Diffusion-limited aggregation on the hyperbolic plane

We consider an analogous version of the diffusion-limited aggregation model defined on the hyperbolic plane. We prove that almost surely the aggregate viewed at time infinity will have a positive density.Comment: Published at http://dx.doi.org/10.1214/14-AOP928 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Extremal points of high dimensional random walks and mixing times of a Brownian motion on the sphere

We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.Comment: 22 Page

The Jacobian Conjecture for the space of all the inner functions

We prove the Jacobian Conjecture for the space of all the inner functions in the unit disc