AG Analysis & Stochastik

Henriette Heinrich (TU Dresden)

" Siegmund Duality and Time Reversal of Lévy-type Processes "

In 1976 Siegmund introduced the notion of ``duality of Markov processes with respect to a function''. This concept has since then served as a powerful tool for analyzing Markov processes in fields like population genetics, risk theory, and queuing models, as it allows to connect long-term behavior and fluctuation theory.
In this talk, we will discuss Siegmund duality of Lévy-type processes, focusing on structural properties of (dual) generators and drawing relations to their adjoint operators. We will also investigate the connection of duality to the time reversal of soutions to Lévy-driven SDEs, with a focus on the example of generalized Ornstein-Uhlenbeck processes. Under certain conditions, we will show that the two concepts coincide.

Yana Mokanu

"On ergodicity of Lévy-type processes in R^d"

Abstract

Yuki Ueda (Hokkaido University of Education)

"On the class of freely quasi-infinitely divisible distributions and an extension of Bercovici-Pata bijection"

Bożejko posed the question of whether the Bercovici-Pata bijection, which is a one-to-one correspondence between the classes of infinitely divisible distributions in classical and free probability, can be extended beyond the class of infinitely divisible distributions.

In this talk, we affirmatively answer this question by investigating the class of (classically or freely) quasi-infinitely divisible distributions. Specifically, we find the common characteristic triplets appeared from the Lévy-Khintchine-type representations of these distributions.

In the first part of this talk, we introduce classical and free quasi-infinite divisibility and explain the similarities and differences in their distributional properties. We then give examples of freely quasi-infinitely divisible distributions from various perspectives, including those obtained via the subordination method for free deconvolution by Arizmendi, Tarrango, and Vargas (2020) and distributions related to Catalan sequences studied by Liszewska and Młotkowski (2020), among others. Finally, by identifying common characteristic triplets, we show that the Bercovici-Pata bijection extends to a subclass of quasi-infinitely divisible distributions.This is a joint-work of Ikkei Hotta, Wojciech M\l otkowski and Noriyoshi Sakuma.

Claudius Lütke Schwienhorst (TU Dresden)
 

"Lévy Langevin Monte Carlo for heavy-tailed target distributions"

We extend the Monte Carlo method of Oechsler 2024 to the setting of a target distribution with heavy tails: We choose a regularly varying distribution and prove the convergence of a solution of a stochastic differential equation to this target. Hereby, the stochastic differential equation is driven by a general Lévy process - unlike in the case of a classical Langevin diffusion. This method is justified, apart from the possibility to sample from non-smooth targets, by the fact that an exponential convergence to the invariant distribution holds, which in general cannot be guaranteed by the classical Langevin diffusion in presence of heavy tails. Advantageous compared to other Langevin Monte Carlo methods is the option of an easy implementation of the method by only using a compound Poisson process as noise term and a numerically manageable drift term.

Shend Thaqi (TU Dresden)

"Regularity of multiplicative processes on infinite-dimensional Lie groups"

We investigate multiplicative processes—stochastic processes with independent, but not necessarily stationary increments—on infinite-dimensional Lie groups. Assuming stochastic continuity, these processes exhibit strong regularity properties reminiscent of real-valued Lévy processes. We show that multiplicative processes admit a modification with càdlàg paths. Furthermore, to analyze the behavior of these processes within the group structure, we introduce a suitable notion of length on Lie groups, allowing us to measure the distance of the process from the identity. If the process has bounded jumps, we prove the existence of moments for this distance.

Winfried Sickel (Uni Jena)

"On the regularity of characteristic functions"

Abstract.

Zbigniew Palmowski (Wrocław University of Science and Technology)

"Stable random walks in cones"

Krzysztof Bogdan (Wrocław University of Science and Technology)

"Stable processes with reflections"

We construct a Hunt process that can be described as the isotropic
stable Lévy processes confined by reflections to a bounded open
Lipschitz set. This is achieved through a direct analytic and
potential-theoretic approach using the transition probability semigroup obtained in our earlier work, along with suitable excessive functions (barriers). Our method, based on nonlocal perturbations of integral kernels, offers a general alternative to the concatenation of Markov
processes, previously applied in the literature and using complex
probabilistic methods. We will also discuss applications to stationary
distributions of the corresponding semigroup. This is a joint work with
Markus Kunze (Universität Konstanz).

Callum Murphy-Barltrop (TU Dresden)

"Modelling multivariate extremes with neural networks - is it possible to prove consistency?"