Warren, Jonathan 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) Cited in 1 ReviewCited in 24 Documents MSC: 60J65 Brownian motion 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) Keywords:Brownian motion; Bessel process; continuous-state branching process; stochastic differential equation PDFBibTeX XMLCite \textit{J. Warren}, Lect. Notes Math. 1655, 1--15 (1997; Zbl 0884.60081) Full Text: Numdam EuDML