Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions

Abstract

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 $L = 1/2 \Sum _{i,j=1}^d a_{i,j (\partial x_i \partial x_j)) = \Sum_ {i=1} ^d b_i (\partial / \partial x_i)$. A path $X(\dot)$ is said to survive if $X(t) \not\equiv 0$, for all $t\geq 0$. Since $\beta> 0, P_\mu (X(\dot)$ survives) $>0$, for all $0\not\equiv \mu \in M(R^d)$, where $M(R^d)$ denotes the space of finite measures on $R^d$. We define transience, recurrence and local extinction for the support of the supercritical measure-valued diffusion starting from a finite meausre as follows. The support is recurrent if $P _ \mu (X(t,B)>0$, for some $t \geq 0 | X(\dot)$ survives) =1, for every $0 \not\equiv \mu \in M(R^d)$ and every open set $B \subset R^d$. For $d\geq 2$, the support is transient if $P_\mu(X(t,B)>0$, for some $t \geq 0 |X (\dot)$ survives) $<1$, for every $\mu \in M(R^d)$ and bounded $B\subset R^d$ which satisfy $\supp(\mu)\bigcap \bar{B} = \emptyset$. A similar definition taking into account the topology of $R^1$ is given for $d=1$. The support exhibits local extinction if for each $\mu \in M(R^d)$ and each bounded $B\subset R^d$, there exists a $P_\mu$-almost surely finite random time $\zeta_B$ such that $X(t,B) = 0$, for all $t\geq \zeta_B$. Criteria for transience, recurrence and local extinction are developed in this paper. Also studied is the asymptotic behavior as $t \to \infty$ of $E_\mu \int_0^t \langle \psi, X(s) \rangle ds$, and of $E_\mu \langle g,X(t) \rangle$, for $0\leq g, \psi \in C_c(R^d), where $\langle f, X(t) \rangle \not\equiv \int_{R^d} f(x) X(t,dx). A number of examples are given to illustrate the general theory.