 |
|
|
| | | | | |
|
|
|
|
|
Almost sure finiteness for the total occupation time of an (d,α,β)-superprocess
|
Xiaowen Zhou, Concordia University
|
Abstract
For $0<α≤ 2$ and $0<β≤ 1$ let $X$ be the $(d,α,β)$-superprocess,
i.e. the superprocess with $α$-stable spatial movement in $bR^d$ and
$(1+β)$-stable branching. Given that the initial measure $X_0$ is Lebesgue
on $bR^d$, Iscoe conjectured in [7] that the total occupational time $int_0^infty
X_t(B)dt$ is a.s. finite if and only if $dβ<α$, where $B$ denotes any
bounded Borel set in $reels^d$ with non-empty interior.
In this note we give a partial answer to Iscoe's conjecture by showing that
$int_0^infty X_t(B)dt<infty$ a.s. if $2dβ<α$ and, on the other hand,
$int_0^infty X_t(B)dt=infty$ a.s. if $dβ> α$.
For $2dβ<α$, our result can also imply the a.s. finiteness of the total
occupation time (over any bounded Borel set) and the a.s. local extinction for
the empirical measure process of the $(d,al,β)$-branching particle system
with Lebesgue initial intensity measure.
|
Full text: PDF
Pages: 22-31
Published on: February 10, 2010
|
Bibliography
- T. Bojdecki, L.G. Gorostiza, and A. Talarczyk. Occupation
times of branching systems with initial inhomogeneous Poisson states and related superprocesses. Elctron. J. Probab. 14
(2009), paper no. 46, 1328-1371.
MR2511286
- D.A. Dawson. Measure-valued Markov processes.
École d'Été de Probabilitiés de Saint Flour 1991, Lect. Notes in Math.1541 (1993), Springer, Berlin.
MR1242575
- D.A. Dawson and K. Fleischmann. Strong clumpimg of critical
space-time branching models in subcritical dimensions. Stochastic Process. Appl.
30 (1988), 193-208.
MR0978354
- D.A. Dawson and E.A. Perkins. Historical processes. Mem. Amer. Math. Soc.
454 (1991).
MR1079034
- S.N. Evans and E.A. Perkins. Absolute continuity results for
superprocesses. Trans. Amer. Math. Soc. 325 (1991),
661-681. MR1012522
- L.G. Gorostiza and A. Wakolbinger. Persistence criteria for a
class of critical branching particle systems in continuous time. Ann. Probab.
19 (1991), 266-288.
MR1085336
- I. Iscoe. A weighted occupation time for a class of measure-valued branching
processes. Probab. Th. Rel. Fields 71
(1986), 85-116.
MR0814663
- Z. Li and X. Zhou. Distribution and propagation properties of
superprocesses with general branching mechanisms. Commun.
Stoch. Anal. 2 (2008), 469-477.
MR2485004
- I. Karatzas and S.E. Shreve. Brownain Motion and
Stochastic Calculus, 2nd edn. Springer (1996).
MR1121940
- L.C.G. Rogers and D. Williams. Diffusions, Markov Processes
,and Martingales. Vol. 1: Foundations. Wiley, New York (1987).
MR1331599
- X. Zhou. A zero-one law of almost sure local extinction for
$(1+beta)$-super-Brownian motion. Stochastic Process. Appl. 118 (2008), 1982-1996.
MR2462283
|
|
|
|
|
|
|
|
| | | | |
Electronic Communications in Probability. ISSN: 1083-589X |
|
Context