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The Wasserstein cost of the Girsanov Theorem

Primarily human-written textHuman understanding: all partsmath.PR — Probabilitymath.ST — Statistics Theory

Contributed by Simon Coste ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

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 Q\mathbb{Q} be the law of the solution of a stochastic differential equation with a bounded Lipschitz drift. Let W1,…,WnW_1,\ldots,W_n be independent Brownian paths, and let ZiZ_i be their Girsanov weights for Q\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 Q\mathbb{Q} when nn is large. We prove that the Wasserstein cost Wpp(μ^n,Q)W_p^p(\widehat\mu_n,\mathbb{Q}) is of order (log⁡n)−p/2(\log n)^{-p/2}, in expectation and almost surely, for every fixed p⩾1p\geqslant1. In the Brownian case where Zi=1Z_i=1 these results show that log⁡nWp(μ^n,Q)\sqrt{\log n}W_p(\widehat\mu_n,\mathbb{Q}) is asymptotically sandwiched between 2λd\sqrt{2\lambda_d} and 42λd4\sqrt{2\lambda_d}, where λd\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=1d=1.

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

Provenance statement

Proof ideas obtained with interactions with GPT-6-Astra. The writing was done by the human author. Some less important parts (the appendix and the related work section) was written by GPT-Astra-6. Numerics was designed by the human author, then implemented by Grok-4.7. The Lean formalization was generated by rounds of audit and development piloted by the human author and implemented by GPT-Astra-6, Grok-4.7, Fable 5.1.

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Version history

  1. v1 · 16 pagesSubmitted by Simon Coste16 pages. Full Lean formalization publicly available at gitlab.com/SimonCoste/wassersteinCurrentNov 12, 2026