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4-dimensional Brownian motion is recurrent with positive capacity - MaRDI portal

4-dimensional Brownian motion is recurrent with positive capacity (Q2266296)

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4-dimensional Brownian motion is recurrent with positive capacity
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    4-dimensional Brownian motion is recurrent with positive capacity (English)
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    1984
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    \textit{M. Fukushima} [J. Math. Soc. Japan 36, 161-176 (1984; Zbl 0535.60068)] proved that for d-dimensional Brownian motion if \(d\geq 5\), then the sample paths are transient quasi everywhere, that is, except on a set of capacity zero. The author has proved as a special case that if \(d\leq 3\), the sample paths are recurrent with positive capacity or equivalently the Ornstein-Uhlenbeck process on the Wiener space hits the set of recurrent Brownian paths with positive probability, actually probability 1 [Propriétés quasi-partout de fonctions aléatoires gaussiennes. Lect. Notes Univ. Strasbourg (1983)]. In this paper the author proves that 4-dimensional Brownian paths are also recurrent with positive capacity.
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    recurrent
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    positive capacity
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    Ornstein-Uhlenbeck process
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    recurrent Brownian paths
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