Markovian reduction and exponential mixing in total variation for random dynamical systems

The paper proves exponential mixing in total variation for discrete-time random dynamical systems by (i) reducing to a Markov process on an extended space and invoking an existing exponential contraction in the dual-Lipschitz metric, and (ii) converting this to total-variation contraction for finite blocks via an image-of-measures theorem that yields Lipschitz densities for the projections; see the statement of Theorem 3.1 and its proof outline, including the reduction and the use of Theorem 4.1 and Corollary 4.2 in the appendix . For Markovian noise, their Section 1 frames the Rec/Cou scheme and proves exponential mixing for the extended chain under dissipativity, controllability, and local minorisation/recurrence hypotheses (Theorem 1.2) . The candidate solution proposes a different Doeblin–Dobrushin coupling for an N-step kernel on an augmented, compact state space, which would be fine in spirit, but it contains two critical faults: (1) it incorrectly deduces a uniform pointwise lower bound on the one-step density on a small ball from a uniform lower bound on its integral over that ball (the inequality direction is reversed), and (2) its one-step binder is constructed only for y in a small set A, yet the binding step is applied at a time when the noise is not ensured to lie in A. These gaps prevent the claimed uniform Doeblin minorisation. The paper’s argument does not rely on such a uniform pointwise lower bound and is logically consistent with the stated hypotheses.