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Asymptotic extinction probability of an absorbing random walk in random environment

A Random Walk in Random Environment (RWRE) is a process in which the transition probabilities of a markov chain on Z are chosen randomly and then the joint probability measure of environment and random walk is considered. Motivated by a model from statistical mechanics we consider a version of RWRE where the process can also die out. In a first step towards describing the distribution of the RWRE conditioned on survival the asymptotical rate with which the probability for survival goes to zero is determined. A number of cases is given in which this can be estimated and it is shown that there are many different possible rates. For this a potential function is introduced on fixed environments and we consider valleys in which the process will spend a long time. Random Walks in Random Environments Let Ω be the set of all transition probabilities on Z: Ω := { ω = (ω− x , ω o x, ω + x )x∈Z ∣∣∀x ∈ Z : ω x , ω x, ω− x ≥ 0, ω x + ω x + ω− x = 1} One can interpret ω ∈ Ω as the transition probabilities of a markov chain (Xn)n∈N on Z started in zero. This is refered to as an environment. When the process (Xn)n∈N is at position x ∈ Z at some point in time, in the next step it moves to the right with probability ω x , to the left with probability ω − x and stays with probability ω x = 1− ω x − ω− x . This probability measure is denoted Pω. The space Ω is then given a suitable σ-Algebra F such that the mapping