Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach

Axel Masó-Puigdellosas, Daniel Campos, and Vicenç Méndez

Phys. Rev. E 106, 034126 – Published 16 September 2022

Abstract

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward $B$ and forward $F$ times being the times since the last and until the next reset, respectively, given that the current state of the system $X (t)$ is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that $φ (t) \sim t^{- 1 - α}$ , significant changes in the properties of the conditioned backward and forward times happen at half-integer values of $α$ due to the composition between the long-time scaling of diffusion $P (x, t) \sim 1 / \sqrt{t}$ and the reset time PDF.

Received 3 May 2022
Accepted 5 September 2022

DOI:https://doi.org/10.1103/PhysRevE.106.034126

Physics Subject Headings (PhySH)

Diffusion Random walks

Master equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Axel Masó-Puigdellosas, Daniel Campos, and Vicenç Méndez

Grup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

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Vol. 106, Iss. 3 — September 2022

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Images

Figure 1
Log-lin plot of the stationary distribution for diffusion with an exponential reset time distribution (red) and two Pareto reset time distributions with different decay exponents $α$ (green and blue). The simulations have been performed for $N = 10^{5}$ trajectories with $D = 0.5$ , $T = 1$ ( $r = 1$ for the exponential), and measurement time $t = 10^{4}$ . The solid curves show the analytical results in Eqs. (21) and (24).
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Figure 2
(a) Propagator of diffusion with Pareto resetting ( $α = 0.25$ ) for the simulation of $N = 10^{5}$ trajectories at different measurement times $t$ . The multiplicative factor in the $y$ -axis is $C_{1} (t) = \frac{2 π Γ (1 - α)}{Γ (1 / 2 - α)} \sqrt{D t}$ . As the time increases, the renormalized propagator tends to be flat as found in Eq. (32). (b) Propagator of diffusion with Pareto resetting ( $α = 0.75$ ) for the simulation of $N = 10^{5}$ trajectories at different measurement times $t$ . The multiplicative factor in the $y$ -axis is $C_{2} (t) = \frac{π Γ (α) {(D t)}^{1 - α}}{sin (π α) Γ (2 α - 1)}$ . As time increases, the propagator approaches this scaling limit. In both panels A and B, the diffusion constant is $D = 0.5$ and $T = 1$ in the Pareto reset distribution. The solid colored lines have been drawn from Eq. (36), while the dots, the triangles, and the squares correspond to the propagator obtained from the simulations. The solid black curves correspond to the asymptotic behavior found in Eq. (32). Both panels have been plotted on the log-lin axis.
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Figure 3
(a) Stationary conditioned backward time distribution for diffusion and exponential resetting with $r = 0.05$ and Pareto resetting with $T = 5$ and two different values of the decay exponent $α$ . The solid lines have been drawn from the behaviors in Eqs. (46) and (47) for the exponential and Pareto cases, respectively. (b) Conditioned backward time PDF for diffusion and Pareto resetting with $T = 5$ and $α = 0.25$ . Three different measurement times have been plotted, showing that $f (B | x, t)$ does not reach a stationary shape. The solid lines correspond to the behavior in Eq. (47) with the time-dependent normalization defined in Eq. (49). Both panels (a) and (b) have been obtained averaging $N = 10^{6}$ different trajectories with diffusion constant $D = 0.1$ .
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Figure 4
(a) Stationary conditioned forward time distribution for diffusion under exponential resetting with $r = 0.05$ (circles) and under Pareto resetting with $T = 5$ and two different values of the decay exponent: $α = 0.75$ (squares) and $α = 1.75$ (triangles). The solid lines have been drawn from the behaviors in Eqs. (58) and (59) for the exponential and Pareto cases respectively. (b) Conditioned forward time PDF for diffusion and Pareto resetting with $T = 5$ and $α = 0.25$ . Three different measurement times have been plotted, showing that $f (B | x, t)$ does not reach a stationary shape. The solid lines correspond to the power-law behavior in Eq. (60) for $α < 1 / 2$ . Both panels (a) and (b) have been obtained averaging $N = 10^{6}$ different trajectories with diffusion constant $D = 0.1$ .
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