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On the distribution of functionals of Brownian motion stopped at the moment inverse the local time. (Russian) Zbl 0742.60073

Let \(W_ x(t)\) be the Brownian motion, starting from the point \(x\), \(\ell(s,y)\) the Brownian local time, \[ {\mathfrak e}=\min\{s:\;\ell(s,u)=v\hbox{ or }\ell(s,z)=v\}. \] The author considers the integral functionals of Brownian motion and Brownian local time and the functionals \(\sup_{y\in R}\ell({\mathfrak e},y)\), \(\inf W_ x(t)\) and \(\sup W_ x(t)\) over the interval \([0,{\mathfrak e}]\). The Laplace transformations for joint and marginal distribution of such functionals are obtained. For some special cases the explicit formulas for the distribution are given, too.

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals
60J25 Continuous-time Markov processes on general state spaces
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