We study a certain discrete-time Markov evolution in a countably infinite state space that describes the motion of a single particle which is confined through an unbounded potential. From the probabilistic point of view it is a Markov chain whose paths are killed with random intensity coming from an external potential. Its (non-conservative) transition semigroup is a counterpart of the classical Feynman–Kac semigroup. We are mainly interested in long-range Markov chains whose generators are nonlocal (in a specific sense) discrete operators.

In the talk we will give a short introduction to this topic. We will first discuss sharp estimates for functions that are (sub-)harmonic in infinite sets with respect to the discrete Feynman–Kac operators and some applications to the decay rates of solutions to equations involving nonlocal discrete Schrödinger operators (e.g. fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators). These results will be compared with respective estimates for the case of the nearest-neighbour random walk which evolves on an infinite graph of finite geometry. Further, we will investigate the behaviour of the heat kernels of the discrete Feynman-Kac semigroups as well as the associated ultracontractivity properties.

Our approach is based on the direct step property (DSP in short) of the underlying Markov chain and it encompasses a fairly general class of processes and operators. We will present a few constructions leading to DSP Markov chains and illustrate them by various examples.

This is a joint work with Kamil Kaleta (WUST), René Schilling (TU Dresden) and Mateusz Śliwiński (WUST).

In this talk, we establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex hull are known explicitly, and satisfactory bounds on the expected value of its diameter can be found in the literature as well. In this talk we investigate circumradius and inradius of the Brownian convex hull and obtain lower and upper bounds on their expected values. Our other goal is to find bounds on the related inverse processes (that correspond to the perimeter, area, diameter, circumradius and inradius of the convex hull) which provide us with some information on the speed of growth of the size of the Brownian convex hull.

In this talk, relying on Foster-Lyapunov drift conditions, we will discuss subexponential upper and lower bounds on the rate of convergence in the $L^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We will further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we will comment on exponential ergodicity in the $L^p$-Wasserstein distance for a class of Ito processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes will be presented, including Langevin tempered diffusion processes, piecewise Ornstein–Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we will provide a sharp characterization of the rate of convergence via matching upper and lower bounds.

In this talk we consider a partially homogeneous random walk on the quadrant with zero drift at the interior. The goal of the talk is to explain how to obtain qualitative and quantitative knowledge on the passage time of the origin and its moments. The focus of the talk is on the adaptation of the classical methods to our set up and the heuristics on how to obtain those classifications from both a probabilistic and a geometrical perspective.

This is a joint work with Mikhail Menshikov and Andrew Wade.