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Supremum distribution of Bessel process of drifting Brownian motion

Pyć A., Żak T., Serafin G.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 2

201--222

Let us assume that (B(1)t, B(2)t, B(3)t + μt) is a three-dimensional Brownian motion with drift μ, starting at the origin. Then Xt = ∥(B(1)t, B(2)t, B(3)t + μt)∥, its distance from the starting point, is a diffusion with many applications. We investigate the supremum of (Xt), give an infinite-series formula for its distribution function and an exact estimate of the density of this distribution in terms of elementary functions.

Hitting hyperbolic half-space

Małecki J., Serafin G.

Demonstratio Mathematica

2012

Vol. 45, nr 2

337-360

Let (…) be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space (…) having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for (…) which enables us to study the process (…) killed when exiting the hyperbolic half space, that is the set (…). We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process (…). Finally, we derive formula for the (…) Poisson kernel of the set D.