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.
The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian α−stable symmetric Lévy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.This measure ν is the so called Lévy jump measure of the Lévy process L t . We also call (b, A, ν) the generating triplet [3,21].In this paper, we consider stochastic differential equations (SDEs) with a special class of Lévy processes, the α-stable symmetric Lévy motions. The corresponding Fokker-Planck equations (for the evolution of the probability density function of the solution) contain a nonlocal term, i.e., the fractional Laplacian term, which quantifies the non-Gaussian effect. It is hardly possible to have analytical solutions for these nonlocal Fokker-Planck equations even for simple systems. We thus consider numerical simulation for these nonlocal equations, either on a bounded domain or a unbounded domain. Due to the nonlocality, however,
The flow in a Hele-Shaw cell with a time-increasing gap poses a unique shrinking interface problem. When the upper plate of the cell is lifted perpendicularly at a prescribed speed, the exterior less viscous fluid penetrates the interior more viscous fluid, which generates complex, time-dependent interfacial patterns through the Saffman-Taylor instability. The pattern formation process sensitively depends on the lifting speed and is still not fully understood. For some lifting speeds, such as linear or exponential speed, the instability is transient and the interface eventually shrinks as a circle. However, linear stability analysis suggests there exist shape invariant shrinking patterns if the gap b(t) is increased more, where τ is the surface tension and C is a function of the interface perturbation mode k. Here, we use a spectrally accurate boundary integral method together with an efficient time adaptive rescaling scheme, which for the first time makes it possible to explore the nonlinear limiting dynamical behavior of a vanishing interface. When the gap is increased at a constant rate, our numerical results quantitatively agree with experimental observations (Nase et al., Phys. Fluids, vol. 23, 2011, pp. 123101). When we use the shape invariant gap b(t), our nonlinear results reveal the existence of k-fold dominant, one-dimensional, web-like networks, where the fractal dimension is reduced to almost one at late times. We conclude by constructing a morphology diagram for pattern selection that relates the dominant mode k of the vanishing interface and the control parameter C.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.