The WKB Method: A User-Guide

Ashcroft, Peter

doi:10.1007/978-3-319-41213-9_6

Cited by 2 publications

(12 citation statements)

References 42 publications

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“…To determine the quasi-stationary distribution of the SIS model we follow the standard approach described for example in [1,[25][26][27][28][29]56]. We write x = n/N for the fraction of infected individuals in the following.…”

Section: S4 Susceptible-infected-susceptible Model and Wkb Methodsmentioning

confidence: 99%

“…This approach relies on a saddlepoint approximation in the limit of weak noise. As a consequence, the WKB instanton provides information about the most likely path by which a systems transits from one long-lived state to another [1,4,[25][26][27][28][29]. Little can be learned from the WKB approach about the statistics of transition paths in stochastic systems with finite noise.…”

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confidence: 99%

“…For some models the quasi-stationary distribution can be approximated analytically, for example with the WKB method (see e.g. [3,25,52,55,56], and also Section S4 of SM ). We now return to the example of a biased random walk.…”

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confidence: 99%

“…An ensemble of extinction paths is shown in Fig. 2(a), along with the WKB instanton to extinction [25,56]. As is illustrated in Fig.…”

mentioning

confidence: 99%

“…2(b). This time scale characterises the duration of the transition towards extinction once the system has left the endemic state, and is not to be confused with the lifetime of the metastable state itself [3,21,25,55,56] (see also Sec. S4 of the SM).…”

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confidence: 99%

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Sampling rare trajectories using stochastic bridges

Aguilar,

Baron,

Galla

et al. 2021

Preprint

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The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rate trajectories. Further, we also compare the stochastic bridges produced using our method to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the paths produced using our method, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that our method can be used to judge the accuracy of the WKB approximation at finite levels of noise.

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Section: S4 Susceptible-infected-susceptible Model and Wkb Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

“…An ensemble of extinction paths is shown in Fig. 2(a), along with the WKB instanton to extinction [25,56]. As is illustrated in Fig.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Sampling rare trajectories using stochastic bridges

Aguilar,

Baron,

Galla

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

Sampling rare trajectories using stochastic bridges

et al. 2022

View full text Add to dashboard Cite

The numerical quantification of the statistics of rare events in stochastic processes is a challenging computational problem. We present a sampling method that constructs an ensemble of stochastic trajectories that are constrained to have fixed start and end points (so-called stochastic bridges). We then show that by carefully choosing a set of such bridges and assigning an appropriate statistical weight to each bridge, one can focus more processing power on the rare events of a target stochastic process while faithfully preserving the statistics of these rare trajectories. Further, we also compare the stochastic bridges we produce to the Wentzel-Kramers-Brillouin (WKB) optimal paths of the target process, derived in the limit of low noise. We see that the generated paths, encoding the full statistics of the process, collapse onto the WKB optimal path as the level of noise is reduced. We propose that the method can also be used to judge the accuracy of the WKB approximation at finite levels of noise.

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The WKB Method: A User-Guide

Cited by 2 publications

References 42 publications

Sampling rare trajectories using stochastic bridges

Sampling rare trajectories using stochastic bridges

Sampling rare trajectories using stochastic bridges

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