### A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We prove that a planar random walk with bounded increments and mean zero which is conditioned to stay in a cone converges weakly to the corresponding Brownian meander if and only if the tail distribution of the exit time from the cone is regularly varying. This condition is satisfied in many natural examples.

The exit distribution for open sets of a path-continuous, strong Markov process in ${\mathbf{R}}^{n}$ is characterized as a weak star limit of successive spherical sweepings of measures, starting with the unit point mass. Then this is used to prove that two path-continuous strong Markov processes with identical exit distributions from balls when starting form the center, have identical exit distributions from all opens sets, provided they both exit a.s. from bounded sets. This implies that the only path-continuous,...

$\mathcal{L}$ denotes a (compact, nonsingular) lamination by hyperbolic Riemann surfaces. We prove that a probability measure on $\mathcal{L}$ is harmonic if and only if it is the projection of a measure on the unit tangent bundle ${T}^{1}\mathcal{L}$ of $\mathcal{L}$ which is invariant under both the geodesic and the horocycle flows.