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We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary…

We study limit laws for return time processes defined on infinite conservative ergodic measure preserving dynamical systems. Especially for the critical cases with purely atomic limiting distribution we derive distorted processes posessing…

Directed topology was introduced as a model of concurrent programs, where the flow of time is described by distinguishing certain paths in the topological space representing such a program. Algebraic invariants which respect this…

For stochastic processes leading to condensation, the condensate, once it is formed, performs an ergodic stationary-state motion over the system. We analyse this motion, and especially its characteristic time, for zero-range processes. The…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

For a Dunford-Schwartz operator in the $L^p-$space, $1\leq p< \infty$ , of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of…

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling-Kac condition. We…

Assuming that a reflected Ornstein-Uhlenbeck state process is observed at discrete time instants, we propose generalized moment estimators to estimate all drift and diffusion parameters via the celebrated ergodic theorem. With the sampling…

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at…

In this note, we present a version of Hoeffding's inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffding's inequality for discrete-time…

The training algorithms for AI systems all introduce far-from-equilibrium dynamical processes, and understanding the irreversibility of these algorithms is a fundamental step towards understanding the learning dynamics of modern AI systems.…

Given an oriented graph $D$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endpoints in $X$. When the subset $X$ is of size $p$ (resp. at most $p$), this operation is called an…

We consider an ergodic process on finitely many states, with positive entropy. Our first main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set…

The fluctuation relations, which characterize irreversible processes in Nature, are among the most important results in non-equilibrium physics. In short, these relations say that it is exponentially unlikely for us to observe a…

Power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in von Neumann's ergodic theorem with continuous time is considered. All possible exponents of the considered power-law convergence are found; for…

In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic splitting is the obvious…

We construct ergodic probability measures with infinite metric entropy for typical continuous maps and homeomorphisms on compact manifolds. We also construct sequences of such measures that converge to a zero-entropy measure.

For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation…

This paper presents different approaches, based on functional inequalities, to study the speed of convergence in total variation distance of ergodic diffusion processes with initial law satisfying a given integrability condition. To this…

We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure…