Hitting Distribution of a Correlated Planar Brownian Motion in a Disk

Abstract

In this article, we study the hitting probability of a circumference CR for a correlated Brownian motion B̲(t)=B1(t),B2(t), ρ being the correlation coefficient. The analysis starts by first mapping the circle CR into an ellipse E with semiaxes depending on ρ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.