of $X$, as described in terms of the density of $E\xi$ and the Palm kernel of $X$ with respect to $\xi$. The talk is based on some work done in stages throughout my career.