On the Distribution Density of the First Exit Point of a Diffusion Process form a Small Neighborhood of Its Initial Position

Abstract

Considering a diffusion process in a d-dimensional space, we study the distribution of the first exit point of a process from a small neighborhood of its initial state. A weak asymptotic expansion in terms of the small scale parameter is obtained for the density of the distribution. In the case of spherical neighborhoods, simple formulas are deduced for the first three coefficients of the expansion, reflecting the probabilistic sense of the coefficients of an elliptic partial differential equation.