Universal criterion for selective outcomes under stochastic resetting (2502.09127v1)

Abstract: Resetting plays a pivotal role in optimizing the completion time of complex first passage processes with single or multiple outcomes/exit possibilities. While it is well established that the coefficient of variation -- a statistical dispersion defined as a ratio of the fluctuations over the mean of the first passage time -- must be larger than unity for resetting to be beneficial for any outcome averaged over all the possibilities, the same can not be said while conditioned on a particular outcome. The purpose of this letter is to derive a universal condition which reveals that two statistical metric -- the mean and coefficient of variation of the conditional times -- come together to determine when resetting can expedite the completion of a selective outcome, and furthermore can govern the biasing between preferential and non-preferential outcomes. The universality of this result is demonstrated for a one dimensional diffusion process subjected to resetting with two absorbing boundaries.

• The paper introduces a novel universal criterion that predicts when stochastic resetting reduces conditional mean first passage times.
• It employs the conditional coefficient of variation and a defined threshold metric to evaluate the benefits of resetting across varied processes.
• Analysis of a one-dimensional diffusion model exemplifies how resetting strategically biases outcomes in complex systems.

Essay: Universal Criterion for Selective Outcomes under Stochastic Resetting

The paper, authored by Suvam Pal et al., explores the complex domain of stochastic processes with a focus on stochastic resetting. The research introduces a universal criterion applicable to selective outcomes under stochastic resetting, advancing our understanding of the nuanced role resetting plays in optimizing completion times across varied outcomes.

Overview of Stochastic Resetting

Stochastic resetting has emerged as a significant concept, influencing fields such as physics, chemistry, biology, and beyond. Resetting dynamics involve returning a process to a starting configuration intermittently, which can truncate long, unproductive paths and potentially expedite completion times in first passage tasks. The established understanding connotes that for an underlying process characterized by large fluctuations in first passage times, the coefficient of variation (CV) must exceed unity for resetting to offer a benefit in optimizing average completion time.

Conditional Versus Unconditional Outcomes

The paper takes a pioneering step by decoding the implications of stochastic resetting when the process is not about averaging outcomes universally but targeting selective outcomes. Until now, the theoretical framework was adept at handling non-selective, unconditional averages; however, the conditional specificity was less explored. The authors propose a universal condition, asserting that the interplay between the mean and CV of conditional times dictates when resetting is advantageous for a particular target outcome.

Theoretical Insights and Universal Criterion

The core innovation resides in deriving the criterion:

$$CV^\sigma\left(\mathbf{\Sigma}\right)>\Lambda^\sigma\left(\mathbf{\Sigma}\right)$$

Here, $CV^\sigma$ and $\Lambda^\sigma$ refer to the conditional coefficient of variation and a newly defined threshold, respectively. This criterion transcends specific models and system dimensionalities, positioning itself as a universal rule, applicable across systems where the underlying dynamics satisfy certain renewal properties.

Illustration: Diffusion Process

To substantiate the theoretical claim, a one-dimensional diffusion process with dual absorption sites is scrutinized under the lens of the proposed criterion. This setup exemplifies the application of the criterion, indicating regions in parameter space where resetting is either beneficial or detrimental for conditional mean times. The analysis reveals non-intuitive domains where resetting optimizes conditional outcomes—eliciting complex trade-offs and preferential biases beyond superficial proximity arguments.

Implications and Future Directions

This research lays the groundwork for further exploration of stochastic resetting in systems characterized by multiple selective outcomes. It suggests resetting as a strategic tool to bias processes toward desired outcomes selectively, fundamentally influencing processes where conditional outcomes bear distinct significances, such as biochemical reactions, ecological foraging, or algorithmic tasks involving multiple exit routes.

Potential future directions might probe deeper into multi-dimensional random searches, complex potential landscapes, or exploration of resetting in fields such as computational optimization and robotics, where selective efficiency is paramount. Additionally, experimental avenues leveraging optical traps or autonomous systems might empirically validate and expand the theoretical findings presented.

In summary, the paper signifies a remarkable contribution by unfurling a universal criterion with broad applicability across stochastic processes. The nuanced perspective on selective outcomes offers fertile ground for both theoretical and practical advancements in understanding and harnessing stochastic resetting processes.