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...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.展开更多
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...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.展开更多
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 rec...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.展开更多
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(...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)).展开更多
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...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.展开更多
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 pr...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.展开更多
基金the Natural Science Foundation of Hunan Province (Grant No. 99JJY2001) Hunan Provincial Foundation for Young and Middleaged People (Grant No. 00JJEY2141) .
文摘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.
基金Supported in part by 985 Project,973 Project(Grant No.2011CB808000)National Natural Science Foundation of China(Grant No.11131003)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20100003110005)the Fundamental Research Funds for the Central Universities
文摘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.
基金Project supported by the National Natural Science Foundation of China (No.10271109).
文摘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.
基金supported by National Natural Science Foundation of China(Grant Nos.11531001 and 11626245)
文摘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)).
文摘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.
基金Research supported in part by Tianyuan Fund ofr Mathematics of NSFC (10526021)A Grant from Ministry of Education
文摘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.
