There are many results giving bounds on the $\beta$-mixing coefficients for diffusion processes (see Proposition 1 in [1] for example). For an homogeneous and stationary process, it means upper bounding the following quantity
$$ \beta(s) = \sup_{t} d_{TV}\left( \mathcal{L}(X_t,X_{t+s}), \mathcal{L}(X_t)\otimes \mathcal{L}(X_{t+s}) \right), $$
where $d_{TV}$ is the total variation distance and $\mathcal{L}(Y)$ denotes the distribution of the random variable $Y$. It is a way to measure "how far" are $X_t$ and $X_{t+s}$ from being independent. For some reason, I'm looking at another mixing coefficient which basically uses the Kullback-Leibler divergence $K$ in place of the total variation distance, i.e.
$$ I(s) = \sup_{t} K\left( \mathcal{L}(X_t,X_{t+s}) || \mathcal{L}(X_t)\otimes \mathcal{L}(X_{t+s}) \right). $$
This is related to the "coefficient of information" and "information regularity" given in [2]. From the Pinsker's inequality we have $\beta(s)\leq \sqrt{I(s)/2}$ so it makes sense to start for cases where we already know a bound on $\beta(s)$. I have already computed this quantity for the Brownian motion and Ornstein-Ulhenbeck processes and we get what we expect but we have access to the densities which are gaussians, so they are quite specific situations.
Some papers get somehow close to that direction, looking at the decay of entropy (see Theorem 2.2 in [3] for instance), but they never really tackle this specific question of bounding the coefficient $I(s)$.
It seems no one ever took interest in this coefficient but maybe I'm just missing some references. Maybe it can be done without much difficulty with existing results, maybe it can't be done. Any indication or help would be greatly appreciated.
Thanks in adavnce.
[1] Pardoux, E., & Veretennikov, Y. (2001). On the Poisson equation and diffusion approximation. I. The Annals of Probability, 29(3), 1061-1085.
[2] Richard C. Bradley. "Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions." Probab. Surveys 2 107 - 144, 2005. https://doi.org/10.1214/154957805100000104
[3] Dominique Bakry, Patrick Cattiaux, Arnaud Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. Journal of Functional Analysis, Volume 254, Issue 3, 2008, Pages 727-759. https://doi.org/10.1016/j.jfa.2007.11.002.