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The behavior of superprocesses near extinction. (English) Zbl 0749.60046

Let \(\{X_ t;\;t\geq 0\}\) be a continuous superprocess in a locally compact separable metric space \(E\) starting with a finite measure. The underlying motion process is assumed to be a conservative Feller process. A martingale problem characterization is used to study the behavior of \(X\) near its extinction time. Roughly speaking, \(X_ t\) normalized to be a probability measure will extinct at a single point distributed according to the motion law. In some cases, there are times near extinction at which the closed support of the superprocess is concentrated near the extinction point (although the closed support of \(X_ t\) at fixed times \(t\) is typically spread throughout space). In the case of a super-\(\alpha\)-stable motion \(X\) in \(E=\mathbb{R}\), \(0<\alpha\leq 2\), a Tanaka-like formula for \((X_ t,f)\) is obtained and discussed for a class of functions \(f\) including the indicator functions \(\mathbf{1}\{x:\;x\leq a\}\), \(a\in\mathbb{R}\), of half axes. For instance, \(\{X_ t(x\leq a);\;t\geq 0\}\) fails to be a semimartingale if \(1<\alpha\leq 2\).

MSC:

60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G44 Martingales with continuous parameter
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