- Date: 15 May. (Thu.)
- Place: West Wing, 6th floor, Colloquium Room and on the Web (zoom)
- Time: 16:50–19:00
- Speaker 1: Ju-Yi Yen (University of Cincinnatti)
- Time: 16:50–17:50
- Title: Excursion-Theoretic Approaches to Limit Theorems for Additive Functionals of Markov Processe
- Abstract:
This talk explores a uni ed excursion-theoretic framework for proving limit theorems of additive functionals associated with various classes of Markov processes, including Brownian motion, null recurrent di usions, and symmetric strong Markov processes. Motivated by classical results such as the Darling Kac theorem and the Ray Knight theorems, we investigate how local time provides a natural time scale for analyzing functionals of processes that do not admit nite invariant measures. We begin with Brownian motion, where the inverse local time at zero enables a strong law of large numbers and central limit theorem for time integrals of functions along sample paths. These results are revisited using Itos excursion theory, highlighting its utility in both deriving moments and capturing uctuation behavior. We then extend these ideas to null recurrent linear di usions by transforming them into time-changed Brownian motions via Zvonkins method. Excursion-based representations again yield central limit theorems, even in the absence of stationary distributions. Finally, we examine a broader setting of symmetric strong Markov processes, where local times are used to de ne regenerative structures. By leveraging generalized RayKnight theorems and Gaussian process techniques, we establish limit theorems under minimal assumptions, unifying previous results under a single probabilistic strategy. This excursion-centric viewpoint not only clari es asymptotic behaviors but also opens paths toward analyzing more complex dynamics such as complex-valued processes and higher-dimensional extensions.
- Speaker 2: Loïc Chaumont (Université d’Angers)
- Time: 18:00–19:00
- Title: Levy processes resurrected in the positive half-line
- Abstract:
Levy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. In this work, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Levy process. First, we shall give a detailed definition of the resurrected process whose law is described in terms of that of the process killed when it reaches the negative half line. In particular, we will specify the explicit form of the resurrection kernel. Then we will see that when the initial Levy process X creeps downward and satisfies certain additional condition, the resurrected process is absorbed at 0 with probability one, independently of its starting
point. Some criteria for absorption and some criteria for non absorption will be given. The most delicate case is when X enters immediately in the negative half line and drifts to -infinity. It is then possible to give a sufficient condition for absorption but up to now, even when X is the negative of a subordinator, we do not know whether this condition can be dropped or not. We shall take a closer look at the case of stable processes. This is a joint work with Victor Rivero (CIMAT, Guanajuato) and Marria Emilia Caballero (Instituto de Matematicas, UNAM, Mexico).
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