Probability of Brownian motion hitting an obstacle

The probability p(x) that Brownian motion with drift, starting at x, hits an obstacle is analyzed. The obstacle Omega is a compact subset of R-n. It is shown that p(x) is expressible in terms of the field U(x) scattered by Omega when it is hit by a plane wave. Therefore results for U(x), and methods for finding U(x), can be used to determine p(x). We illustrate this by obtaining exact and asymptotic results for p(x) when Omega is a slit in R-2, and asymptotic results when Omega is a disc in R-3.