# gst: Lévy flights and Lévy walks under stochastic resetting.

<< Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. (AA)  demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, Lévy-type, α

-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index α determining the power-law decay of the jump length distribution. For heavier (smaller α) distributions, the domain becomes narrower in comparison to lighter tails. >>

<< Additionally, (AA) explore connections between Lévy flights (LFs) and Lévy walks (LWs) in the presence of stochastic resetting. First of all, (They) show that for Lévy walks, the stochastic resetting can also be beneficial in the domain where the coefficient of variation is smaller than 1. Moreover, (They) demonstrate that in the domain where LWs are characterized by a finite mean jump duration (length), with the increasing width of the interval, the LWs start to share similarities with LFs under stochastic resetting. >>️

Bartosz Żbik, Bartłomiej Dybiec. Lévy flights and Lévy walks under stochastic resetting. Phys. Rev. E 109, 044147. April 22, 2024.

https://journals.aps.org/pre/abstract/10.1103/PhysRevE.109.044147

Also: keyword Lévy in FonT

https://flashontrack.blogspot.com/search?q=Lévy&m=1

Also: escape, noise, in https://www.inkgmr.net/kwrds.html 

Keywords: gst, escape, noise, stochastic resetting, Lévy

# gst: wandering domains: a hypothetical scenario with bounded orbits.

<< A significant open question in transcendental dynamics asks if it is possible for a point, and thus all points, of a wandering domain to have a bounded orbit. >>

<< In other words, is there a transcendental entire function f with a wandering domain U such that its forward orbit, U_n≥0 f^n(U), is bounded? >>

AA << give a partial answer to this question by constructing an example of a such a wandering domain U which has a nearly bounded orbit; >>

<< In other words, the set of natural numbers n for which f^n(U) is contained in D has upper (and lower) natural density one. >>

<< This is in particularly strong contrast to all existing examples of wandering domains, for which the quantity (..) is equal to zero for any choice of bounded domain D. >>

Leticia Pardo-Simon, David J. Sixsmith. Wandering domains with nearly bounded orbits. arXiv: 2307.16682v1 [math.DS]. Jul 31, 2023. 

https://arxiv.org/abs/2307.16682

Also: Unbounded fast escaping wandering domains. https://arxiv.org/abs/2210.13350 ; Escaping sets of continuous functions. https://arxiv.org/abs/1601.04010 

Also: 'escape', in FonT: https://flashontrack.blogspot.com/search?q=escape

Also: 'fuga', in Notes (quasi-stochastic poetry): https://inkpi.blogspot.com/search?q=fuga 

Keywords: gst, wandering domain, escape 

# gst: apropos of transitions, escape inside a noisy layered system

<< Noisy layered systems can exhibit amplified fluctuation patterns depending on their connectivity. Here (AA) showed that noise originally injected in one layer may induce faster basin escape in connected layers. This is both due to the amplification of the noise amplitude and the system specific correlations that the noise acquire while going through the first layer. Indeed, (..) one sees that the noise in the second is correlated in both space and time with clear dependence on the network structure. For networks with low algebraic connectivity, (AA) numerically showed that the first escape time is shorter in the two cases where (i) fluctuations are amplified in the second layer and (ii) noise in the second layer is rescaled in order to have the same variance in both layers. While point (i) is rather intuitive, i.e. larger fluctuations lead to shorter first escape times, point (ii) is more involved. Indeed, this indicates that noise with spatial and temporal correlations (..) selects directions that enable faster exits from the initial basin of attraction. >>

Melvyn Tyloo. Faster network disruption from layered oscillatory dynamics. arXiv: 2210.01180v1 [nlin.AO]. Oct 3, 2022. 

https://arxiv.org/abs/2210.01180

Also

keyword 'escape' in FonT

https://flashontrack.blogspot.com/search?q=escape

keyword 'escape' | 'fuga' in Notes 

(quasi-stochastic poetry)

https://inkpi.blogspot.com/search?q=escape

https://inkpi.blogspot.com/search?q=fuga

Keywords: gst, transitions, escape, noise, noisy layered systems