26 Citations
Exit time asymptotics for dynamical systems with fast random switching near an unstable equilibrium
- MathematicsStochastics and Dynamics
- 2019
We consider the exit problem for a one-dimensional system with random switching near an unstable equilibrium point of the averaged drift. In the infinite switching rate limit, we show that the exit…
Malliavin calculus approach to long exit times from an unstable equilibrium
- MathematicsThe Annals of Applied Probability
- 2019
For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. Using Malliavin calculus tools, we obtain precise…
Normal forms approach to diffusion near hyperbolic equilibria
- Mathematics
- 2010
We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighborhood of a hyperbolic critical point. We show that if the distribution of the…
Noisy heteroclinic networks
- Mathematics
- 2007
We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distribution in a…
Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial Jordan block case
- Mathematics, PhysicsStochastics and Dynamics
- 2019
We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the…
Small noise limit for diffusions near heteroclinic networks
- Mathematics
- 2010
This is a nontechnical exposition of the theory on vanishing noise limit for random perturbations of dynamical systems admitting heteroclinic networks developed by the author [Y. Bakhtin, Noisy…
On Gumbel limit for the length of reactive paths
- Mathematics
- 2013
We give a new proof of the vanishing noise limit theorem for exit times of one-dimensional diffusions conditioned on exiting through a point separated from the starting point by a potential wall. We…
Weak noise and non-hyperbolic unstable fixed points: Sharp estimates on transit and exit times
- Mathematics
- 2015
We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable…
Scaling Limit for the Diffusion Exit Problem, a Survey
- Mathematics
- 2014
In this review, an outline of the so called Freidlin-Wentzell theory and its recent extensions is given. Broadly, this theory studies the exponential rate at which the probabilities of rare events…
Spectral Theory for Random Poincaré Maps
- MathematicsSIAM J. Math. Anal.
- 2017
A discrete-time, continuous-space Markov chain is constructed, called a random Poincare map, which encodes the metastable behaviour of the system, and provides expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighbourhoods of periodic orbits.
References
SHOWING 1-10 OF 26 REFERENCES
The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point
- Mathematics
- 1981
We consider the Markov diffusion process ξ∈(t), transforming when ɛ=0 into the solution of an ordinary differential equation with a turning point ℴ of the hyperbolic type. The asymptotic behevior as…
Random perturbations of heteroclinic attractors
- Mathematics
- 1990
Estimates are derived for the mean recurrence time of orbits in the neighborhood of an attracting homoclinic orbit or heteroclinic cycle in an ordinary differential equation, subject to small…
Random perturbations of 2-dimensional hamiltonian flows
- Mathematics, Physics
- 2004
Abstract.We consider the motion of a particle in a periodic two dimensional flow perturbed by small (molecular) diffusion. The flow is generated by a divergence free zero mean vector field. The long…
Random perturbations of dynamical systems and diffusion processes with conservation laws
- Mathematics
- 2004
Abstract.In this paper we consider random perturbations of dynamical systems and diffusion processes with a first integral. We calculate, under some assumptions, the limiting behavior of the slow…
Noisy heteroclinic networks
- Environmental Science
- 2003
The influence of small noise on the dynamics of heteroclinic networks is studied, with a particular focus on noise-induced switching between cycles in the network. Three different types of switching…
Stochastic Differential Equations
- Mathematics
- 1985
We now return to the possible solutions X t (ω) of the stochastic differential equation
(5.1)
where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of…
Introduction to the Modern Theory of Dynamical Systems
- Mathematics
- 1998
self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with all areas of develop the interested fundamental tools and…
Ordinary Differential Equations.
- Mathematics
- 1958
together with the initial condition y(t0) = y0 A numerical solution to this problem generates a sequence of values for the independent variable, t0, t1, . . . , and a corresponding sequence of values…