The speed of a branching system of random walks in random environment

Abstract

We consider a branching system of random walks in random environment in Z, for which extinction is possible. We study the speed of the rightmost particle, conditionally on the survival of the branching process.

A random walk with a branching system in random environments

“…This result was shown by Devulder [7] in the case where the time environment ξ is deterministic under a second moment condition.…”

“…As remarked by Devulder [7] in the special case where the time environment ξ is constant, in our model there is a competition between the walking mechanism and the branching system. When log ρ 0 (ζ)η(dζ) > 0, the walking mechanism pushes the particles to go to −∞ (the usual random walk in the random environment goes to ∞); when the branching process {Z n : n 0} is supercritical, the existence of many particles makes it possible that some particles may go to +∞.…”

“…Different models can be defined according to different dependences of P and P on the environment. In [1][2][3][4][5][6], the environment affects only P ; in [7], the environment affects only P ; in [8,9], the environment affects both P and P . In all the above-mentioned papers, the only environment is indexed by locations.…”

On graded irreducible representations of Leavitt path algebras

“…As mentioned in [16], branching systems arise in the study of various disciplines such as random walk, symbolic dynamics and scientific computing (see e.g. [15], [22], [28]). Given an arbitrary directed graph E, Gonçalves and Royer defined in [16] and [17] a branching system using a measure space (X, µ) and indicated a method of constructing a large number of representations of the graph C *algebra C * (E) in the space of bounded linear operators on L 2 (X, µ).…”

Unitary equivalence of representations of graph algebras and branching systems

“…Similarly, branching systems arise in neighboring disciplines, such as random walks, symbolic dynamics, and scientific computing (see, e.g., [10], [15], [5], [3], and [18]). …”