On the Study of Transience and Recurrence of the Markov Chain Defined by Directed Weighted Circuits Associated with a Random Walk in Fixed Environment

By using the cycle representation theory of Markov processes, we investigate proper criterions regarding transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps in a fixed environment. 1. Introduction A systematic research has been developed (Kalpazidou [1], MacQueen [2], Minping and Min [3], Zemanian [4], and others) in order to investigate representations of the finite-dimensional distributions of Markov processes (with discrete or continuous parameter) having an invariant measure, as decompositions in terms of the cycle (or circuit) passage functions: for any directed sequence (or ) of states called a cycle (or a circuit), , of the corresponding Markov process. The representations are called cycle (or circuit) representations while the corresponding discrete parameter Markov processes generated by directed circuits , , are called circuit chains. Following the context of the theory of Markov processes’ cycle-circuit representation, the present work arises as an attempt to investigate proper criterions regarding the properties of transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps (having one elastic left barrier) in a fixed ergodic environment (Kalpazidou [1], Derriennic [5]). The paper is organized as follows. In Section 2, we give a brief account of certain concepts of cycle representation theory of Markov processes that we will need throughout the paper. In Section 3, we present some auxiliary results in order to make the presentation of the paper more comprehensible. In particular, in Section 3, a random walk with jumps (having one elastic left barrier) in a fixed ergodic environment is considered, and the unique representations by directed cycles (especially by directed circuits) and weights of the corresponding Markov chain are investigated. These representations will give us the possibility to study proper criterions regarding transience and recurrence of the abovementioned Markov chain, as it is described in Section 4. Throughout the paper, we will need the following notations: 2. Preliminaries Let us consider a denumerable set S. Then the directed sequence modulo the cyclic permutations, where , , completely defines a directed circuit in . The ordered sequence associated with the given directed is called a directed cycle in . A directed circuit may be considered as , if there exists an , such that , , ,