I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ and $l \in \mathbb{Z}^{+}$ such that for any $\phi, \psi \in C_2^{\infty}(X)$ and for any $t \geq 0$ one has

$|(g_t\phi,\psi)-\int_{X}\psi d\mu \int_{X}\phi d\mu| \leq e^{-\gamma t}||\phi||_{l,2}||\psi||_{l,2}$ where $||h||_{l,p} =\sum_{|\alpha|\leq l}||D^{\alpha}h||_{p}$ and $C_2^{\infty}(X) = \{h \in C^{\infty}(X) : ||h||_{l,2} < \infty \ \ \text{for any}\ l \in \mathbb{Z}+\}$.

My question is if there is any analogue of this mixing and rate of mixing in infinite volume surfaces other than lattices w.r.t the Patterson Sullivan measure. If yes, any reference will be very useful.