  Stein's method for measuring convergence to a continuous target distribution
relies on an operator characterizing the target and Stein factor bounds on the
solutions of an associated differential equation. While such operators and
bounds are readily available for a diversity of univariate targets, few
multivariate targets have been analyzed. We introduce a new class of
characterizing operators based on Ito diffusions and develop explicit
multivariate Stein factor bounds for any target with a fast-coupling Ito
diffusion. As example applications, we develop computable and
convergence-determining diffusion Stein discrepancies for log-concave,
heavy-tailed, and multimodal targets and use these quality measures to select
the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare
random and deterministic quadrature rules, and quantify bias-variance tradeoffs
in approximate MCMC. Our results establish a near-linear relationship between
diffusion Stein discrepancies and Wasserstein distances, improving upon past
work even for strongly log-concave targets. The exposed relationship between
Stein factors and Markov process coupling may be of independent interest.
