A regularity method for lower bounds on the Lyapunov exponent for stochastic differential equations

Abstract

We put forward a new method for obtaining quantitative lower bounds on the top Lyapunov exponent of stochastic differential equations (SDEs). Our method combines (i) an (apparently new) identity connecting the top Lyapunov exponent to a Fisher information-like functional of the stationary density of the Markov process tracking tangent directions with (ii) a novel, quantitative version of Hörmander's hypoelliptic regularity theory in an L 1 framework which estimates this (degenerate) Fisher information from below by an W 1,s loc Sobolev norm. This method is applicable to a wide range of systems beyond the reach of currently existing mathematically rigorous methods. As an initial application, we prove the positivity of the top Lyapunov exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we prove that this class includes the Lorenz 96 model in any dimension, provided the additive stochastic driving is applied to any consecutive pair of modes.

Lower bounds on the Lyapunov exponents of stochastic differential equations

“…In view of the sign-indefinite formula (2.1) and its time-infinitesimal version, the Furstenberg-Khasminskii formula, it is reasonable to hope that a time-infinitesimal version of Proposition 2.4 might exist. The authors establish such a formula in our recent work [16].…”

“…In this article we will review existing work and our recent contributions [16,21] in proving that a given system of interest modeled by a stochastic differential equation is chaotic as in high sensitivity to initial conditions for trajectories initiated at Lebesgue-typical points in phase space. The specific systems we apply our methods to are the Lorenz-96 system [67] and Galerkin truncations of the 2d Navier-Stokes equations in a rectangular, periodic box (of any aspect ratio), provided they are subjected to sufficiently strong stochastic forcing 1 (equivalently, sufficiently weak damping) and are sufficiently high dimensional.…”

“…Section 2 concerns formulae of Lyapunov exponents through the stationary statistics of tangent directions and contains both classical results and our recent results from [16] which connects Lyapunov exponents to a certain Fisher information-type quantity. We discuss in Section 3 how to connect the Fisher information to regularity using ideas from hypoellipticity theory (also original work from [16]), and in Section 4 we discuss applications to a class of weakly-driven, weakly-dissipated SDE with bilinear nonlinear drift term (original work in [16] for Lorenz-96 and for Galerkin Navier-Stokes in [21]). In Section 5 we briefly discuss our earlier related work on Lagrangian chaos in the (infinite-dimensional) stochastic Navier-Stokes equations [17].…”

“…For simplicity, in this note we restrict our attention to SDE on R n , however, our more general results apply to SDE posed on orientable, geodesically complete, smooth manifolds; see [16]. Let X 0 , X 1 , .…”

“…Proposition 2.7 (Theorem A in [16]). Assume (x t , v t ) has a unique stationary measure ν with density f = dν dq on SR n .…”