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Potential Analysis

, Volume 44, Issue 3, pp 497–541 | Cite as

Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

  • Kôhei Uchiyama
Article

Abstract

We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.

Keywords

Harmonic measure for heat operator Brownian hitting time Caloric measure Space-time distribution 

Mathematics Subject Classification (2010)

Primary 60J65 Secondary 60J45 60J60 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyMeguroJapan

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