3 On L 1 - distance between first exit times from two regions ∗ †

First exit times from regions and their dependence on variations of boundaries are discussed for diffusion processes. The paper presents an estimate of L1-distance between exit times from two regions via expectations of exit times. Abbreviated title: On L1-distance between first exit times It is known that first exit times from a region for smooth solutions of ordinary equations does not depend continuously on variations of the initial data or on the boundary of the region. But first exit times for non-smooth trajectories of diffusion processes have some path-wise regularity with respect to these variations (some related results can be found in author’s papers (1987),(1992)). This paper studies path-wise dependence on fluctuations of the boundary for first exit times of diffusion processes. We present an effective estimate of L1-distance between exit times from two regions for a diffusion process via expectations of exit times. Let (Ω,F ,P) be a standard probability space. Consider a n-dimensional diffusion process y(t) such that dy(t) = f(y(t))dt + β(y(t))dw(t), t > 0, (1) y(0) = a. (2) Here w(t) is a standard d-dimensional Wiener process, f and β are non-random functions with respective values in R and R. The random vector a with values in R does not depend on w(·). All vectors and matrices are real, and D̄ denotes the closure of a region D. MSC 2000 subject classifications. Primary60G17, 60G40 ; secondary 60J50, 60J60, 60J65.