Asymptotic behaviour of a Brownian motion on exterior domains
Let \(p_t(x,y)\) denote the transition density of the Brownian motion of \(\mathbb{R}^d\) killed at the boundary \(\partial D\) of a domain \(D\) having non-polar compact complement. It is shown that, as \(t\to+\infty\), \(p_t(x,y)\) behaves as \({2\over\pi} u(x)u(y) t^{-1}(\log t)^{-2}\) for \(d=2\) and as \((2\pi t)^{-d/2}u(x)u(y)\) for \(d\geq 3\), where \(u(x)\) is the unique positive harmonic function vanishing on \(\partial D\) which is at infinity equivalent to \(\log|x|\) for \(d=2\) and to 1 for \(d\geq 3\). The proof is based on Harnack's inequality and a comparison with the particular case of \(D\) being the complement of the unit ball. \(u(x)\) takes place naturally, since it equals the limit as \(t\to\infty\) of \(\mathbb{P}_x\)(lifetime \(>t\)) for \(d\geq 3\) and of \({1\over 2}\log t \mathbb{P}_x\)(lifetime \(>t\)) for \(d=2\).
scientific article; zbMATH DE number 1454832