The Two-Dimensional Narrow Escape Problem with Deterministic Initial Condition: Mean Escape Time and Escape location probabilities

Source abstract

This paper presents a rigorous asymptotic analysis of the two-dimensional narrow escape problem for a Brownian particle initiated at a deterministic position x. While we address the mean escape time, the main novelty lies in our derivation of the asymptotic escape distribution. We derive explicit asymptotic formulas that characterize the dependence of the escape behavior on the relative configuration of the starting point and the window centers. These results remain valid as the starting position approaches the boundary windows and include refined error estimates. The formulas are obtained with two independent methods, each of them having their own interest. A central feature of this work is the rigorous treatment of the underlying mixed boundary value problems in the absence of a contact angle at the escape window boundaries. By leveraging recent results on the maximal regularity of the gradient of the solutions of such mixed boundary value problems, we ensure the analytical validity of our approach despite the inherent loss of regularity of these solutions at the escape window boundaries. Finally, we apply these general results to provide a full sharp asymptotic equivalents for regimes where window sizes scale at specific relative rates.