We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set A. We derive the asymptotic form of the density, say p A t (x, y), for large times t and for x and y in the exterior of A valid uniformly under the constraint |x|∨|y|is the transition kernel of the Brownian motion (without killing) and e A is the Green function for the 'exterior of A' with a pole at infinity normalized so that e A (x) ∼ lg |x|. We also provide fairly accurate upper and lower bounds of p A t (x, y) for the case |x| ∨ |y| > t as well as corresponding results for the higher dimensions.