We focus on invariant measures of an interacting particle system in the case when the set of sites, on which the particles move, has a structure different from the usually considered set ${\mathbb{Z}}^d$. We have chosen the tree structure with the dynamics that leads to one of the classical particle systems, called the zero range process. The zero range process with the constant speed function corresponds to an infinite system of queues and the arrangement of servers in the tree structure is natural in a number...

We consider an infinite system of hard balls in ${}^d$ undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also show that Gibbs measures are reversible measures.

In this work infinitely divisible cylindrical probability measures on arbitrary Banach spaces are introduced. The class of infinitely divisible cylindrical probability measures is described in terms of their characteristics, a characterisation which is not known in general for infinitely divisible Radon measures on Banach spaces. Further properties of infinitely divisible cylindrical measures such as continuity are derived. Moreover, the classification result enables us to deduce new results on...

This second part of our two part work on i.d. process has four main goals:(1) To develop a potential operator for recurrent i.d. (infinitely divisible) processes and to use this operator to find the asymptotic behavior of the hitting distribution and Green’s function for relatively compact sets in the recurrent case.(2) To develop the appropriate notion of an equilibrium measure and Robin’s constant for Borel sets.(3) To establish the asymptotic behavior questions of a potential theoretic nature...

We show that associated with every i.d. (infinitely divisible) process on a locally compact, non-compact 2nd countable Abelian group is a corresponding potential theory that yields definitive results on the behavior of the process in both space and time. Our results are general, no density or other smoothness assumptions are made on the process. In this first part of two part work we have four main goals.(1) To lay the probabilistic foundation of such processes. This mainly consists in giving the...

We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.

The article considers the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. The paper considers some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination will be considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method...

The so-called ϕ-divergence is an important characteristic describing "dissimilarity" of two probability distributions. Many traditional measures of separation used in mathematical statistics and information theory, some of which are mentioned in the note, correspond to particular choices of this divergence. An upper bound on a ϕ-divergence between two probability distributions is derived when the likelihood ratio is bounded. The usefulness of this sharp bound is illustrated by several examples of...

The admissibility of spaces for Itô functional difference equations is investigated by the method of modeling equations. The problem of space admissibility is closely connected with the initial data stability problem of solutions for Itô delay differential equations. For these equations the $p$-stability of initial data solutions is studied as a special case of admissibility of spaces for the corresponding Itô functional difference equation. In most cases, this approach seems to be more constructive...

We consider a family of nonlinear stochastic heat equations of the form ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, where $\dot{W}$ denotes space–time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$, and $\sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $ℒf=c{f}^{\text{'}\text{'}}$ for some $cgt;0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is...

Sensitivity analysis of irreducible Markov chains considers an original Markov chain with transition probability matrix $P$ and modified Markov chain with transition probability matrix $P$. For their respective stationary probability vectors $\pi ,\tilde{\pi }$, some of the following charactristics are usually studied: $\parallel \pi -\tilde{\pi }{\parallel }_p$ for asymptotical stability [3], $|{\pi }_i-{\tilde{\pi }}_i|,\frac{|{\pi }_i-{\tilde{\pi }}_i|}{{\pi }_i}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P=P\left(t\right)$ and stationary probability vector $\pi \left(t\right)$, derivatives are also used for studying...