Some Topics On Two-parameter Random Walk

In this thesis the probability problems about two-parameter random walks in different situations are explored mainly. The escape probability problems about two-parameter random walks in horizontal direction, in diagonal direction and within the rectangle time domain are discussed.In Chapter 1, some background knowledge needed in this thesis is provided. The source and development evolution of two-parameter random walk theory are presented. Moreover, the necessity of this thesis and the main conclusions of our work are stated.In Chapter 2, the definition of stochastic processes including one-parameter random walks,some preliminaries and theorems needed in the following are pre-sented.In Chapter 3,two-parameter random walks are introduced firstly, then the probabilities of the event that two-parameter random walk which does not come back at the origin move over the integer lattices in the horizontal direction, are considered.In Chapter 4, following the results in [20] given by Khoshnevisan.D and Pal Revesz, the " escape probability" different from Chapter 3 is considered. We study on the probability that two-parameter random walk,which docs not back at the origin along the diagonal line in multi-dimensions.In Chapter5, based on the asympototic results about Rm,n =#{Si,j,1≤i≤m,1<≤j≤n} the number of the points visited by a two-parameter random walk in [9] [10], a result about the " escape probability " of the two-parameter random walks within time " rectangle domain " is obtained, which generalizes the " escape probability " results of one-parameter random walks to that of two-parameter cases.