Statistical Properties for Flows with Unbounded Roof Function, Including the Lorenz Attractor

Abstract

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors.We also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two. such as expansivity [9], the central limit theorem (CLT) [17], moment estimates [28], and exponential decay of correlations [6].Recently, [31] (see also previous work of [14]) gave an analytic proof of existence of geometric Lorenz attractors in the extended Lorenz modelA consequence of the current paper, in conjunction with [8], is that CLTs and moment estimates hold for these attractors. Exponential decay of correlations remains an open question: the proof in [6] relies on the existence of a smooth stable foliation for the flow, which holds for the classical Lorenz attractor [7] but not for the examples of [14,31]. (As far as the results on the CLT and moment estimates go, our results for the extended Lorenz model are completely analytic, whereas results for the classical Lorenz attractor rely on the computer-assisted proof in [35].)More generally, our results apply to all singular hyperbolic attractors for threedimensional flows. By [30], this incorporates all nontrivial robustly transitive attractors for three-dimensional flows (including nontrivial Axiom A attractors as well as classical and geometric Lorenz attractors). Our results apply also to codimension two singular hyperbolic attractors in arbitrary dimension.A standard step in the analysis of Lorenz attractors and singular hyperbolic attractors is to consider a suitable Poincaré map with good nonuniform hyperbolicity properties. The return time function (roof function) is generally unbounded due to the presence of steady-states for the flow. Due to this unboundedness, the original proof of the CLT for the classical Lorenz equations [17] is quite complicated, relying on an inducing scheme that involves phase-space exclusion arguments of the type in [12] to remove points that return too quickly to a vicinity of the steady-states.As we pointed out in an earlier unpublished preprint version of this paper, it is possible to give a much simpler proof of the CLT than the one in [17]. In this paper, we provide the details. As in [17], we prove also the functional version, namely the weak invariance principle (WIP). We also provide moment estimates as advertised in [28, Section 5.3] and our results apply to the general class of singular hyperbolic attractors in [8] (it is not clear that the argument in [17] applies in this generality). In addition, we prove iterated versions of the WIP and moment estimates as advertised in [18, Section 10.3]. By [18,19] it is therefore possible to prove homogenisation results wh...

Mixing properties and statistical limit theorems for singular hyperbolic flows without a smooth stable foliation

“…In the Axiom A case, the CLT and associated invariance principles are well-known [17,38,46] and we extend these results to general (codimension two) singular hyperbolic attractors. In particular, as described in Section 9.1, the (functional) CLT and related results follow from [13] using the results in this paper. Moreover, many strong limit laws are obtained for the associated Poincaré maps in Theorem 8.7.…”

“…When Λ contains equilibria, the CLT and its functional version still holds by [27] at least for geometric Lorenz attractors. As pointed out in [13], a simpler argument than in [27] applies in general situations where the roof function is unbounded and includes the entire class of singular hyperbolic attractors analysed in this paper. We refer to the introduction of [13] for a more comprehensive list of statistical limit laws, with precise statements, that can be obtained in this way.…”

Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

“…Remark 2. 12. In what follows it may be necessary to increase T 1 , leading to changes to f, τ , Ŵ and {S i } (and the constant C in Lemma 2.9).…”

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

“…However, the assumptions in [27] do not cover unbounded coefficients, while the common noise coefficient may depend on both the state of the solution and its law. In equation (11) considered below, the coefficient P only depends on the law of X, which simplifies the problem to some extent, but also allows us to use a different approach, where we treat the stochastic and the rough integrals in two separated steps. As it will become clear from the proofs in Section 4.4, there is no need to create a joint rough path (or rough driver).…”