Assume a group G acts on a set. Given a subgroup H of G , by an //-selector we mean a selector of the set of all orbits of H . We investigate measurability properties of //-selectors with respect to G-invariant measures. Let us fix a set A and a group G acting on it. By p we denote a G-invariant countably additive measure on A . The most common example of such a situation is an invariant measur...

We consider homogeneous random walks in the quarter-plane. The necessary conditions which characterize random walks of which the invariant measure is a sum of geometric terms are provided in Chen et al. (arXiv:1304.3316, 2013, Probab Eng Informational Sci 29(02):233–251, 2015). Based on these results, we first develop an algorithm to check whether the invariant measure of a given random walk is...

We present a method of approximating the unique invariant measure and associated Lyapunov exponents of a random dynamical system deened by the iid composition of a family of maps in situations where only one Lyapunov exponent is observed. As a corollary, our construction also provides a method of estimating the top Lyapunov exponent of an iid random matrix product. We develop rigorous numerical...

This paper presents a Lie-Trotter splitting for inertial Langevin equations (Geometric Langevin Algorithm) and analyzes its long-time statistical properties.The splitting is defined as a composition of a variational integrator with an Ornstein-Uhlenbeck flow. Assuming the exact solution and the splitting are geometrically ergodic, the paper proves the discrete invariant measure of the splitting...

Let A; B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure , there exists a projection onto a torus T r of dimension r dim , that maps-almost every B-orbit to a uniformly distributed sequence in T r. As a corollary we obtain that the Hausdorr dimension of any bi-invariant measure, as well as...

Unipotent flows are very well-behaved dynamical systems. In particular, Marina Ratner has shown that every invariant measure for such a flow is of a nice algebraic form. This talk will present some consequences of this important theorem, and explain a few of the ideas of the proof. In general, algebraic technicalities will be pushed to the background as much as possible. Eg. Let X = torus T = R...

Durret t (1984) proved the existence of an invariant measure for the critical and supercritical contact process seen from the right edge. Galves and Presutti (1987) proved, in the supercritical case, that the invariant measure was unique, and convergence to it held starting in any semi-infinite initial state. We prove the same for the critical contact process. We also prove that the process sta...

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$int_{S}f(sigma(y)xt)dmu(t)-int_{S}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin S,$$ where $S$ is a semigroup, $sigma$ is an involutive morphism of $S$, and $mu$ is a complex measure that is linear combinations of Dirac measures $(delta_{z_{i}})_{iin I}$, such that for all $iin I$, $z_{...

We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of onedimensional surjective cellular automata. We also discuss ...