Narrow escape problem in the presence of the force field

Let us consider a Brownian particle confined to a bounded domain by a reflecting boundary, except for a small absorbing part which is thought of as a window, through wich it can escape. The narrow escape problem deals with computing the mean sojourn time of the aforementioned Brownian particle in the domain. In our work we consider this problem in a three-dimensional Riemannian manifold under the influence of the force field. We compute an asymptotic expansion of mean sojourn time for Brownian particles, as window shrinking to the point. (This is joint work with William Trad and Leo Tzou)