Single Birth Chain In Random Environment And The Generalized Random Swimming

The birth and death chain in a random environment and the single birth chain in a random environment are two important models of Markov chain in a random environment. Torrez(1979) calculated the extinction probabilities for the birth and death chain in a random environment with its both boundary points as absoption barriers. In Chapter One of this thesis, using the method appearing in ref.[7], author calculates the extinction probabilities for the single birth chain in a random environment with a reflection barrier, and obtains the vector difference equations about them. Extinction probabilities for the birth and death chain in a random environment - a special case of the single birth chain in a random environment are also obtained and the results are applied into examples. Solomon(1975) introduced the concept of random walk in a random environment, proved the existence of the model on the set of integers, and obtained its limit theorems. Wang Rongming(1993) calculated the extinction probabilities for a class of random walk in a random environment with the set of non-negative integers as state space. Generalied random walk in random environments is the natural generalization of random walk in random environments. In Chapter Two, author constructs the model of generalized random walk in random environments on the set of non-negative integers, proves its existence, and obtains explicit formula of its extinction probabilities. The main results are calculated as the following:1. Extinction probabilities for the single birth chain in Marko-vian environment with an absorption barrier 0.2. Extinction probabilities for the single birth chain in Marko-vian environment with an absorption barrier 0 and a reflection barrier M.3. Extinction probabilities for the birth and death chain in Markovian environment with an absorption barrier 0 and a reflection barrier M.4. Extinction probabilities for the homogeneous birth and death chain in Markovian environment with an absorption barrier 0 and a reflection barrier M.5. Extinction probabilities for the generalized random walk in random environments with the set of non-negative integers as state space.