Asymmetric Stochastic Resetting: Modeling Catastrophic Events

TL;DR

This paper introduces a novel asymmetric resetting mechanism for stochastic processes, modeling catastrophic events where resets only occur from certain directions, leading to new steady-state behaviors and optimal resetting conditions.

Contribution

It develops a general framework for asymmetric resetting, deriving exact steady states and first passage times, and applies it to specific models, revealing significant differences from symmetric resetting.

Findings

Existence of an optimal resetting rate is altered by asymmetry.

Explicit solutions for steady states and first passage times are obtained.

Numerical simulations confirm analytical results with high accuracy.

Abstract

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then, the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then, the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. We present a general framework to obtain the exact non-equilibrium steady state of the system and the mean first passage time for the system to reach the origin. Employing this framework, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement.