## Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions.(English)Zbl 0854.60087

Summary: 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 $L = {1 \over 2} \sum^d_{i,j = 1} a_{ij} \bigl( \partial^2 / (\partial x_i \partial x_j) \bigr) + \sum^d_{i = 1} b_i (\partial/ \partial x_i).$ A path $$X (\cdot)$$ is said to survive if $$X(t) \not \equiv 0$$, for all $$t \geq 0$$. Since $$\beta > 0$$, $$P_\mu (X (\cdot)$$ survives) $$> 0$$, for all $$0 \not \equiv \mu \in {\mathcal M} (R^d)$$, where $${\mathcal M} (R^d)$$ 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 measure as follows. The support is recurrent if $P_\mu \bigl( X(t,B) > 0, \text{ for some } t \geq 0 \mid X (\cdot) \text{ survives} \bigr) = 1,$ for every $$0 \not \equiv \mu \in {\mathcal M} (R^d)$$ and every open set $$B \subset R^d$$. For $$d > 2$$, the support is transient if $P_\mu \bigl( X(t,B) > 0, \text{ for some } t \geq 0 \mid X (\cdot) \text{ survives} \bigr) < 1,$ for every $$\mu \in {\mathcal M} (R^d)$$ and bounded $$B \subset R^d$$ which satisfy $$\text{supp} (\mu) \cap \overline B = \emptyset$$. A similar definition taking into account the topology of $$R^1$$ is given for $$d = 1$$. The support exhibits local extinction if for each $$\mu \in {\mathcal M} (R^d)$$ 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. Also studied is the asymptotic behavior as $$t \to \infty$$ of $$E_\mu \int^t_0 \langle \psi, X(s) \rangle ds$$, and of $$E_\mu \langle g, X(t) \rangle$$, for $$0 \leq g$$, $$\psi \in C_c (R^d)$$, where $$\langle f, X(t) \rangle \equiv \int_{R^d} f(x) X(t, dx)$$. A number of examples are given to illustrate the general theory.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J60 Diffusion processes
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### References:

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