[Paper Review] An intermediate regime for exit phenomena driven by non-Gaussian Levy noises

Zhihui Yang, Jinqiao Duan|ArXiv.org|Aug 7, 2008

Complex Systems and Time Series Analysis4 references3 citations

TL;DR

This paper investigates the first exit time of a stochastic dynamical system driven by non-Gaussian Lévy noise with small intensity. It establishes that for a class of Lévy processes with heavy-tailed but not strictly stable jumps, the mean exit time scales asymptotically as $ O\left(\frac{|\ln \varepsilon|}{\varepsilon^\alpha}\right) $, representing an intermediate regime between polynomial exit times (stable Lévy noise) and exponential exit times (Gaussian noise).

ABSTRACT

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.

Motivation & Objective

• To analyze the first exit time of a stochastic dynamical system from a bounded domain under small noise intensity.
• To characterize the asymptotic behavior of the mean exit time when driven by non-Gaussian Lévy processes.
• To identify a new intermediate scaling regime between polynomial and exponential exit time behaviors.
• To establish conditions on the Lévy jump measure under which this intermediate regime emerges.
• To extend existing exit time theory beyond Gaussian and strictly stable Lévy noise cases.

Proposed method

• Analyzes a scalar stochastic differential equation (SDE) driven by a Lévy process with drift, diffusion, and a Lévy jump measure $ \nu $, under small noise intensity $ \varepsilon $.
• Uses the generator of the Lévy process and applies large deviation principles to estimate the mean exit time $ E_x[\sigma(\varepsilon)] $ from an interval $ [-b,a] $ containing the attracting equilibrium.
• Imposes conditions on the Lévy measure $ \nu $ such that $ \nu(dy) \sim \frac{dy}{|y|^{1+\alpha}(|1 + \frac{\ln|y|}{-\ln \varepsilon}| + \frac{1}{-\ln \varepsilon})} $ for small $ \varepsilon $, introducing a logarithmic correction to the power-law tail.
• Applies the Lebesgue convergence theorem and asymptotic analysis to verify convergence of normalized exit time distributions.
• Derives the asymptotic mean exit time via a limit argument involving the inverse of the integral of the jump measure over exit boundaries.
• Compares the derived scaling $ O\left(\frac{|\ln \varepsilon|}{\varepsilon^\alpha}\right) $ with known cases: $ O(\varepsilon^{-\alpha}) $ for stable Lévy noise and $ \exp(C/\varepsilon^2) $ for Gaussian noise.

Experimental results

Research questions

• RQ1 What is the asymptotic scaling of the mean first exit time for a stochastic system driven by non-Gaussian Lévy noise with small intensity?
• RQ2 Can a regime exist between the polynomial exit times of stable Lévy noise and the exponential exit times of Gaussian noise?
• RQ3 How does the structure of the Lévy jump measure influence the exit time scaling?
• RQ4 Under what conditions on the Lévy measure does the exit time scale as $ O\left(\frac{|\ln \varepsilon|}{\varepsilon^\alpha}\right) $?
• RQ5 Is this intermediate regime robust under perturbations of the Lévy measure near power-law tails?

Key findings

• For a class of non-Gaussian Lévy processes with a logarithmic correction in the Lévy measure, the mean exit time scales as $ E_x[\sigma(\varepsilon)] \sim \alpha \left( \frac{1}{a^\alpha} + \frac{1}{b^\alpha} \right)^{-1} \frac{|\ln \varepsilon|}{\varepsilon^\alpha} $.
• This scaling represents an intermediate regime: faster than the polynomial $ O(\varepsilon^{-\alpha}) $ of stable Lévy noise but slower than the exponential $ \exp(C/\varepsilon^2) $ of Gaussian noise.
• The logarithmic correction in the Lévy measure, $ \nu(dy) \sim \frac{dy}{|y|^{1+\alpha}(|1 + \frac{\ln|y|}{-\ln \varepsilon}| + \frac{1}{-\ln \varepsilon})} $, is essential to achieve this intermediate scaling.
• The result holds for any initial condition $ x \in [-b + \gamma, a - \gamma] $, with $ \gamma > 0 $, ensuring the process starts away from the boundary.
• The asymptotic behavior is derived rigorously using large deviation techniques and convergence theorems applied to the normalized exit time.
• The analysis confirms that the intermediate regime emerges when the Lévy measure decays faster than power law but slower than exponential, with the logarithmic factor introducing a critical transition.