Random search with resetting in heterogeneous environments

TL;DR

This work addresses mean first-passage times for stochastic searches with resetting in a bounded 1D environment with space-dependent diffusivity $D(x)$. It develops a backward-Fokker-Planck framework to obtain MFPT expressions, delivering an exact closed form for arbitrary $D(x)$ in the Stratonovich interpretation and asymptotic results for Itô and anti-Itô, complemented by closed forms for linear $D(x)$. The authors also analyze non-monotonic diffusivity (e.g., oscillatory $D(x)$) and extend the treatment to two absorbing boundaries and the semi-infinite limit, revealing how heterogeneity and the interpretation of multiplicative noise shape resetting efficiency. Overall, resetting can substantially reduce the MFPT when diffusivity is higher near the target, while highly oscillatory heterogeneity can diminish resetting benefits, with the effects further modulated by reset location and domain size.

Abstract

We investigate random searches under stochastic position resetting at rate $r$, in a bounded 1D environment with space-dependent diffusivity $D(x)$. For arbitrary shapes of $D(x)$ and prescriptions of the associated multiplicative stochastic process, we obtain analytical expressions for the average time $T$ for reaching the target (mean first-passage time), given the initial and reset positions, in good agreement with stochastic simulations. For arbitrary $D(x)$, we obtain an exact closed-form expression for $T$, within Stratonovich scenario, while for other prescriptions, like Itô and anti-Itô, we derive asymptotic approximations for small and large rates $r$. Exact results are also obtained for particular forms of $D(x)$, such as the linear one, with arbitrary prescriptions, allowing to outline and discuss the main effects introduced by diffusive heterogeneity on a random search with resetting. We explore how the effectiveness of resetting varies with different types of heterogeneity.

Figures (7)

• Figure 1: Linear diffusivity $D(x)=1+\alpha(x-1/2)$, with $\alpha = + 1.5$ (light orange), $\alpha = -1.5$ (red), and $\alpha=0$ (black), depicted in the inset of the lower panel. (a) $T$ vs. $r$, provided by Eq. (\ref{['eq:mfpt-r-L']}), for $x_0 = 0.4$ (filled symbols), and $x_0 = 0.7$ (hollow), considering the Stratonovich interpretation ($A=1$). (b) Optimal resetting rate $r^*$ (that minimizes the MFPT) vs. $x_0$, for different values of $A$ indicated in the legend. The inset shows a magnification of the critical region in linear scale. (c) Corresponding optimal MFPT $T^*$ vs. $x_0$. Symbols correspond to numerical simulations (average over $10^4$ trajectories) of Eq. (\ref{['eq:process1b']}), and lines to theoretical results.
• Figure 2: Oscillating diffusivity$D(x) = 1 + d\cos\left( 2\pi x \right)$, with $A=1$ , depicted in the inset of the lower panel. (a) Optimal resetting rate, $r^*$, and (b) corresponding optimal MFPT, $T^*$ vs. $x_0$, for $d = 0.9$ (orange), $d =- 0.9$ (red) and $d=0$ (black). The inset in (a) is a magnification around the critical $x_0$.
• Figure 3: Oscillating diffusivity$D(x) = 1 + d\cos\left( 6\pi x \right)$, with $A=1$, depicted in the inset of the lower panel. (a) Optimal resetting rate, $r^*$, and (b) corresponding optimal MFPT, $T^*$ vs. $x_0$, for $d = 0.9$ (orange), $d =- 0.9$ (red) and $d=0$ (black). The inset in (a) is a magnification around the critical $x_0$. The homogeneous case is also plotted for comparison (black lines).
• Figure 4: (a) and (d): Phase diagrams given by ${\rm sign}(T_1) \equiv dT/dr|_{r=0}$ vs. $x_0$, for the oscillating diffusivity $D(x)=1+d\cos(2\pi x)$ with $d = -0.9$ (orange) and $d = 0.9$ (red). The curves for $d=0$ (black) are also plotted. The profiles are depicted in the respective insets. MFPT vs. $r$ for $(d,x_0)=$$(-0.9,0.15)$ (b) and $(-0.9,0.71)$ (c), $(d,x_0)=$$(0.9,0.15)$ (e) and $(0.9,0.71)$ (f). The approximations for small $r$, given by Eq. (\ref{['eq:serie']}), and for large $r$, given by Eq. (\ref{['eq:wkb-sol']}) are plotted. In (e), the inset is a magnification of the main graph, to display the minima. The symbols correspond to stochastic simulations of Eq. (\ref{['eq:process1b']}).
• Figure 5: Linear diffusivity $D(x)=1+1.9(x-1/2)$, depicted in the inset of the lower panel, with absorbing boundaries at $x=0$ and $x=L=1$ (a) Optimal resetting rate $r^*$ vs. $x_0$, for different values of $A$ indicated in the legend. (b) Corresponding optimal MFPT $T^*$ vs. $x_0$.