Hitting distribution of a correlated planar Brownian motion in a disk

TL;DR

This paper analyzes the probability distribution of a correlated planar Brownian motion hitting a circle by transforming it into an elliptical problem and deriving the distribution using elliptic coordinates and Poisson kernels.

Contribution

It introduces a novel approach to compute hitting distributions for correlated Brownian motions via geometric transformations and elliptic coordinate methods.

Findings

Derived the hitting distribution as a series of Poisson kernels.

Mapped the circle to an ellipse to simplify the differential operator.

Provided explicit formulas for the distribution in the correlated case.

Abstract

In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $\rho$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $\rho$ 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.