Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Stojkoski, Viktor; Sandev, Trifce; Kocarev, Ljupco; Pal, Arnab

doi:10.48550/arxiv.2107.11686

Cited by 4 publications

(4 citation statements)

References 81 publications

(121 reference statements)

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“…The relaxation dynamics to this stationary state turns out to be rather unusual [58]. Furthermore, since the BM with resetting reaches a stationary state at long times, the two-time correlation function between positions x(t 1 )x(t 2 ) decays exponentially with the time difference |t 1 − t 2 | [71,72]. These 'weak correlations' differ strongly from the standard BM without resetting where the two-time correlation function…”

Section: Introductionmentioning

confidence: 91%

Record statistics for random walks and Lévy flights with resetting

Majumdar¹,

Mounaix²,

Sabhapandit³

et al. 2021

J. Phys. A: Math. Theor.

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We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length $\eta$ drawn independently from a symmetric and continuous distribution $f(\eta)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelated time-series (for $(1-r) \ll 1$). Remarkably, we found that for every fixed $r \in [0,1[$ and any $N$, the mean number of records $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

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Section: Introductionmentioning

confidence: 91%

Record statistics for random walks and Lévy flights with resetting

Majumdar¹,

Mounaix²,

Sabhapandit³

et al. 2021

J. Phys. A: Math. Theor.

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“…At this point, we make an assumption that for large system size N → ∞, x should be same as the stationary state ensemble average of a single process x = ∞ 0 dx x P s (x), to the leading order. This ergodicity hypothesis-that the stationary state attained due to stochastic resetting renders ergodicity, has been recently shown in [37] for diffusive systems under stochastic resetting.…”

mentioning

confidence: 81%

Effect of tax dynamics on linearly growing processes under stochastic resetting: A possible economic model

Santra¹

2022

EPL

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We study a system of $N$ agents, whose wealth grows linearly, under the effect stochastic resetting and interacting via a tax-like dynamics---all agents donate a part of their wealth, which is in turn redistributed equally among all others. This mimics a socio-economic scenario where people have fixed incomes, suffer individual economic setbacks and pay taxes to the state. The system always reaches a stationary state, which shows a trivial exponential wealth distribution in the absence of tax dynamics. The introduction of the tax dynamics leads to a number of interesting features in the stationary wealth distribution. In particular, we analytically find that for an ordered system (where all agents are alike), increase in taxation results in a transition from a society where agents are most likely poor to another where rich agents are more common. We also study disordered systems, where the growth rates of the agents are chosen from a distribution and the taxation is proportional to the individual growth rates. We find an optimal taxation, which produces a complete economic equality (average wealth is independent of the individual growth rates), beyond which there is a reverse disparity, where agents with low growth rates are more likely to be rich. We consider three income distributions observed in real world and show that they exhibit same qualitative features. Though a simple, minimalistic model, this model provides some good analytical insight into the subject and has scope for step by step addition of more complexity observed in the real world. All our analytical results are in the $N\to\infty$ limit and backed by numerical simulations.

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“…In the stationary regime of srGBM the EE can be analytically quantified by knowing that the logarithm of income follows a standard diffusion with stochastic resetting that has a drift [38,39]. In particular, because of the stationarity, it follows that var [log (x (t))] = var [log (x (t + ∆))] and therefore b t,∆ is simply given by the autocorrelation function of diffusion with stochastic resetting.…”

Section: Earnings Elasticitymentioning

confidence: 99%

Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

Stojkoski,

Jolakoski,

Pal

et al. 2021

Preprint

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We explore the role of non-ergodicity in the relationship between income inequality, the extent of concentration in the income distribution, and mobility, the feasibility of an individual to change their position in the income distribution. For this purpose, we explore the properties of an established model for income growth that includes "resetting" as a stabilising force which ensures stationary dynamics. We find that the dynamics of inequality is regime-dependent and may range from a strictly non-ergodic state where this phenomenon has an increasing trend, up to a stable regime where inequality is steady and the system efficiently mimics ergodic behaviour. Mobility measures, conversely, are always stable over time, but the stationary value is dependent on the regime, suggesting that economies become less mobile in non-ergodic regimes. By fitting the model to empirical data for the dynamics of income share of the top earners in the United States, we provide evidence that the income dynamics in this country is consistently in a regime in which non-ergodicity characterises inequality and immobility dynamics. Our results can serve as a simple rationale for the observed real world income dynamics and as such aid in addressing non-ergodicity in various empirical settings across the globe.

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Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Cited by 4 publications

References 81 publications

Record statistics for random walks and Lévy flights with resetting

Record statistics for random walks and Lévy flights with resetting

Effect of tax dynamics on linearly growing processes under stochastic resetting: A possible economic model

Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity

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