Continuous-time Branching Random Walk In A Catalytic Medium

In this paper, we mainly discuss some problems of continuous-time branching random walk in a catalytic medium, including(1) the limiting extinction probability of the continuous-time branching random walk on the integer lattice Z~d(d≥1) with two catalysts.(2) one-dimensional continuous-time branching random walk with a catalyst and an absorbing state on the non-negative integer lattice.We divide the thesis into three chapters. The first Chapter is the introduction of this thesis. We introduce the branching random walk on the integer lattice Z~d(d≥1) with a single source , and the latest development, the problems discussed in this thesis and main results.In Chapter 2, we mainly give a complete proof for an important theory derived by Albeverio, Bogachev and Yarovaya, and further study the limiting extinction probabilityof the continuous-time branching random walk on the integer lattice Z~d(d≥1) with two catalysts. when d=1, 2, we discuss the extinction probability to the case that f_i(μ),i=1, 2 is special; when d≥3, we first explain the existence and uniqueness of q(x) for special f_i(μ), i=1,2, then for general f_i(μ),i=1,2.In Chapter 3, we study the extinction probability about one-dimensional continuous-time branching random walk with a catalyst and an absorbing state on the non-negative integer lattice.