The Brownian Excursion multi - dimensional local time density ∗
نویسندگان
چکیده
Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for Galton-Watson trees.
منابع مشابه
The Sde Solved by Local times of a Brownian Excursion or Bridge Derived from the Height Proole of a Random Tree or Forest
Let B be a standard one-dimensional Brownian motion started at 0. Let Lt;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (Lt;v(jBj);v 0) conditionally given Bt = 0 and Lt;0(jBj) = ` is shown to be that of the unique strong solution X of the It^ o SDE
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Let B be a standard one-dimensional Brownian motion started at 0. Let Lt;v(jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (Lt;v(jBj); v 0) conditionally given Bt = 0 and Lt;0(jBj) = ` is shown to be that of the unique strong solution X of the Itô SDE dXv = n 4 X2 v t R v 0 Xudu 1 o dv + 2 p XvdBv on the interval [0; Vt(X)), where Vt...
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The purpose of this note is to present an example of a family of “exceptional points” on Brownian paths which cannot be constructed using an entrance law. Watanabe (1984, 1987) proved that various families of exceptional points on Brownian paths may be constructed using entrance laws. Special cases include excursions of onedimensional Brownian motion within square root boundaries (see Watanabe ...
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