We study two different versions of the simple exclusion process on augmented Galton-Watson trees, the constant speed model and the varying speed model. In both cases, the simple exclusion process starts from an equilibrium distribution with non-vanishing particle density. Moreover, we assume to have initially a particle in the root, the tagged particle. We show for both models that the tagged particle has a positive linear speed and we give explicit formulas for the speeds.
Studying the behavior of the tagged particle in an exclusion process is a classical problem @cite_2 . When the underlying graph is @math , many results were obtained. In the case of translation invariant transition probabilities, a law of large numbers for the position of the tagged particle is known in all regimes @cite_15 @cite_17 . For the @math -dimensional ladder graph, the speed of a tagged particle was studied by Zhang @cite_0 . For the exclusion process on regular trees, @ established a law of large numbers @cite_5 . We obtain their results as a special case. For random environments of the exclusion process, less results are known. In @cite_10 , Chayes and Liggett consider the case of the exclusion process in a one-dimensional i.i.d. random environment. For @math when the transition probabilities are symmetric and not concentrated on the nearest neighbors in the one-dimensional case, Kipnis and Varadhan established a central limit theorem for the tagged particle in their famous paper @cite_18 . Their result was the starting point for a sequence of papers showing central limit theorems for the position of the tagged particle, see @cite_7 @cite_16 for an overview.
@cite_18 @cite_7 @cite_0 @cite_2 @cite_5 @cite_15 @cite_16 @cite_10 @cite_17
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We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk. We investigate a tagged particle in the exclusion processes on 1, ..., N ×ℤ d , with different densities in different levels k ×ℤ d , ∀ k. Ignoring the level the tagged particle lying in, we only concern its position in ℤ d , denoted by X t . Note that the whole space is not homogeneous. We define the environment process viewed from the tagged particle, of which X t can be expressed as a functional. It is called the tagged particle process. We show the ergodicity of the tagged particle process, then prove the strong law of large numbers. Furthermore, we show the central limit theorem of X t provided the zero-mean condition. For a gentle approach to the problems connected with interacting Markov processes we review first what is known in the absence of interaction. For simplicity let S be a countable (or finite) set, and consider a countable (or finite if S is finite) collection of independent Markov processes on S,with common transition function P t (x,y), x, y ∈ S. It is natural to assume constant invariant measure, i.e., @math (0.1) We consider exclusion processes on a rooted @math -regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For @math , we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure. The processes described in the title always have reversible stationary distributions. In this paper, we give sufficient conditions for the existence of, and for the nonexistence of, nonreversible stationary distributions. In the case of an i.i.d. environment, these combine to give a necessary and sufficient condition for the existence of nonreversible stationary distributions. On montre l'extremalite d'une mesure invariante pour le processus «n» a partir d'une particule etiquetee et on utilise les proprietes d'ergodicite du processus initial pour montrer que la position suit une loi des grands nombres