decay of correlations and statistical limit laws are by now classical for uniformly hyperbolic diffeomorphisms [@Bowen75; @Ruelle78; @Sinai72]. In particular, if $f:X\to X$ is uniformly hyperbolic and $\mu$ is an equilibrium measure corresponding to a Hölder potential, then the correlation function $\int_X v \,w\circ f^n\,d\mu-\int_X v\,d\mu\int_X w\,d\mu$ decays exponentially quickly as $n\to\infty$ for Hölder observables $v,w:X\to{{\mathbb R}}$.

Young [@Young98] extended this result to a large class of nonuniformly hyperbolic systems, including planar dispersing billiards, and also established polynomial decay of correlations for systems that are more slowly mixing [@Young99]. The results were then shown to be optimal by Sarig [@Sarig02] and Gouëzel [@Gouezel-sharp]. Turning to the infinite measure case, the fundamental difference is that $\lim_{n\to\infty}\int_X v \,w\circ f^n\,d\mu=0$ for reasonably well-behaved observables $v,w$. Hence there arises the preliminary problem of showing that $a_n \int_X v \,w\circ f^n\,d\mu\to
\int_X v\,d\mu\int_X w\,d\mu$ for a suitable normalising sequence $a_n\to\infty$ and for sufficiently well-behaved $v,w$. The secondary problem is to estimate the speed of convergence (rate of mixing). Definitive results on the preliminary problem, and first results on the rate of mixing, were obtained recently in [@MT12].

In the continuous time situation, decay of correlations is less well understood. Exponential decay of correlations has 