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

Abstract

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.

Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications

“…So far, numerical methods for the integral fractional Laplacian (1.1) still remain very limited, especially in high dimensions (i.e., d > 1), and the main challenges come from its strong singularity. Recently, several finite difference methods have been proposed to discretize the one-dimensional (1D) fractional Laplacian; see [12,18,16] and references therein. Among them, the method in [12] is the current state-of-the-art finite difference method for the 1D fractional Laplacian -it can achieve the second order of accuracy uniformly for any α ∈ (0, 2).…”

Most probable dynamics of a genetic regulatory network under stable Lévy noise

“…Comparing with the experimental data, they demonstrated the reasonability and validity of the genetic regulatory model and considered a set of excursion trajectories in an excitable case under Gaussian noise. Instead of using the mean first exit time and first escape probability to study the bistable case of this model driven by α-stable Lévy noise [41], we examine the most probable trajectories rather than stochastic trajectory sample paths to characterize the dy- Lévy noise can be numerically simulated [42][43][44][45][46][47][48]. The noise induced most probable trajectories, which describe the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state), are obtained by examining the solution of the nonlocal Fokker-Planck equation.…”

Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric Lévy motion

“…First, we validate our equations (3.25) and (3.26) and the numerical implementation by comparing with analytical result for the special case of f ≡ 0, d = 0, ε = 1 and β = 0 [8] (−1, 1), exiting the domain and landing first to the right E = (1, ∞). Here, we consider the pure jump Lévy process: d = 0, f ≡ 0 and ε = 1.…”