Hitting the blinking target under stochastic resetting

TL;DR

This work addresses first-passage problems where a target intermittently switches between active and inactive states, modeled as a two-state gate with rates $α$ and $β$, and analyzes hitting times to the target under stochastic resetting at rate $r$. The authors develop a Laplace-transform framework to obtain closed-form expressions for the hitting-time distributions with and without resetting, including explicit results for Brownian motion in one dimension with symmetric switching ($α=β=γ$). They show that resetting can render the mean first-passage time finite even when the target gating would otherwise cause divergence, while also revealing a residual memory effect due to the target dynamics that prevents full Markovian simplification. Numerical Langevin simulations corroborate the analytical results, illustrating the interplay between reset rate, switching rate, and hitting probabilities and densities in gated-search scenarios.

Abstract

The first hitting times of a stochastic process, i.e., the first time a process reaches a particular level, are of significant interest across various scientific disciplines, including biology, chemistry, and economics. We modify the standard setup by allowing the target to spontaneously switch between two states, either active or inactive, and investigate the distribution of first hitting times accrued while the target is active. For this setup, we provide closed formulas for the distribution of the first hitting time. Additionally, we can introduce stochastic resetting to the underlying process and, utilizing our results, derive the formulas for the first time the active target is hit by the process under stochastic resetting. Interestingly, we show that resetting in this setup still leaves some memory; the system is no longer Markovian, which prevents a straightforward application of standard techniques. The analytical results are accompanied by computer simulations of Langevin dynamics.

Figures (6)

• Figure 1: Experimental setup corresponding to hitting an active target (top left panel --- (a)) and passing over inactive target (top right panel --- (b)). The bottom panel (c) presents a sample trajectory, green area show when target is in active state.
• Figure 2: Densities $g^R(t|x_,J(0))$ of $\tau^R$ with $J(0)=A$ and $J(0)=I$ for various switching rates $\gamma$ (top panel -- (a)) and densities for the equilibrium initial condition $g^R(t|x_,E)$ (bottom panel -- (b)). Solid lines corresponds to the analytical solution (see Sec. \ref{['sec:results-BM']}) while points represent results obtained from simulation of Langevin equation (see Sec. \ref{['sec:langevin']}). Model parameters: $z=0, x_0 = 1, \sigma = \sqrt{2}$ and $r=10$.
• Figure 3: The MFPT $\langle \tau^R \rangle_E$ (top panel -- (a)) and ratio of theoretical $\langle \tau^R \rangle_E^{\mathrm{th}}$ and simulation $\langle \tau^R \rangle_E^{\mathrm{sim}}$ results (bottom panel -- (b)). In the top panel: points represent results of computer simulations, see Sec. \ref{['sec:langevin']}, solid lines are analytical formula ( Eqs. (\ref{['eq:mean-tauR-A']}), (\ref{['eq:mean-tauR-I']})), and dashed line is the case without blinking (Eq. \ref{['eq:mfpt-sr']}). Model and simulation parameters: $\Delta t = 10^{-4}$, $N=10^5$, $\sigma = \sqrt{2}$, $z=0, x_0 = 1$.
• Figure 4: The same as in Fig. \ref{['fig:MFPT1']} for $z=0, x_0 = 0.5$.
• Figure 5: The probability $\pi_{ic}$ of hitting the target from the initial condition side under stochastic resetting. Various points represents different switching rates $\gamma$. In the top panel (a) $x_0=1$, while in the bottom panel (b) $x_0=0.5$.