CRITERIA OF STRONG TRANSIENCE FOR OPERATOR-SELF-SIMILAR MARKOV PROCESSES

Yamamuro in [1] defines strong and weak transience of Markov processes; gives a criterion for strong transience of Feller processes; and further, discusses strong and weak transience of Ornstein-Uhlenbeck type processes. In this article, the authors weaken the Feller property of the result in [1] to weak Feller property and discuss the strong transience of operator-self-similar Markov processes.

On the transience and recurrence of Lamperti’s random walk on Galton-Watson trees

In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ < m or λ > m, respectively;(2) when λ = m, 0 < α < 1,(Xn)n≥ 0 is transient/recurrent according to whether A < 0 or A > 0, respectively.In particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).

AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS OF THE LAST EXIT TIME OF AN ELLIPTIC DIFFUSION PROCESS

Let Ls be the last exit time from a compact set B of an elliptic diffusion process X. A moderate estimate for the distribution of Ls is obtained, and the sufficient and necessary condition for E^x（L^κB） 〈∞ is proved.

Some Limit Results for Transient Subsequence of Brownian Motion on Line

Let {W(t), 0≤t<∞} be a standard, one dimensional Brownian motion, and {t n, n≥1} be a sequence of positive constans with t n+1 ≥C 2t n (C>1). We obtain that liminf n→∞ inf k≥n|W(t k)|t n 1logn =1e a.s.and the set of the limit points of inf k≥n|W(t k)|t n 1logn is 1e, 1 almost surely.

Recurrence and invariant measure of Markov chains in double-infinite random environments

The concepts of π-irreduciblity, recurrence and transience are introduced into the research field of Markov chains in random environments.That a π-irreducible chain must be either recurrent or transient is proved, a criterion is shown for recurrent Markov chains in double-infinite random environments, the existence of invariant measure of π-irreducible chains in double-infinite environments is discussed,and then Orey's open-questions are partially answered.

Spectral Gap and Convergence Rate for Discrete-time Markov Chains

Abstract Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L2（π）-space, denoted by gap（P）, is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap（P） and the convergence rate of P^n. When P is transient, the convergence rate of pn is equal to 1 - gap（P）. When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of pn in terms of gap（P）. These results are extended to L^∞ （π）-space.

POLYNOMIAL RECURRENCE FOR LEVY PROCESSES

In this paper, the authors study the ω-transience and ω-recurrence for Levy processes with any weight function ω, give a relation between ω-recurrence and the last exit times. As a special case, the polynomial recurrence and polynomial transience are also studied.

A REMARK ON EXTINCTION OF A CLASS OF SUPERPROCESSES

The extinction of a class of superprocesses associated with general branching characterstics and underlying Markov processes is investigated. The extinction is closely associated with the branching characteristics and the recurrence and transience of underlying processes.