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Branching processes, the Ray-Knight theorem, and sticky Brownian motion. (English) Zbl 0884.60081

Azéma, J. (ed.) et al., Séminaire de probabilités XXXI. Berlin: Springer. Lect. Notes Math. 1655, 1-15 (1997).
Let \((\Omega,({\mathcal F}_t)_{t \geq 0}, P)\) be a filtered probability space and \((X_t;t\geq 0)\) a sticky Brownian motion, i. e. a continuous, adapted process taking values in \([0,\infty)\) which satisfies the stochastic differential equation \[ X_t=x+\theta \int_0^t I_{\{X_s=0\}}ds+\int_0^t I_{\{X_s>0\}}dW_s, \] where \((W_t;t\geq 0)\) is a real valued \(({\mathcal F}_t)\)-Brownian motion with parameter \(\theta\), started from \(x\). By Chitashvili [Technical report BS–R8901, Center for Mathematics and Computer Science (Amsterdam, 1989)] the joint law of \(X\) and \(W\) is unique, but \(X\) is not measurable with respect to \(W\), what is interpreted as ‘extra randomness’. This extra randomness is identified in terms of killing in a branching process. The main result is Theorem 1: Let \(X\) and \(W\) start from zero and let \(L_t=\sup_{s\leq t}(-W_s)\). Then \(P(X_t\leq x\mid \sigma(W))=\exp (-2\theta (W_t+L_t-x))\) a.s. for \(x\in [0,W_t+L_t]\).
For the entire collection see [Zbl 0864.00069].
Reviewer: V.Topchij (Omsk)

MSC:

60J65 Brownian motion
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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