Mean first exit times of Ornstein–Uhlenbeck processes in high-dimensional spaces

Abstract

The d-dimensional Ornstein–Uhlenbeck process (OUP) describes the trajectory of a particle in a d-dimensional, spherically symmetric, quadratic potential. The OUP is composed of a drift term weighted by a constant and a diffusion coefficient weighted by σ > 0. In the absence of drift (i.e. θ = 0), the OUP simply becomes a standard Brownian motion (BM). This paper is concerned with estimating the mean first-exit time (MFET) of the OUP from a ball of finite radius L for large . We prove that, asymptotically for , the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by ) has asymptotically no effect on its MFET. This finding might be surprising because, for small , the OUP exit time is significantly larger than BM by a margin that depends on θ. As it allows for the drift to be ignored, it might simplify the analysis of high-dimensional exit-time problems in numerous areas.Finally, our short proof for the non-asymptotic MFET of OUP, using the Andronov–Vitt–Pontryagin formula, might be of independent interest.