Long exit times near a repelling equilibrium

@article{Bakhtin2019LongET,
  title={Long exit times near a repelling equilibrium},
  author={Yuri Bakhtin and Hong-Bin Chen},
  journal={arXiv: Probability},
  year={2019}
}
For a smooth vector field in a neighborhood of a critical point with all positive eigenvalues of the linearization, we consider the associated dynamics perturbed by white noise. Using Malliavin calculus tools, we obtain polynomial asymptotics for probabilities of atypically long exit times in the vanishing noise limit. 
View PDF on arXiv
5 Citations
Background Citations
3

5 Citations

Citation Type

Citation Type

Atypical exit events near a repelling equilibrium
We consider exit problems for small white noise perturbations of a dynamical system generated by a vector field, and a domain containing a critical point with all positive eigenvalues of…
THE CONCENTRATION OF LIMITING INVARIANT MEASURES FOR STOCHASTIC DYNAMICAL SYSTEMS
In this paper, we study the concentration phenomena of invariant measures for stochastic differential equations with locally Lipschitz continuous coefficients and more than one ergodic state in Rd.…
THE CONCENTRATION OF INVARIANT MEASURES FOR STOCHASTIC DYNAMICAL SYSTEMS WITH LOCALLY LIPSCHITZ CONTINUOUS COEFFICIENTS IN R
In this paper, we study the concentration phenomena of the invariant measures for stochastic differential equations with locally Lipschitz continuous coefficients and more than one ergodic state in…
The concentration of zero-noise limits of invariant measures for stochastic dynamical systems
In this paper, we study concentration phenomena of zero-noise limits of invariant measures for stochastic differential equations defined on Rd with locally Lipschitz continuous coefficients and more…
Anti-concentration for subgraph counts in random graphs
Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational work…

References

SHOWING 1-10 OF 16 REFERENCES
Malliavin calculus approach to long exit times from an unstable equilibrium
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…
Tails of exit times from unstable equilibria on the line
TLDR
A revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus is given and precise vanishing noise asymptotics are obtained for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
Normal forms approach to diffusion near hyperbolic equilibria
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…
The exit distributions for small random perturbations of dynamical systems with a repulsive type stationary point
The stochastic differential equations with a small parameter e, which for e=0 degenerate into a dynamical system with a repulsive type stationary point, are considered. The asymptotic behavior of the…
Small noise limit for diffusions near heteroclinic networks
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…
Scaling limit for escapes from unstable equilibria in the vanishing noise limit: Nontrivial Jordan block case
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…
ON THE EXIT LAW FROM SADDLE POINTS
Random Perturbations of Dynamical Systems
1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.-…
RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS (Grundlehren der mathematischen Wissenschaften, 260)
INTRODUCTION TO THE MODERN THEORY OF DYNAMICAL SYSTEMS (Encyclopaedia of Mathematics and its Applications 54)
...
...