Abstracts

Jacek Banasiak and Adam Błoch 
— Network transport with node dynamics I and II.

The talk’s starting point is a linear metapopulation model, where the individuals of subpopulations can migrate between the nodes. Acknowledging that individuals appear at the target some time after they depart from home, we arrive at a system of ordinary differentia equations with delays. We aim to rephrase the model as a transport model on a digraph, where the delays result from the flow of individuals along the edges between the connected nodes. We show that such a model requires introducing specific dynamic conditions, linear or nonlinear, at the nodes relating the incoming and outgoing flows. The developed framework is an extension of the well-known dynamic network transport model and can be used to build a range of models describing many metapopulation phenomena, e.g., disease spread through migrations or cell maturation. We present a preliminary qualitative analysis of the general model using the theory of semigroups of strongly continuous operators and present its applications to a classical migration model and an epidemiological model with migrations.

Krzysztof Bogdan 
— Self-similar solutions via stationary distributions.

I will report on an approach to the existence of self-similar solutions of the heat equation for homogeneous Markov generators via stationary distributions of the related Ornstein–Uhlenbeck semigroups. We will focus on results for the fractional Laplacian with Dirichlet conditions in cones, obtained jointly with Piotr Knosalla, Łukasz Leżaj and Dominika Pilarczyk, and for the fractional Laplacian with Hardy-type potential, obtained jointly with Tomasz Jakubowski, Panki Kim and Dominika Pilarczyk. We may also discuss related large-time asymptotics, Yaglom limits, entrance laws, and the asymptotics of operator norms of the corresponding semigroups.

Tomasz Komorowski 
— Invariant measure and asymptotic stability for the solution of the stochastic Burgers equation with a drift.

In my talk I will discuss the question of the existence and uniqueness of an invariant probability measure for the stochastic Burgers equation (SBE) on a torus. This equation plays an important role e.g. in fluid dynamics, for describing the evolution of shocks in turbulence, or as a universal object in fluctuating hydrodynamics. It is also intimately connected with the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation. The aforementioned equations appear in statistical mechanics; the former e.g. in the problem of directed random polymers, the latter e.g. in the description of growth of random surfaces. I will present first our joint results with prof. Yu Gu (Univ. of Maryland) concerning the existence, uniqueness and asymptotic stability of the invariant measure for the SBE in the Wasserstein metric. Only in some special cases this invariant measure is known.

Our recent result shows that in the case the noise has a deterministic, sufficiently regular drift, the SBE has a unique invariant probability measure. This measure is absolutely continuous with respect to the invariant measure of the equation without the drift, with finite relative entropy. In addition, the asymptotic stability holds in this case in the total variation metric.

References:

[1] Y. Gu, T. Komorowski, KPZ on torus: Gaussian fluctuation, Ann. Inst. H. Poincare, Prob. and Stat. 2024, Vol. 60, No. 3, 1570-1618, https://doi.org/10.1214/23-AIHP1392; 
[2] Y. Gu, T. Komorowski, Asymptotic stability of the stochastic Burgers equation on a torus, in preparation.

Markus Kunze 
— The fractional Laplacian with reflections.

In contrast to the situation of differential operators on a domain D ⊂ R^d, the notion of a boundary condition for nonlocal operators, such as the fractional Laplacian −(−Δ)^$\alpha$ for $\alpha \in (0,2)$ that we consider here, is much less understood. In this talk, we consider a boundary condition which corresponds to a reflection of the isotropic -stable process, the stochastic process associated to −(−Δ)^$\alpha$, upon leaving D. We construct the transition semigroup of the resulting process and discuss important properties of it. In particular, we characterize its generator. In order to give a probabilistic interpretation of the boundary condition, we also consider an extension of the semigroup to the ladder space D = N₀× D.

This talk is based on joint work with Krzysztof Bogdan.

Andrey Pilipenko
— Limit theorems for one-dimensional diffusions.

We prove general results on the convergence in distribution of one-dimensional diffusions with various boundary conditions (reflecting, sticky, etc.). As an application, we consider homogenization problems where a diffusion with a semi-permeable membrane is obtained as the limit of diffusions with multiple membranes or with drifts that converge in some sense to a delta function. Our approach is based on the Itô–McKean representation of a one-dimensional diffusion as a space-time transformation of a Brownian motion [1]. This approach appears to be significantly simpler than the analytic one proposed in [2].

References:

[1] Itô, K. and McKean, H. P. (1965). Diffusion processes and their sample paths.
[2] Freidlin, M. I. and Wentzell, A. D. (1994). Necessary and sufficient conditions for weak convergence of one-dimensional Markov processes. In The Dynkin Festschrift: Markov Processes and Their Applications (pp. 95-109).

Yuri Tomilov 
— From Beurling—Kato to Ritt Operators.

Classical results of Beurling and Kato show that holomorphy of a C₀ - semigroup T (t)_t≥0 can be detected from the behaviour of T (t) near t­ = 0. I will discuss generalisations of this phenomena and its discrete-time analogues, where the role of holomorphic semigroups is played by Ritt operators. The main tool is a Hardy—Sobolev functional calculus on either sectorial or Stolz domains. I will then explain how these calculi connect with Beurling—Kato type defect conditions. This leads to direct and converse criteria, together with examples showing the sharpness of the results.

This is joint work with A. Borichev and A. Gomilko.