On the exit law from saddle points
type: Item
instance of: scholarly article
title: On the exit law from saddle points
zbMATH Open document ID: 0845.60051
DOI: 10.1016/0304-4149(95)00063-1
author: Martin V. Day
published in: Stochastic Processes and their Applications
publication date: 1996-03-24T00:00:00Z
review text: The author considers the effects of adding an asymptotically small random Brownian perturbation to a planar dynamical system with a saddle point equilibrium. He obtains a new limit law for the exit time \(\tau^\varepsilon_D\) from a neighborhood of the saddle, assuming the initial point is on the stable manifold. More precisely, if the first-order system in the plane is \(\dot x_t = b(x_t)\), with \(b(0) = 0\) and \(\Lambda = {\partial b\over \partial x} (0)\) has eigenvalues \(\lambda_2 < 0 < \lambda_1\), and the perturbated system is \[ dx^\varepsilon_t = b(x^\varepsilon_t) dt + \varepsilon^{1/2} \sigma(x^\varepsilon_t) dW_t,\qquad \sigma \text{ non singular,} \] then \(\lambda_1 \tau^\varepsilon_D - \text{Log} (\varepsilon^{-1/2})\) converges weakly to the law of the sum of two random independent variables \(H\) and \(C_v\); the law of \(H\) has a density \(k(t) = {2\over \sqrt{\pi}} e^{-(t + e^{-2t})}\), \(C_{\pm1}\) are a pair of constants and the law of \(C_v\) is \(P(C_v = C_1) = P(C_v = C_{-1}) = 1/2\). Finally the author notices that this result can be generalized to higher dimensions, but only if the unstable manifold has dimension 1.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 34E20
zbMATH DE Number: 859318
zbMATH Keywords: small noise
zbMATH Keywords: Brownian perturbation
zbMATH Keywords: dynamical system with a saddle point equilibrium
zbMATH Keywords: stable manifold
zbMATH Keywords: unstable manifold
reviewed by: J.-P. Lepeltier
MaRDI profile type: Publication
full work available at URL: 0304-4149(95)00063-1
OpenAlex ID: W2018394408
cites work: Q3331506
cites work: Q3794956
cites work: On the exponential exit law in the small parameter exit problem
cites work: Conditional exits for small noise diffusions with characteristic boundary
cites work: Regularity of boundary quasi-potentials for planar systems
cites work: Q3672830
cites work: Q5330830
cites work: The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point
cites work: Large deviations for the first exit time on small random perturbations of dynamical systems with a hyperbolic equilibrium point
description: scientific article; zbMATH DE number 859318