Escape probability and mean residence time in random flows withunsteady drift

Brannan, James R.; Duan, Jinqiao; Ervin, Vincent J.

doi:10.1155/s1024123x01001521

Cited by 10 publications

(17 citation statements)

References 19 publications

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“…Next, we compare the numerical solution with the analytical solution for the mean exit time [13] in the special case of (7) Figure 2 shows the numerical solutions (the dashed lines) obtained by solving the discretized equations (16) and (17) or (19) with the fixed resolution J = 80, (a, b) = (−1, 1) and different values of α = 0.5, 1, 1.5, while the corresponding analytical solutions are shown with the solid lines. The comparison shows that the numerical solutions are very accurate as one can hardly distinguish the numerical solution from the corresponding analytical one.…”

Section: Comparing With Analytical Solutionsmentioning

confidence: 99%

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

Gao

Duan

et al. 2014

SIAM J. Sci. Comput.

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The mean first exit time and escape probability are utilized to quantify dynamical behaviors of stochastic differential equations with non-Gaussian α-stable type Lévy motions. An efficient and accurate numerical scheme is developed and validated for computing the mean exit time and escape probability from the governing differential-integral equation. An asymptotic solution for the mean exit time is given when the pure jump measure in the Lévy motion is small. From both the analytical and numerical results, it is observed that the mean exit time depends strongly on the domain size and the value of α in the α-stable Lévy jump measure. The mean exit time and escape probability could become discontinuous at the boundary of the domain, when the value of α is in (0, 1). Introduction.Random fluctuations in complex systems in engineering and science are often non-Gaussian [33,12,11]. For instance, it has been argued that diffusion by geophysical turbulence [29] corresponds, loosely speaking, to a series of "pauses," when the particle is trapped by a coherent structure, and "flights" or "jumps" or other extreme events, when the particle moves in the jet flow. Paleoclimatic data [13] also indicate such irregular processes.Lévy motions are thought to be appropriate models for non-Gaussian processes with jumps [26]. Recall that a Lévy motion L(t), or L t , is a stochastic process with stationary and independent increments. That is, for any s, t with 0 ≤ s < t, the distribution of L t − L s only depends on t − s, and for any 0 ≤ t 0 < t 1 < · · · < t n , L ti − L ti−1 , i = 1, . . . , n, are independent. Without loss of generality, we may assume that the sample paths of L t are almost surely right continuous with left limits.This generalizes the Brownian motion B(t), which satisfies all three conditions. But additionally, (i) almost all sample paths of the Brownian motion are continuous in time in the usual sense and (ii) Brownian motion's increments are Gaussian distributed.

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Section: Comparing With Analytical Solutionsmentioning

confidence: 99%

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

Gao

Duan

et al. 2014

SIAM J. Sci. Comput.

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“…In order to solve the numerical optimization problems (5) and (8), we need a numerical scheme to simulate the solutions of (3) and (6) for given initial guesses α, γ, ǫ, respectively. In this Appendix, we only recall a finite difference scheme [12] for solving (3), as a similar scheme works for (6).…”

Section: Appendixmentioning

confidence: 99%

“…for x ∈ (a, b); and u(x) = 0 for x / ∈ (a, b). Numerical approaches for the mean exit time and escape probability in the SDEs with Brownian motions were considered in [4,5], among others. In the following, we describe the numerical algorithms for the special case of (a, b) = (−1, 1) for clarity of the presentation.…”

Section: Appendixmentioning

confidence: 99%

Quantifying model uncertainty in dynamical systems driven by non-Gaussian Lévy stable noise with observations on mean exit time or escape probability

Gao

Duan

2016

Communications in Nonlinear Science and Numerical Simulation

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Complex systems are sometimes subject to non-Gaussian α−stable Lévy fluctuations. A new method is devised to estimate the uncertain parameter α and other system parameters, using observations on mean exit time or escape probability for the system evolution. It is based on solving an inverse problem for a deterministic, nonlocal partial differential equation via numerical optimization. The existing methods for estimating parameters require observations on system state sample paths for long time periods or probability densities at large spatial ranges. The method proposed here, instead, requires observations on mean exit time or escape probability only for an arbitrarily small spatial domain. This new method is beneficial to systems for which mean exit time or escape probability is feasible to observe.PACS Numbers: 05.40. 95.75.Pq, 89.90.+n

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“…This equation is solved via a finite element code [2,3]. The (spatial) average mean residence time U is defined by In particular, we consider a cellular domain which surrounds the fixed point, in order to investigate more detailed behavior about this stochastic system.…”

Section: Mean Residence Time For Circuit Modelmentioning

confidence: 99%

Dynamical behavior of the activator–repressor circuit model under random fluctuations

Lee

Çınar

Duan

2011

Communications in Nonlinear Science and Numerical Simulation

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Escape probability and mean residence time in random flows withunsteady drift

Cited by 10 publications

References 19 publications

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

Mean Exit Time and Escape Probability for Dynamical Systems Driven by Lévy Noises

Quantifying model uncertainty in dynamical systems driven by non-Gaussian Lévy stable noise with observations on mean exit time or escape probability

Dynamical behavior of the activator–repressor circuit model under random fluctuations

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