Asymmetric stochastic resetting: Modeling catastrophic events

Carlos A. Plata, Deepak Gupta, and Sandro Azaele

Phys. Rev. E 102, 052116 – Published 11 November 2020

Abstract

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. An approach to obtain the exact nonequilibrium steady state of such systems and the mean first passage time to reach the origin is presented. Furthermore, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting, such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement.

Received 24 July 2020
Accepted 22 October 2020

DOI:https://doi.org/10.1103/PhysRevE.102.052116

Physics Subject Headings (PhySH)

Ecological population dynamics Econophysics Nonequilibrium statistical mechanics Stochastic processes

Fokker–Planck equation

Statistical Physics & Thermodynamics

Authors & Affiliations

Carlos A. Plata , Deepak Gupta , and Sandro Azaele

Dipartimento di Fisica “G. Galilei,” INFN, Università di Padova, Via Marzolo 8, 35131 Padova, Italy

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Vol. 102, Iss. 5 — November 2020

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Images

Figure 1
Sketch of asymmetric stochastic resetting process. The system evolves over time (zigzag curve) with interrupting events (horizontal arrows) which bring the former to a certain value $x_{r} > 0$ with rate $r (x) = r Θ (x - x_{r})$ .
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Figure 2
Relaxation to the steady-state distribution $p_{ss} (x)$ . (a) (I) Pure diffusion. (b) (II) Ecological model. The solid curve stands for the stationary theoretical prediction while circles, squares, and triangles are obtained from the numerical simulation at three different times. The parameters for panel (a) are $x_{0} = x_{r} = 2$ , $r = 0.5$ , and $D = 1$ and for panel (b) $b = 0.9$ , $x_{0} = x_{r} = 0.25$ , and $r = 0.5$ . In each case, the vertical dashed line corresponds to the resetting location $x_{r}$ .
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Figure 3
Asymptotic of $[τ (x) - τ (x_{r})]$ with respect to $x$ . Clearly, we can see that $[τ (x) - τ (x_{r})]$ approaches $r^{- 1}$ (horizontal red dashed line) as $x \to \infty$ . The parameters used in the above plots for diffusion model (a) are $D = 1, x_{r} = 5, r = 0.1$ and for ecological one (b) are $x_{r} = 1.0, b = 0.8, r = 2.0$ .
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Figure 4
Mean first passage time $τ (x)$ . (a) (I) Pure diffusion. (b) (II) Ecological model. It is observed that $τ (x)$ reaches a constant value for large $x$ , and it increases with the initial location of the system. As it is reasonable, the mean first passage time decreases with resetting rate $r$ for given $x$ . The vertical dashed line indicates the resetting location $x_{r}$ . The parameters for panel (a) are $x_{r} = 5$ and $D = 1$ and for panel (b) $b = 0.5$ and $x_{r} = 5$ . In each case, the vertical dashed line corresponds to the resetting location $x_{r}$ .
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Figure 5
Mean first passage time $τ (x_{r})$ as a function the resetting rate $r$ . (a) (I) Pure diffusion. (b) (II) Ecological model. In both cases, solid curve is the analytical prediction given by Eqs. (14) and (15), whereas the squares are obtained from numerical simulations. The parameters for the panel (a) are $x_{r} = x_{0} = 0.1$ and $D = 1$ and for the panel (b) $x_{r} = x_{0} = 0.1$ and $b = 0.5$ .
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Figure 6
Mean first passage time $τ (x)$ [given in (14) and (15)] with respect to resetting rate $r$ for given $x$ . Herein we show $τ (x)$ for diffusion model in panel (a) and the ecological model in panel (b). As it is reasonable, the mean first passage time decreases with resetting rate $r$ for given $x$ . For $r \to 0$ , $τ (x)$ diverges only for the diffusion model while it stays finite [indicated by filled circles in panel (b)] for the ecological setting, and is in agreement with the mean first passage time in the absence of resetting. The parameters for panel (a) are $D = 1, x_{r} = 5$ and for panel (b) $b = 0.5, x_{r} = 5$ .
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