Resetting optimized competitive first-passage outcomes in non-Markovian systems

Abstract

We investigate the role of stochastic resetting in non-Markovian systems, where memory effects arise due to slow relaxation, rugged energy landscapes, disordered environments, and molecular crowding. Using the celebrated continuous-time random walk (CTRW) framework, we analyze first-passage processes with multiple competing outcomes and examine how resetting can selectively enhance desired events. We characterize the efficiency of resetting through conditional mean first-passage times (MFPTs) and demonstrate that its impact is highly sensitive to the underlying waiting-time statistics. Furthermore, we derive an inequality that quantifies how resetting controls fluctuations in conditional first-passage times (FPTs), revealing regimes where variability is significantly suppressed. Our results provide a systematic understanding of how long-term memory influences competitive first-passage outcomes and establish resetting as a powerful control mechanism beyond the conventional Markovian setting.

Figures (4)

• Figure 1: A schematic representation of the CTRW in a one-dimensional arena with two absorbing boundaries with(out) resetting (see panel--(a)). Panel (b) plots the escape probabilities i.e.,$\epsilon^-_0=1-u$ and $\epsilon^+_0=u$, which is independent of waiting time statistics. Here we have taken $\tau=0.1$ with different classes of WTD i.e., ML ($\beta=0.8$) and Pareto ($\beta=1.5,3$). However, resetting can reshape these probabilities as shown in panel (c), also breaking the universality.
• Figure 2: Variation of the conditional MFPTs with resetting rate. Panels (a) and (b) represent the resetting-induced conditional MFPT for WTD, which belongs to class I with different exponents $\beta$. While other parameters are fixed to $\sigma=0.01,~\tau=1$ and domain length $l=1$. The solid lines indicate the theoretical results, whereas the numerical results are indicated with markers. Consequently, following the inequalities in Eqs. \ref{['app-deriv-cond-ml-minus']} and \ref{['app-deriv-cond-ml-plus']}, here we find resetting to be useful in optimizing the MFPTs. Similarly, with class II WTD, we observe that intermittent restarts become beneficial in regulating MFPTs as captured in panels (c) and (d), and the corresponding discussions have been made in Eqs. \ref{['slope-pareto-minus']} and \ref{['slope-pareto-plus']}. Note that here we consider Mittag-Leffler (ML) kind PDF with exponent $0<\beta<1$ as class I WTD, and the Pareto distribution with exponent $1<\beta<2$ as class II WTD.
• Figure 3: Illustration of the system-parameter domains obtained from the criteria defined in Eqs. \ref{['cv0-lam0-left']} and \ref{['kappa']} for realizing the outcome '$\varsigma=-$’, with a Pareto WTD of exponent $\beta=3.5$. Panel (a) shows the regions where $CV_0^- <\Lambda_0^-$ (shaded in violet) and $\kappa^- < 0$ (shaded in green). Panel (b) demonstrates that resetting has a negligible effect on both the conditional MFPT and the exit-time fluctuations through the left boundary for the initial condition $u=0.3$ and domain length $l=1$. In contrast, for the initial condition $u=0.8$, resetting significantly modifies the dynamics and becomes effective in optimizing these metrics, as illustrated in panel (c).
• Figure S1: Illustration of the system-parameter domains associated with the realization of the outcome '$\varsigma=+$’, based on the criteria defined in Eqs. \ref{['cv0-lam0-right']} and \ref{['kappa']}, for a Pareto WTD with exponent $\beta=3.5$. Panel (a) presents the regions of parameter space where the condition $CV_0^+ < \Lambda_0^+$ is satisfied (shaded in violet), along with the regime $\kappa^+ < 0$ (shaded in green), which together signal parameter ranges where resetting-induced optimization is anticipated. Panel (b) shows that, for the initial condition $u=0.2$ and a fixed domain length $l=1$, stochastic resetting exerts a pronounced influence on both the conditional MFPT and the corresponding exit-time fluctuations through the right boundary. In contrast, for the initial condition $u=0.7$, as illustrated in panel (c), the impact of resetting on the transport properties is significantly reduced, rendering resetting comparatively less effective in this regime.