On the survival probability of a random walk in random environment with killing

TL;DR

This paper analyzes the decay rate of survival probability for one-dimensional random walks in random environments with killing, revealing three regimes based on environmental law, and providing explicit formulas for decay rates.

Contribution

It derives explicit formulas for the decay rate of survival probability in random walks with killing, identifying three distinct regimes depending on the environment law.

Findings

Three regimes of decay depending on environment law

Explicit formulas for survival probability decay rates

Survival probability tends to zero as time increases

Abstract

We consider one dimensional random walks in random environment where every time the process stays at a location, it dies with a fixed probability. Under some mild assumptions it is easy to show that the survival probability goes to zero as time tends to infinity. In this paper we derive formulas for the rate with which this probability decays. It turns out that there are three distinct regimes, depending on the law of the environment.