- Stochastic Stability for Flows with Smooth Invariant Measures
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- AMS :: Bulletin of the American Mathematical Society

We study the notion of stochastic stability with respect to diffusive perturbations for flows with smooth invariant measures. We investigate the question fully for non-singular flows on the circle. We also show that volume-preserving flows are stochastically stable with respect to perturbations that are associated with homogeneous diffusions.

The notion of stochastic stability of the invariant measures of a continuous dynamical system was formulated as early as by Pontryagin, Andronov and Vitt [ 17 ] and Bernstein [ 4 ]. They solved the problem completely for flows on the real line that possess a globally attracting set of equilibrium points.

## Stochastic Stability for Flows with Smooth Invariant Measures

Such flows of course are the gradients of some potential function. In the case of a unique equilibrium, stochastic stability is proved and a formula for the perturbed invariant measure was produced in terms of the quasi-potential.

Stochastic stability for flow on a compact manifold was discussed in [ 14 ], [ 16 ] and extended to the case that the maximal invariant set of the flow consists of a finite number of hyperbolic basic sets. Since then, it seems that very little attention has been given to this problem.

In the meantime, stochastic stability of invariant measures has been a topic of continuing interest and surprising difficulty for discrete time dynamical systems theory. The main directions where progress has been made are: In this note we study the question of stochastic stability for the overlooked case of non-singular flow on the circle and for a volume preserving flow on a manifold.

We will let h: All equations in this note will be assumed to be written in local coordinates.

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Consider the stochastic differential equation. For general background on diffusions on a manifold see for instance [ 12 , Chp. A class of perturbations refers to a collection of equations of the form 1. By stochastic stability with respect to a class of small stochastic perturbations, we mean that each flow defined by 1. It is well-known that all zero-noise limit measures are invariant under h [ 14 ]. See also [ 6 ] for a recent related work using zero-noise limits.

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We can associate with 1. Probability density functions evolve under 1.

## AMS :: Bulletin of the American Mathematical Society

However, this is a stronger assumption than we need and may impose unnecessary constraints. In one dimension one can solve directly for u , but in higher dimensions a solution seems unlikely. In this section we point out that stochastic stability with respect to standard diffusion follows easily from known results in two extreme cases, volume preserving flows and gradient flows. First suppose that h is volume preserving and the perturbation is by standard diffusion.

According to [ 12 , Theorem 5. This gives us the following:.

If h is volume-preserving and the volume, m, is ergodic, then m is stochastically stable under standard diffusion. Secondly, consider the case that h is a gradient flow, i. This case is the direct generalization of the early results [ 17 ], [ 4 ] and a special case of result in [ 11 ]. In [ 12 , Theorem 5. In the simplest case we have:.

If h is a gradient flow with potential H, and H has a unique minimum point, then the delta measure on this point is stochastically stable. If there is not a unique minimum, then the delta measure at each of the minimum points is ergodic and any linear combination of them is nonergodic invariant. However, there is also an elementary proof of these facts in this case. Rearranging and successively integrating by parts we obtain:. Then using the Schwartz inequality, we obtain:. An elementary proof of this also exists in this case.

If volume is also ergodic for the flow defined by h , then the constant multiples of the volume are the only absolutely continuous invariant measures.

In particular, exit problems, metastable states, optimal stabilization, and asymptotics of stationary distributions are also carefully considered. The authors' main tools are the large deviation theory the centred limit theorem for stochastic processes, and the averaging principle - all presented in great detail. The results allow for explicit calculations of the asymptotics of many interesting characteristics of the perturbed system. Most of the results are closely connected with PDEs, and the authors' approach presents a powerful method for studying the asymptotic behavior of the solutions of initial-boundary value problems for corresponding PDEs.

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Freidlin , Alexander D.