Stochastic resetting by a random amplitude

Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. Here we generalize the resetting step by introducing a random resetting amplitude such that the diffusing particle may be only partially reset towards the trajectory origin or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) and population dynamics or financial markets, as well as generic search processes.

Figure 1

Flowchart of the two main concepts, independent and dependent random amplitude stochastic resetting with specific choices of the resetting and propagation statistics.

Figure 2

RASR sample paths with ballistic displacement ($v=0.5$) and (a) and (b) independent (Poissonian with mean $\zeta =1.6$) and (c) and (d) dependent (uniformly distributed) resetting amplitudes. Resetting events ($\times$) occur (a) and (c) at a constant pace and (b) and (d) with Poissonian waiting times, both with mean rate $r=\frac{1}{4}$.

Figure 3

Height profile PDF $P(x,t)$ as a function of $x$ for six different $t$ for ballistic motion with Poissonian resetting times and exponential resetting amplitudes. The probability of no reset until $t$ is represented by the vertical line at $x=vt$; it is shown in log-lin scale for different $t$ in the inset. Simulations results are shown by points and the analytical results are shown by solid lines. The parameters are $v=0.5, r=0.125$, and $\zeta =2$.

Figure 4

(a) Mean and (b) variance of the height $x$ as a function for $t$ for exponential resetting amplitudes and ballistic motion, with Poissonian and constant pace resetting times. Points represent simulations results and solid lines are the analytical results. The parameters are $v=0.5, r=0.125$, and $\zeta =2$.

Figure 5

Height profile $P(x,t)$ as a function of $x$ for four different $t$, for ballistic motion with exponential resetting amplitudes and two different resetting scenarios: (a) constant pace resetting and (b) Poissonian resetting. The probability of no reset until $t$ is represented by the vertical line at $x=vt$; it is shown in log-lin scale for different $t$ in the inset of (b). Points show the results of simulations and solid lines the analytical results. The parameters are $v=0.5, r=0.125$, and $\zeta =2$.

Figure 6

(a) Mean and (b) variance of the height $x$ as a function of time $t$ for exponential resetting amplitudes and ballistic motion, with Poissonian and constant pace resetting. Points represent simulations and solid lines the analytical results. For all realizations, $r=0.125$.

Figure 7

(a) Mean and (b) variance of the height profile for dependent stochastic resetting with Poissonian [$r=0.125$, Eqs. (72) and (73)] and constant pace [Eqs. (62) and (65)] resetting times for a uniform resetting amplitude and two different initial heights $x_0$, in comparison with classical resetting (SR). We plot both quantities according to the normalization in Eqs. (74) and (75). The propagating process is ballistic ($v=0.5$) in all cases. Numerical results are shown by points and the analytical results by solid lines.

Figure 8

PDF $P(x,t;x_0)$ of the height profile for different initial heights and ballistic motion with uniform resetting: (a) and (c) constant pace resetting and (b) and (d) Poissonian resetting, compared to the classical resetting scenario with enforced resets to the origin, for (a) and (b) $t=1/r$ and (c) ad (d) $t=10/r$. Numerical results are shown by points and analytical results by solid lines. The parameters are $v=0.5$ and $r=0.125$.

Figure 9

(a) Mean $\langle x\left(t\right)|x_0\rangle$ and (b) variance $\mathrm{Var}\left\{x\left(t\right)\right|x_0\}$ of the height profile for dependent stochastic resetting with Poissonian and constant pace resetting times for uniform resetting amplitude and two different initial heights $x_0$, in comparison with classical resetting. The propagating process is ballistic ($v=0.5$) in all cases. For both types of resetting the resetting rate is $r=0.125$. Numerical results are shown by points and the analytical results by lines.

Figure 10

PDF $P(x,t;x_0)$ of the height profile for different initial heights and uniform resetting amplitude, compared to the classical resetting scenario with enforced resets to the origin, for ballistic propagation ($v=0.5$) and Poissonian resetting times ($r=0.125$) (a) $t=1/r$ and (b) $t=10/r$. Numerical results are shown by points and analytical results by lines.