Accelerated first-passage dynamics in a non-Markovian feedback Ornstein--Uhlenbeck process

TL;DR

This paper analyzes how time-averaged feedback in a non-Markovian Ornstein-Uhlenbeck process influences first-passage times, revealing that feedback can accelerate dynamics by lowering energy barriers and altering rare event statistics.

Contribution

It introduces a novel analysis of feedback-modified first-passage dynamics using large deviation theory, showing how memory feedback can control and accelerate stochastic processes.

Findings

Feedback lowers the effective energy barrier for first passage.

Optimal first-passage time shifts from infinity to finite due to feedback.

Alternative trajectories are sub-optimal and do not contribute to acceleration.

Abstract

We study the first-passage dynamics of a non-Markovian stochastic process with time-averaged feedback, which we model as a one-dimensional Ornstein--Uhlenbeck process wherein the particle drift is modified by the empirical mean of its trajectory. This process maps onto a class of self-interacting diffusions. Using weak-noise large deviation theory, we calculate the leading order asymptotics of the time-dependent distribution of the particle position, derive the most probable paths that reach the specified position at a given time and quantify their likelihood via the action functional. We compute the feedback-modified Kramers rate and its inverse, which approximates the mean first-passage time, and show that the feedback accelerates dynamics by storing finite-time fluctuations, thereby lowering the effective energy barrier and shifting the optimal first-passage time from infinite to finite. Although we identify alternative mechanisms, such as slingshot and ballistic trajectories, we find that they remain sub-optimal and hence do not accelerate the dynamics. These results show how memory feedback reshapes rare event statistics, thereby offering a mechanism to potentially control first-passage dynamics.