Local time of an Ornstein-Uhlenbeck particle

TL;DR

This paper analyzes the local time of an Ornstein-Uhlenbeck particle using quantum formalism, deriving explicit statistical measures, large deviation functions, and stationary distributions with analytical and numerical validation.

Contribution

It introduces a quantum perturbation approach to compute local time statistics of Ornstein-Uhlenbeck particles, including cumulants, correlations, and large deviations, extending to non-conditioned stationary distributions.

Findings

Explicit formulas for mean, variance, and covariance of local times.

Large deviation functions for local time in absence of boundaries.

Stationary distribution and approach dynamics of local time.

Abstract

In this paper, we study the local time spent by an Ornstein-Uhlenbeck particle at some location till time t. Using the Feynman-Kac formalism, the computation of the moment generating function of the local time can be mapped to the problem of finding the eigenvalues and eigenfunctions of a quantum particle. We employ quantum perturbation theory to compute the eigenvalues and eigenfunctions in powers of the argument of the moment generating function which particularly help to directly compute the cumulants and correlations among local times spent at different locations. In particular, we obtain explicit expressions of the mean, variance, and covariance of the local times in the presence and in the absence of an absorbing boundary, conditioned on survival. In the absence of absorbing boundaries, we also study large deviations of the local time and compute exact asymptotic forms of the associated large deviation functions explicitly. In the second part of the paper, we extend our study of the statistics of local time of the Ornstein-Uhlenbeck particle to the case not conditioned on survival. In this case, one expects the distribution of the local time to reach a stationary distribution in the large time limit. Computations of such stationary distributions are known in the literature as the problem of first passage functionals. In this paper, we study the approach to this stationary state with time by providing a general formulation for evaluating the moment generating function. From this moment generating function, we compute the cumulants of the local time exhibiting the approach to the stationary values explicitly for a free particle and a Ornstein-Uhlenbeck particle. Our analytical results are verified and supported by numerical simulations.