Wendelin Werner

On the occupation times of cones by Brownian motion

SummaryWe study some features concerning the occupation timeAt of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A1<u) asu→0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inft→∞At/(tf(t))=0 or ∞ for a decreasing functionf.

Asymptotics for occupation times of half-lines by stable processes and perturbed reflecting brownian motion

We derive the integral tests for the sample path behaviour of the occupation times of half-lines by reflecting Brownian motion perturbed by its local time at zero (i.e. , where B; is a linear Brownian motion, l its local time at 0 and μ positive constant) and stable processes, which correspond to the generalized Arcsine laws which have been derived for such processes

On windings of multidimensional reflected brownian motion

We investigate the asymptotic behaviour of winding numbers of two-dimensional reflected Brownian motion and windings around lines of three-dimensional transient reflected Brownian motion in a cone, in the case where the reflection vector field is radially homogeneous. We also derive some other properties of the latter process