Narrow Escape, Part III: Riemann surfaces and non-smooth domains

We consider Brownian motion in a bounded domain $\Omega$ on a two-dimensional Riemannian manifold $(\Sigma,g)$. We assume that the boundary $\p\Omega$ is smooth and reflects the trajectories, except for a small absorbing arc $\p\Omega_a\subset\p\Omega$. As $\p\Omega_a$ is shrunk to zero the expected time to absorption in $\p\Omega_a$ becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted $E\tau$, as $\epsilon=|\partial \Omega_a|_g/|\partial \Omega|_g\to0$. We derive a leading order asymptotic approximation $E\tau = \ds{\frac{|\Omega|_g}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)]$. The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is small. If the absorbing window is located at a corner of angle $\alpha$, then $E\tau = \ds{\frac{|\Omega|_g}{D\alpha}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ if near a cusp, then $E\tau$ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is $E\tau = \ds{\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\epsilon} + O(1))$.