math.PR — Probability

The Wasserstein cost of the Girsanov Theorem

Contributed by Simon Coste

This is an expository note on the asymptotics of the Wasserstein distance between the empirical measure of infinite-dimensional random paths and their common distribution. We present the classical proof for these asymptotics in the case of the Wiener measure, unifying quantization arguments from Dereich and coauthors or Boissard and Le Gouic, and we adapt it for It\^o diffusions sampled through the Girsanov theorem. More precisely, let $\mathbb{Q}$ be the law of the solution of a stochastic differential equation with a bounded Lipschitz drift. Let $W_1,\ldots,W_n$ be independent Brownian paths, and let $Z_i$ be their Girsanov weights for $\mathbb{Q}$. The reweighted empirical measure $$\widehat\mu_n=\frac{\sum_{i=1}^n Z_i\delta_{W_i}}{\sum_{i=1}^n Z_i}$$ approximates $\mathbb{Q}$ when $n$ is large. We prove that the Wasserstein cost $W_p^p(\widehat\mu_n,\mathbb{Q})$ is of order $(\log n)^{-p/2}$, in expectation and almost surely, for every fixed $p\geqslant1$. In the Brownian case where $Z_i=1$ these results show that $\sqrt{\log n}W_p(\widehat\mu_n,\mathbb{Q})$ is asymptotically sandwiched between $\sqrt{2\lambda_d}$ and $4\sqrt{2\lambda_d}$, where $\lambda_d$ is the first eigenvalue of the Laplacian operator in the unit ball. We provide numerical experiments to estimate the (conjectured) limiting constant in dimension $d=1$.

Author comments: 16 pages. Full Lean formalization publicly available at gitlab.com/SimonCoste/wasserstein