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

Abstract

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>θ</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:math> and a diffusion coefficient weighted by σ &gt; 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>≫</mml:mo> <mml:mn>0</mml:mn> </mml:math> . We prove that, asymptotically for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> , the OUP takes (on average) no longer to exit than BM. In other words, the mean-reverting drift of the OUP (scaled by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>θ</mml:mi> <mml:mo>⩾</mml:mo> <mml:mn>0</mml:mn> </mml:math> ) has asymptotically no effect on its MFET. This finding might be surprising because, for small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> , 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.