Wasserstein approximations of the L\'evy area random walk via polynomial perturbations of Gaussian distributions

We construct a coupling between the random walk composed of L\'evy area increments from a $d$-dimensional Brownian motion and a random walk composed of quadratic polynomials of Gaussian random variables. This coupling construction is used to produce a new pathwise approximation scheme for stochastic differential equations in the preprint [Flint-Lyons-2015]. The coupling arguments of the present paper are based extensively on the recent coupling results of Davie concerning a multidimensional variant of the Koml\'os-Major-Tusn\'ady theorem and Wasserstein estimates for polynomial perturbations of Gaussian measures.