Dimension-free Wasserstein contraction for nonlinear filters

For a class of partially observed diffusions, sufficient conditions are given for the map from initial condition of the signal to filtering distribution to be contractive with respect to Wasserstein distances, with rate which has no dependence on the dimension of the state-space and is stable under tensor products of the model. The main assumptions are that the signal has affine drift and constant diffusion coefficient, and that the likelihood functions are log-concave. Contraction estimates are obtained from an $h$-process representation of the transition probabilities of the signal reweighted so as to condition on the observations.