Local time of an Ornstein-Uhlenbeck particle

In this paper, we study the local time spent by an Ornstein-Uhlenbeck (OU) particle at some location till time t. Using the Feynman-Kac formalism, the computation of the moment generating function (MGF) 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 MGF 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 OU 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 MGF. From this MGF, we compute the cumulants of the local time exhibiting the approach to the stationary values explicitly for a free particle and a OU particle. Our analytical results are verified and supported by numerical simulations.