Continuity of Lyapunov exponents for stochastic differential equations

TL;DR

This paper proves the continuity and regularity properties of Lyapunov exponents for stochastic differential equations under various convergence and regularity conditions, filling a gap in the continuous-time stochastic dynamics literature.

Contribution

It establishes the first known results on the continuity and Lipschitz/Hölder regularity of Lyapunov exponents for SDEs under coefficient convergence and regularity assumptions.

Findings

Top Lyapunov exponent is continuous under almost uniform coefficient convergence.

All Lyapunov exponents are continuous if invariant measures exist and coefficients converge pointwise.

Lyapunov exponents are Lipschitz continuous under strict monotonicity and $L^{p,p}$ norm conditions.

Abstract

For non-autonomous linear stochastic differential equations (SDEs), we establish that the top Lyapunov exponent is continuous if the coefficients "almost" uniformly converge. For autonomous SDEs, assuming the existence of invariant measures and the convergence of coefficients and their derivatives in pointwise sense, we get the continuity of all Lyapunov exponents. Furthermore, we demonstrate that for autonomous SDEs with strict monotonicity condition, all Lyapunov exponents are Lipschitz continuous with respect to the coefficients under the $L^{p,p}$ norm ($p>2$). Similarly, the H\"older continuity of Lyapunov exponents holds under weaker regularity conditions. It seems that the continuity of Lyapunov exponents has not been studied for SDEs so far, in spite that there are many results in this direction for discrete-time dynamical systems.