Diffusion with doubly stochastic resetting

TL;DR

This work examines diffusion with resetting where the resetting rate $\lambda_t$ itself is a stochastic process with relaxation time $\tau_r$, constituting a doubly stochastic (non-renewal) resetting protocol. By formulating the propagator in Fourier space and introducing the accumulated rate $\Lambda_t$, the authors derive a general expression for the propagator $\hat{P}_r(k,t)$ and, upon averaging, a broad framework $\mathbb{E}[\hat{P}_r(k,t)]$ that remains valid for general spatial operators $L(k)$, including Lévy flights and convection-diffusion. They show that the long-time NESS obeys $\hat{P}_r(k) = 1 - L(k) \int_0^\infty e^{-L(k)u} \mathbb{E}[e^{-\Lambda_u}] du$, reducing to the known exponential profile for constant resetting and revealing a crossover controlled by $\tau_r$ between self-averaged and super-statistical regimes, governed by two scales $\ell_D = \sqrt{D\tau_r}$ and $\ell_{\text{reset}} = \sqrt{D/\bar{\lambda}}$. An exactly solvable plateaus model yields a closed form for $\hat{P}_r(k)$ in terms of the double Laplace transform of the rate distribution, with a tunable parameter $\gamma = \sqrt{\bar{\lambda}/\mu_r}$ that interpolates between annealed and quenched limits, and exact asymptotics showing a transition from exponential to algebraic tails. Altogether, the results illuminate how temporal fluctuations in resetting control the structure of the non-equilibrium steady state and offer broad applicability to systems where the reset mechanism is noisy or environment-driven.

Abstract

Diffusion with stochastic resetting, instantaneous returns of a diffusing particle to a reference point, creates a stationary probability distribution. The paradigm is extended here to a doubly stochastic protocol in which the resetting rate itself fluctuates in time and relaxes on its own timescale. An exact steady-state solution reveals three spatial regimes: a fluctuation-dominated core near the origin, a power-law regime at intermediate distances, and a far-field exponential decay fixed by the rate-relaxation time. These results expose how the instantaneous rate, mean rate, and relaxation time come together to determine the non-equilibrium steady state.

Figures (3)

• Figure 1: Realization of diffusion with doubly stochastic resetting.Top panel: Time series of the resetting rate $\lambda_t$, held constant over independent and identically distributed successive intervals with exponentially distributed lengths $\tau_i\sim\mathrm{exp}(\tau_r^{-1})$ and independently redrawn from the distribution $f_\lambda(\lambda)$. Each horizontal line represents one such interval; its color encodes the instantaneous value of $\lambda_t$, as indicated by the vertical color-bar on the right. Bottom panel: A single diffusive trajectory $x_t$ subject to resetting at rate $\lambda_t$, colored by the same color as the instantaneous resetting rate. Resetting events are highlighted by red dashed vertical lines, thereby emphasizing the impact of rate fluctuations on the timing of returns.
• Figure 2: Probability density function $P(|x|)$ for the absolute displacement of a one-dimensional diffusive particle subject to doubly stochastic resetting, computed numerically from simulations. The resetting rate is initially drawn from an exponential distribution with mean $\bar{\lambda} = 1.0$, and subsequently relaxes after a characteristic relaxation time $\tau_r$ to a fixed value $\bar{\lambda}$. Curves correspond to different values of $\tau_r$, color-coded according to the logarithmic scale indicated by the colorbar. Special analytical limits are shown for comparison: fixed-rate resetting ($\tau_r = 0$, black dashed line) and fully super-statistical resetting ($\tau_r = \infty$, red dot-dashed line). Simulation parameters: diffusion constant $D = 0.5$, number of independent trajectories $N_{\mathrm{sim}} = 10^{7}$.
• Figure 3: Comparison of simulated and analytical probability densities $P(x)$ for the exactly solvable doubly stochastic resetting process. Each subplot displays the probability density across position $x$, computed using both simulation and analytical approaches for different values of the mean resetting rate $\mu_r$ and the mean resetting intensity $\bar{\lambda}$. The simulation results were obtained by running 10,000 realizations of a diffusion process with a diffusion coefficient $D = 1.0$, a total simulation time of $t_{\text{max}} = 100$, a time step of $\Delta t = 0.01$, plotted on a spatial range $x \in [-5, 5]$ divided into 100 bins. In the simulations, the resetting rate $\lambda_t$ is sampled from an exponential distribution with mean $\bar{\lambda}$, and it changes randomly at an average rate $\mu_r$. The diffusion process undergoes resetting to the origin with probability $\lambda_t \Delta t$ at each time step. For each subplot, the simulated probability density (represented by the bar histograms) is shown alongside the analytical solution (black dashed line), computed through numerical inversion of $\hat{P}_r(k)$. The panels are organized as follows: the top, middle, and bottom rows correspond to mean resetting rates $\mu_r = 0.1$, $\mu_r = 1$, and $\mu_r = 10$, respectively. The columns from left to right represent mean resetting intensities $\bar{\lambda} = 0.5$, $\bar{\lambda} = 1$, and $\bar{\lambda} = 10$. Each subplot title indicates the specific values of $\mu_r$ and $\bar{\lambda}$ used. The y-axis is plotted on a logarithmic scale.