Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions
成果类型：
Article
署名作者：
Pinsky, RG
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
发表日期：
1996
页码：
237-267
关键词：
摘要：
We consider the supercritical finite measure-valued diffusion, X(t), whose log-laplace equation is associated with the semilinear equation u(t) = Lu + beta u - alpha u(2), where alpha, beta > 0, and L = 1/2 Sigma(i,j=1)(d) alpha(ij) (partial derivative(2)/(partial derivative x(i) partial derivative x(j))) + Sigma(i=1)(d) b(i) (partial derivative/partial derivative x(i)). A path X(.) is said to survive if X(t) not equal 0, for all t greater than or equal to 0. Since beta > 0, P-mu(X(.) survives) > 0, for all 0 not equal mu is an element of M(R(d)), where M(R(d)) denotes the space of finite measures on R(d). We define transience, recur rence and local extinction for the support of the supercritical measure-valued diffusion starting from a finite measure as follows. The support is recurrent if P-mu(X(t, B) > 0, for some t greater than or equal to 0 IX(.) survives) = 1, for every 0 not equal mu is an element of M(R(d)) and every open set B subset of R(d). For d greater than or equal to 2, the support is transient if P-mu(X(t, B) > 0, for some t greater than or equal to 0 I X(.) survives) < 1, for every mu is an element of M(R(d)) and bounded B subset of R(d) which satisfy supp(mu) boolean AND (B) over bar = . A similar definition taking into account the topology of R(1) is given for d = 1. The support exhibits local extinction if for each mu is an element of M(R(d)) and each bounded B subset of R(d), there exists a P-mu-almost surely finite random time zeta(B) such that X(t, B) = 0, for all t greater than or equal to zeta(B). Criteria for transience, recurrence and local extinction are developed in this paper. Also studied is the asymptotic behavior as t --> infinity of E(mu) integral(0)(t) [psi, X(s)] ds, and of E(mu)[g, X(t)], for 0 less than or equal to g, psi is an element of C-c(R(d)), where [f, X(t)] = integral(Rd) f(x)X(t, dx). A number of examples are given to illustrate the general theory.