Universal criterion for selective outcomes under stochastic resetting

TL;DR

This work extends the well-known CV criterion for resetting from unconditional exits to conditional (selective) exits in stochastic search processes with multiple targets. By deriving a universal condition $CV^{\sigma}(\Sigma) > \Lambda^{\sigma}(\Sigma)$, where $\Lambda^{\sigma}$ depends on both unconditional and conditional first-passage statistics and their fluctuations, the authors establish when resetting accelerates a chosen outcome. The approach uses renewal theory to relate conditional and unconditional mean first-passage times via $\langle T_r^{\sigma}(\mathbf{\Sigma})\rangle = \langle T_r(\mathbf{\Sigma})\rangle + \frac{\partial}{\partial r} \ln \left[ \frac{\widetilde{T}_0(\mathbf{\Sigma},r)}{\widetilde{T}_0^{\sigma}(\mathbf{\Sigma},r)} \right]$, and demonstrates the criterion in a 1D diffusion problem with two absorbing boundaries, revealing phase diagrams and biasing strategies for selective outcomes. The results are universal, not tied to a specific dynamics or dimension, and have potential implications for biomolecular searches, chemical kinetics, and engineered stochastic systems. The work also cautions about scenarios where the underlying process lacks finite moments, in which resetting may be pathological. Overall, the study provides a principled framework to control and bias selective outcomes via resetting.

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.

Figures (4)

• Figure 1: Resetting mediated universal criterion for selective and non-selective outcomes for a first-passage process with multiple exits or outcomes labeled as $E_1, E_2$, $E_3$ etc. Panel (a) illustrates the unconditional completion of the process (regardless of the outcome) averaged over all the possibilities. In contrast, panel (b) shows the conditional completion associated with a specific outcome (indicated by their respective color). Panel (c) uses a Venn diagram to illustrate the universal criteria and their domain of validity marking, in particular, the trade-off boundary as in Eq. (\ref{['lambda-c']}) and various regions of competing outcomes (as explained in Sections \ref{['formalism:universal_conditional']} and \ref{['formalism:trade-off']}).
• Figure 2: Phase diagram demonstrating the various regions of resetting based optimization for the conditional and unconditional mean exit times in the diffusion problem of Sec. \ref{['sec:diffusion']}. The control parameter is $u$ and we compare $\langle T_r^+\rangle$ (conditional exit through the right) and $\langle T_r\rangle$ (unconditional exit averaged over both boundaries). The solid blue curve indicates $CV^+-\Lambda^+$ and the green one $CV-1$ as a function of $u$. Depending on the intersecting points with the zero line (dashed horizontal line), four distinct regions are defined as discussed in Sec. \ref{['formalism:trade-off']}. Region I spans the domain of $u$ for which $CV^+>\Lambda^+~\&~CV>1$ and thus resetting optimizes both conditional and unconditional mean exit times, as can also be seen from panel (a) where we have plotted $\langle T_r^+\rangle$ and $\langle T_r\rangle$ as a function of $r$ (for a value of $u$ indicated by a plot-marker in this region). In Region II, $u$ is such that $CV^+>\Lambda^+$ but $CV<1$, hence a finite resetting rate optimizes $\langle T_r^+\rangle$ but not $\langle T_r\rangle$, see panel (b). Region III represents the case where both $CV^+<\Lambda^+$ and $CV<1$, hence resetting is detrimental for both exit times, see panel (c). Finally, in Region IV, $CV^+<\Lambda^+$ but $CV>1$, therefore the unconditional process is benefited with the introduction of resetting, but not the conditional exit through the right boundary. This is confirmed from panel (d).
• Figure 3: Illustration of the preferential biasing between the left and right exit for the diffusion process by leveraging the criterion [see Eq. (\ref{['eq:preferential']}) and Eq. (\ref{['eq:preferential_diffusion']})]. We plot $CV^+-\Lambda^+$ (blue solid line) and $CV^- -\Lambda^-$ (green solid line) as a function of $u$. For the region of $u$ where $CV^+>\Lambda^+$ but $CV^-<\Lambda^-$, resetting expedites the right exit but not the left, as can be seen from inset (a) showing $\langle T_r^+\rangle$ vs. $r$ (which can be optimized with $r$) and $\langle T_r^-\rangle$ (which cannot). In the symmetric region of $u$, we have $CV^+<\Lambda^+$ but $CV^->\Lambda^-$ so that resetting becomes beneficial for the left exit but not for the right exit. This is also confirmed by the variations of the mean conditional times with $r$ in inset (b).
• Figure 4: A comparison between conditional escape from left boundary and the unconditional escape. Treating $u$ as a control parameter, we illustrate the behavior of $\langle T_r^-\rangle$ and $\langle T_r\rangle$ in the presence of resetting. The solid blue curve indicates $CV^--\Lambda^-$ and the green one represents $CV-1$ as a function of $u$. Similar to Fig. \ref{['fig2']}, we explore four distinct regions of optimization as discussed in Sec. \ref{['formalism:trade-off']}. Region I showcases $CV^->\Lambda^-$and $CV>1$ and thus resetting optimizes both conditional and unconditional first-passage time. It is demonstrated in panel (a), where we have illustrated $\langle T_r^-\rangle$ and $\langle T_r\rangle$ as a function of $r$. Next we explore Region II, where $CV^->\Lambda^-$ but $CV<1$ so that resetting optimizes $\langle T_r^-\rangle$ but not $\langle T_r\rangle$, as shown in panel (b). Region III elaborates a scenario where $CV^-<\Lambda^-$ and $CV<1$ so that resetting prolongs both $\langle T_r^-\rangle$ and $\langle T_r\rangle$ (as shown in panel (c)). Finally, we arrive at the region IV for which $CV^-<\Lambda^-$ but $CV>1$ so that the unconditional process is benefited with the introduction of resetting, but not the conditional exit from the left boundary. This can also be confirmed from panel (d).