English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 $U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its densityis obtained when either the hitting time or the starting site $B_0$ becomeslarge. Our results entail that if Brownian motion is started at ${\bf x}$ andconditioned to hit $U(a)$ at time $t$ for the first time, the distribution ofthe hitting site approaches the uniform distribution or the point mass at$a{\bf x}/|{\bf x}|$ according as $|{\bf x}|/t$ tends to zero or infinity; ineach case we provide a precise asymptotic estimate of the density. In the casewhen $|{\bf x}|/t$ tends to a positive constant we show the convergence of thedensity and derive an analytic expression of the limit density.Comment: 48 pages, 2 figure

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