TRANSIENCE, RECURRENCE AND LOCAL EXTINCTION PROPERTIES OF THE SUPPORTFOR SUPERCRITICAL FINITE MEASURE-VALUED DIFFUSIONS

We consider the supercritical finite measure-valued diffusion, X(t), w hose 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 Sig ma(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/pa rtial derivative x(i)). A path X(.) is said to survive if X(t) not equ al 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 transi ence, recur rence and local extinction for the support of the supercri tical measure-valued diffusion starting from a finite measure as follo ws. The support is recurrent if P-mu(X(t, B) > 0, for some t greater t han 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 i s an element of M(R(d)) and bounded B subset of R(d) which satisfy sup p(mu) boolean AND (B) over bar = <empty set>. A similar definition tak ing 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 finit e 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 extin ction are developed in this paper. Also studied is the asymptotic beha vior 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 exam ples are given to illustrate the general theory.