One-dimensional finite range random walk in random medium and invariant measure equation

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables f! x g (taking values in the unit interval), which serve as an environment. This environment deenes a random walk fX k g (called a RWRE) which, when at x, moves one step to the right with probability ! x , and one step to the left with probability 1 ? ! x. Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) ...

Erosion by a one-dimensional random walk.

We consider a model introduced by Baker et al. [Phys. Rev. E 88, 042113 (2013)] of a single lattice random walker moving on a domain of allowed sites, surrounded by blocked sites. The walker enlarges the allowed domain by eroding the boundary at its random encounters with blocked boundary sites: attempts to step onto blocked sites succeed with a given probability and convert these sites to allo...

O < Random Walk , Scatterino and Invariant Imbeddino — I : One - Dimensional Discrete Case

Random Walk, Scattering, and Invariant Imbedding I. One-dimensional Discrete Case.

Vertex-reinforced random walk on Z has finite range

A stochastic process called Vertex-Reinforced Random Walk (VRRW) is defined in Pemantle (1988a). We consider this process in the case where the underlying graph is an infinite chain (i.e., the one-dimensional integer lattice). We show that the range is almost surely finite, that at least 5 points are visited infinitely often almost surely, and that with positive probability the range contains e...