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 $\theta \geq 0$ and a diffusion coefficient weighted by $\sigma > 0$. In the absence of drift (i.e. $\theta = 0$), 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 $d \gg 0$. We prove that, asymptotically for $d \to \infty$, the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by $\theta \geq 0$) has asymptotically no effect on its MFET. This finding might be surprising because, for small $d \in \mathbb{N}$, the OUP exit time is significantly larger than BM by a margin that depends on $\theta$. 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.