# 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.

## 5 Citations

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## References

SHOWING 1-10 OF 16 REFERENCES

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…

Tails of exit times from unstable equilibria on the line

- MathematicsJ. Appl. Probab.
- 2020

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

- Mathematics
- 2010

We consider the exit problem for small white noise perturbation of a smooth dynamical system on the plane in the neighbourhood 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

- Mathematics
- 1984

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

- 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…

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…

Random Perturbations of Dynamical Systems

- Mathematics, Physics
- 1984

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.-…