Let \(M\) be a manifold, and \(f: M\to M\) be a \(C^1\) diffeomorphism with a nontrivial uniformly hyperbolic basic set \(X\subset M\) (meaning mainly that it obeys axiom A and possesses a dense orbit with moreover some maximality). Let also \(Y\) be an equilibrium measure corresponding to a Hölder potential, and \(\varphi: X\to \mathbb{R}^d\) be a mean zero Hölder observable. The main result asserts that, as \(N\to\infty\),
\[ \sum^N_{n=1} \varphi_0 f^n= W_N+O(N^{{2d+3\over 4d+ 7}+\varepsilon}), \]
almost everywhere for any \(\varepsilon> 0\), \((W_t)\) being some \(d\)-dimensional Brownian motion.
This result is extended then to a non-uniformly hyperbolic setting, under the assumption that the return time function (in \(X\)) lies in \(L^p\), for some \(p> 2\).
This general result is applied to a planar periodic Lorentz gas, studied by Sinaï, i.e. a 3-dimensional flow \((q_t,v_t)\in (\mathbb{R}^2\setminus E)\times\mathbb{S}^1\), where \(E\) is a periodic array of disjoint convex \(C^3\) regions, such that the time between collisions with \(\partial E\) is uniformly bounded: then there exists a nondegenerate 2-dimensional Brownian motion \((W_t)\) , such that for almost every initial condition, \(q_t= W_t+ O(t^{7/15+\varepsilon})\) as \(t\to\infty\), for any \(\varepsilon> 0\).
An almost sure functional law of the iterated logarithm and refinement of it, for example, can also be deduced from the general main result got by the authors.
The probabilistic proof proceeds by a careful study and use of a mixing property, and by blocking arguments, to reduce the case of sums of weakly dependent random variables to the martingale case.