Moment stability and large deviations for random dynamical systems on non-compact manifolds

TL;DR

This paper investigates the large deviations and stability properties of random dynamical systems on non-compact manifolds, extending known results from compact cases and establishing new growth conditions and limit theorems.

Contribution

It extends the eigenvalue characterization of Lyapunov exponents to non-compact manifolds under growth conditions and proves new limit theorems for finite-time Lyapunov exponents.

Findings

Eigenvalue characterization holds under growth conditions on non-compact manifolds.

Central limit theorem established for finite-time Lyapunov exponent.

Moderate deviation estimates derived involving second derivative of moment exponent.

Abstract

The rate function for large deviations of the finite time Lyapunov exponent for the derived process in TM corresponding to a stochastic differential equation in M is related, via the Gartner-Ellis theorem, to the p-th moment Lyapunov exponent. When M is compact there is a characterization of the p-th moment Lyapunov exponent in terms of an eigenvalue problem for an associated differential operator acting on functions on the unit sphere bundle SM. For the non-compact case, we formulate growth conditions which, together with standard assumptions of hypoellipticity and positivity, ensure that the eigenvalue characterization remains valid so long as the corresponding eigenfunction is restricted to be in a suitable function space. There is a central limit theorem for finite time Lyapunov exponent, and there are moderate deviation estimates for the finite time Lyapunov exponent, both involving the second derivative of the moment exponent function at 0.