Strong and weak quantitative estimates in slow-fast diffusions using filtering techniques

The behavior of slow-fast diffusions as the separation of scale diverges is a well-studied problem in the literature. In this short paper, we revisit this problem and obtain a new proof of existing strong quantitative convergence estimates (in particular, $L^2$ estimates) and weak convergence estimates in terms of $n$ (the parameter associated with the separation of scales). In particular, we obtain the rate of $n^{-\frac{1}{2}}$ in the strong convergence estimates and the rate of $n^{-1}$ for weak convergence estimate which are already known to be optimal in the literature. We achieve this using nonlinear filtering theory where we represent the evolution of fast diffusion in terms of its conditional distribution given the slow diffusion. We then use the well-known Kushner-Stratanovich equation which gives the evolution of the conditional distribution of the fast diffusion given the slow diffusion and establish that this conditional distribution approaches the invariant measure of the ``frozen" diffusion (obtained by freezing the slow variable in the evolution equation of the fast diffusion). At the heart of the analysis lies a key estimate of a weighted Lipschitz distance like function between a generic one-parameter family of measures and the family of unique invariant measures (of the ``frozen" diffusion parametrized by a path). This estimate is in terms of the operator norm of the dual of the infinitesimal generator of the ``frozen" diffusion.