A spectral approach to the narrow escape problem in the disk

We study the narrow escape problem in the disk, which consists in identifying the first exit time and first exit point distribution of a Brownian particle from the ball in dimension 2, with reflecting boundary conditions except on small disjoint windows through which it can escape. This problem is motivated by practical questions arising in various scientific fields (in particular cellular biology and molecular dynamics). We apply the quasi-stationary distribution approach to metastability, which requires to study the eigenvalue problem for the Laplacian operator with Dirichlet boundary conditions on the small absorbing part of the boundary, and Neumann boundary conditions on the remaining reflecting part. We obtain rigorous asymptotic estimates of the first eigenvalue and of the normal derivative of the associated eigenfunction in the limit of infinitely small exit regions, which yield asymptotic estimates of the first exit time and first exit point distribution starting from the quasi-stationary distribution within the disk.