Ornstein-Uhlenbeck process and generalizations: particle's dynamics under comb constraints and stochastic resetting

TL;DR

This paper explores generalized Ornstein-Uhlenbeck processes, focusing on particle dynamics in comb geometries and under stochastic resetting, revealing how these factors influence stationary states and dynamical properties.

Contribution

It introduces and analyzes two generalizations of the Ornstein-Uhlenbeck process: one on a comb structure and another with stochastic resetting, highlighting their effects on particle dynamics.

Findings

Dynamical characteristics are derived for the comb model using Langevin and Fokker-Planck equations.

Stochastic resetting leads to non-equilibrium stationary states in both standard and comb geometries.

The interplay of resetting and drift results in unique stationary behaviors in complex geometries.

Abstract

The Ornstein-Uhlenbeck process is interpreted as Brownian motion in a harmonic potential. This Gaussian Markov process has a bounded variance and admits a stationary probability distribution, in contrast to the standard Brownian motion. It also tends to a drift towards its mean function, and such a process is called mean-reverting. Two examples of the generalized Ornstein-Uhlenbeck process are considered. In the first one, we study the Ornstein-Uhlenbeck process on a comb model, as an example of the harmonically bounded random motion in the topologically constrained geometry. The main dynamical characteristics (as the first and the second moments) and the probability density function are studied in the framework of both the Langevin stochastic equation and the Fokker-Planck equation. The second example is devoted to the study of the effects of stochastic resetting on the Ornstein-Uhlenbeck process, including stochastic resetting in the comb geometry. Here, the non-equilibrium stationary state is the main question in task, where the two divergent forces, namely the resetting and the drift towards the mean, lead to compelling results both in the case of the Ornstein-Uhlenbeck process with resetting and its generalization on the two dimensional comb structure.