scholarly journals AN INTERMEDIATE REGIME FOR EXIT PHENOMENA DRIVEN BY NON-GAUSSIAN LÉVY NOISES

2008 ◽  
Vol 08 (03) ◽  
pp. 583-591 ◽  
Author(s):  
ZHIHUI YANG ◽  
JINQIAO DUAN
Keyword(s):  
Dynamical System ◽  
Noise Intensity ◽  
Exit Time ◽  
First Exit Time ◽  
Small Noise ◽  
Intermediate Regime ◽  
Mean Exit Time ◽  
Small Intensity ◽  
The Mean ◽  
Non Gaussian

A dynamical system driven by non-Gaussian Lévy noises of small intensity is considered. The first exit time of solution orbits from a bounded neighborhood of an attracting equilibrium state is estimated. For a class of non-Gaussian Lévy noises, it is shown that the mean exit time is asymptotically faster than exponential (the well-known Gaussian Brownian noise case) but slower than polynomial (the stable Lévy noise case), in terms of the reciprocal of the small noise intensity.

scholarly journals MEAN EXIT TIME AND ESCAPE PROBABILITY FOR A TUMOR GROWTH SYSTEM UNDER NON-GAUSSIAN NOISE

2012 ◽  
Vol 22 (04) ◽  
pp. 1250090 ◽  
Author(s):  
JIAN REN ◽  
CHUJIN LI ◽  
TING GAO ◽  
XINGYE KAN ◽  
JINQIAO DUAN
Keyword(s):  
Tumor Growth ◽  
Exit Time ◽  
Laplacian Operator ◽  
Escape Probability ◽  
Tumor Growth Model ◽  
Bifurcation Phenomena ◽  
Mean Exit Time ◽  
Growth System ◽  
The Mean ◽  
Non Gaussian

Effects of non-Gaussian α-stable Lévy noise on the Gompertz tumor growth model are quantified by considering the mean exit time and escape probability of the cancer cell density from inside a safe or benign domain. The mean exit time and escape probability problems are formulated in a differential-integral equation with a fractional Laplacian operator. Numerical simulations are conducted to evaluate how the mean exit time and escape probability vary or bifurcates when α changes. Some bifurcation phenomena are observed and their impacts are discussed.

The Estimates of the Mean First Exit Time of a Bistable System Excited by Poisson White Noise

2017 ◽  
Vol 84 (9) ◽  
Author(s):  
Yong Xu ◽  
Hua Li ◽  
Haiyan Wang ◽  
Wantao Jia ◽  
Xiaole Yue ◽  
...  
Keyword(s):  
White Noise ◽  
Infinite Order ◽  
Exit Time ◽  
Theoretical Solution ◽  
Bistable System ◽  
Order Partial Differential Equation ◽  
First Exit Time ◽  
Poisson White Noise ◽  
The Mean ◽  
Non Gaussian

We propose a method to find an approximate theoretical solution to the mean first exit time (MFET) of a one-dimensional bistable kinetic system subjected to additive Poisson white noise, by extending an earlier method used to solve stationary probability density function. Based on the Dynkin formula and the properties of Markov processes, the equation of the mean first exit time is obtained. It is an infinite-order partial differential equation that is rather difficult to solve theoretically. Hence, using the non-Gaussian property of Poisson white noise to truncate the infinite-order equation for the mean first exit time, the analytical solution to the mean first exit time is derived by combining perturbation techniques with Laplace integral method. Monte Carlo simulations for the bistable system are applied to verify the validity of our approximate theoretical solution, which shows a good agreement with the analytical results.

scholarly journals A variation on the Donsker–Varadhan inequality for the principal eigenvalue

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160877
Author(s):  
Jianfeng Lu ◽  
Stefan Steinerberger
Keyword(s):  
Lower Bound ◽  
Exit Time ◽  
Principal Eigenvalue ◽  
Short Paper ◽  
First Eigenvalue ◽  
Drift Diffusion ◽  
First Exit Time ◽  
The First Eigenvalue ◽  
Order Elliptic Operator ◽  
The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

scholarly journals First exit time of a Lévy flight from a bounded region in ℝN

2015 ◽  
Vol 52 (3) ◽  
pp. 649-664 ◽  
Author(s):  
Yoora Kim ◽  
Irem Koprulu ◽  
Ness B. Shroff
Keyword(s):  
Exit Time ◽  
Length Distribution ◽  
Step Length ◽  
Upper And Lower Bounds ◽  
Bounded Region ◽  
Lévy Flight ◽  
First Exit Time ◽  
Levy Flight ◽  
Distribution Parameters ◽  
The Mean

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region inN-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

scholarly journals First exit time of a Lévy flight from a bounded region in ℝN

2015 ◽  
Vol 52 (03) ◽  
pp. 649-664 ◽  
Author(s):  
Yoora Kim ◽  
Irem Koprulu ◽  
Ness B. Shroff
Keyword(s):  
Exit Time ◽  
Length Distribution ◽  
Step Length ◽  
Upper And Lower Bounds ◽  
Bounded Region ◽  
Lévy Flight ◽  
First Exit Time ◽  
Levy Flight ◽  
Distribution Parameters ◽  
The Mean

In this paper we characterize the mean and the distribution of the first exit time of a Lévy flight from a bounded region in N-dimensional spaces. We characterize tight upper and lower bounds on the tail distribution of the first exit time, and provide the exact asymptotics of the mean first exit time for a given range of step-length distribution parameters.

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