The Annals of Probability

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

Ross G. Pinsky

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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.

Article information

Ann. Probab., Volume 24, Number 1 (1996), 237-267.

First available in Project Euclid: 15 January 2003

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60: Diffusion processes [See also 58J65]

Measure-valued diffusion supercritical local extinction transience recurrence


Pinsky, Ross G. Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24 (1996), no. 1, 237--267. doi:10.1214/aop/1042644715.

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  • 1 DAWSON, D. A. 1993. Measure-valued Markov processes. Ecole d'Ete de Probabilites de ´ ´ Saint Flour XXI. Lecture Notes in Math. 1541 1 260. Springer, Berlin.
  • 2 FREIDLIN, M. 1985. Functional Integration and Partial Differential Equations. Princeton Univ. Press.
  • 3 GILBARG, D. and TRUDINGER, N. 1983. Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin.
  • 4 IKEDA, N. and WATANABE, S. 1980. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
  • 5 ISCOE, I. 1986. A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85 116.
  • 6 ISCOE, I. 1988. On the supports of measure-valued critical branching Brownian motion. Ann. Probab. 16 200 221.
  • 7 PAZY, A. 1983. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin.
  • 8 PERSSON, A. 1960. Bounds for the discrete part of the spectrum of a semi-bounded Schrodinger operator. Math. Scand. 8 143 153. ¨
  • 9 PINSKY, R. 1995. Positive Harmonic Functions and Diffusion. Cambridge Univ. Press.
  • 11 PROTTER, M. and WEINBERGER, H. 1984. Maximum Principles in Differential Equations. Springer, Berlin.
  • 13 SATTINGER, D. H. 1973. Topics in Stability and Bifurcation Theory. Lecture Notes in Math. 309. Springer, Berlin.