Workshop on Stable Processes and Related Topics
  • Krzysztof Bogdan (WUST, Poland)
  • Zbigniew Palmowski (WUST, Poland)
  • Leif Döring (University of Mannheim, Germany)
  • Tomasz Grzywny (WUST, Poland)
  • Andreas Kyprianou (Bath University, UK)
  • Mateusz Kwaśnicki (WUST, Poland)
  • Marcin Magdziarz (WUST, Poland)
  • Jacek Małecki (WUST, Poland)
  • Juan Carlos Pardo (CIMAT, Mexico)
  • Nicolas Privault (Nanyang Technological University, Singapore)
  • Tomasz Żak (WUST, Poland)

Thursday, December 20th (Room: 2.17, Building C-13, campus of WUST)

13:10–13:15   Opening
13:15–14:00   Andreas Kyprianou       Stable Lévy processes in a cone
(Bath University, UK)
14:00–14:45   Mateusz Kwaśnicki       Lévy processes with completely monotone jumps
(WUST, Poland)
Coffee break
15:00–15:45   Tomasz Żak       Space and time inversions of stochastic processes and Kelvin transform
(WUST, Poland)
Coffee break
16:15–17:00   Marcin Magdziarz       Lamperti transformation - Cure for ergodicity breaking
(WUST, Poland)
17:00–17:45   Leif Döring       Self-Similarity and Stable SDEs
(University of Mannheim, Germany)

Friday, December 21st (Room: 2.17, Building C-13, campus of WUST)

9:30–10:15   Juan Carlos Pardo       Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes
(CIMAT, Mexico)
10:15–11:00   Jacek Małecki       Universality classes for general random matrix flows
(WUST, Poland)
Coffee break
11:30–12:15   Tomasz Grzywny       Transition densities for Lévy processes
(WUST, Poland)
12:15–13:00   Nicolas Privault       Moments of k-hop counts in the random-connection model
(Nanyang Technological University, Singapore)

Leif Döring
Self-Similarity and Stable SDEs
We discuss existence questions for stable stochastic differential equations without drift. Results are on entrance from infinity, finite time explosion and weak existence of solutions. Proofs use recent results for stable process obtained using the Lamperti transformations for self-similar Markov processes.

Tomasz Grzywny
Transition densities for Lévy processes

Andreas Kyprianou
Stable Lévy processes in a cone
Bañuelos and Bogdan (2004) and Bogdan, Palmowski and Wang (2016) analyse the asymptotic tail distribution of the first time a stable (Lévy) process in dimension d\ge 2 exists a cone. We use these results to develop the notion of a stable process conditioned to remain in a cone as well as the the notion of a stable process conditioned to absorb continuously at the apex of a cone (without leaving the cone). As self-similar Markov processes we examine some of their fundamental properties through the lens of its Lamperti-Kiu decomposition. In particular we are interested to understand the underlying structure of the Markov additive process that drives such processes. As a consequence of our interrogation of the underlying MAP, we are able to provide an answer by example to the open question: If the modulator of a MAP has a stationary distribution, under what conditions does its ascending ladder MAP have a stationary distribution? We show how the two forms of conditioning are dual to one another. Moreover, we construct the null-recurrent extension of the stable process killed on exiting a cone, showing that it again remains in the class of self-similar Markov processes.

Mateusz Kwaśnicki
Lévy processes with completely monotone jumps
Lévy processes with completely monotone jumps were introduced by L.C.G. Rogers in 1983. During my talk I will focus on fluctuation theory for these processes, and mention several other directions of current research.
Rogers proved that spatial Wiener--Hopf factors of a Lévy process with completely monotone jumps themselves have completely monotone jumps. A similar result for temporal Wiener--Hopf factors is proved in an unpublished paper arXiv:1312.1866, under additional technical assumptions. Extension to the general case is given in the recent arXiv:1811.06617, and it required a number of peculiar auxiliary lemmas.
In my talk I will discuss the result mentioned above, together with a few applications, some of which are still work in progress.

Marcin Magdziarz
Lamperti transformation - Cure for ergodicity breaking
Recent results in single-particle experiments show that many complex systems display ergodicity breaking. In this talk we demonstrate how to transform a non-ergodic anomalous diffusion process in order to recover the ergodicity property. In the introduced method we use the so-called Lamperti transformation. Our approach enables us to perform statistical inference using only one recorded trajectory of the analyzed process even in the case when the original process displays ergodicity breaking. The method can be applied to examine many important examples of non-ergodic anomalous diffusion models, including space-time fractional diffusion equations, Lévy walks or subordinated fractional Brownian motions.

Jacek Małecki
Universality classes for general random matrix flows
We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed for the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its sub-sequences as solutions to the integro-differential equation. We use the result to study universality classes of random matrix flows. These generalize the classical results related to Dyson Brownian motion and squared Bessel particle systems, but even in these cases we show new phenomenons as the lack of uniqueness of solution and existence of the generalized Marchenko-Pastur distributions supported on the real line. Assuming the uniqueness of the solution to the integro-differential equation we characterize the limiting process as a free diffusion solving free SDE.
This is based on a joint work with José Luis Pérez.

Juan Carlos Pardo
Cut-off phenomenon for Ornstein-Uhlenbeck processes driven by Lévy processes
In this talk, we study the cut-off phenomenon of d-dimensional Ornstein- Uhlenbeck processes which are driven by Lévy processes under the total variation distance. To be more precise, we study the abrupt convergence under the total variation distance of the aforementioned process to its equilibrium. Despite that the invariant distribution is not explicit, its distributional properties allow us to deduce that a profile function always exists in the reversible cases and it may exist in the non-reversible cases. The cut-off phenomenon for the average and superposition processes is also determined.
This is a joint work with Gerardo Barrera.

Nicolas Privault
Moments of k-hop counts in the random-connection model
We derive moment identities for the stochastic integrals of multiparameter processes in a random-connection model based on a point process admitting a Papangelou intensity. Those identities are written using sums over partitions, and they reduce to sums over non-flat partition diagrams in the case of multiparameter processes vanishing on diagonals. As an application we obtain general identities for the moments of k-hop counts in the random-connection model, which simplify the derivations available in the literature.

Tomasz Żak
Space and time inversions of stochastic processes and Kelvin transform