Adapted Wasserstein distance between continuous Gaussian processes

Adapted Wasserstein distance is a generalization of the classical Wasserstein distance for stochastic processes. It captures not only the spatial information but also the temporal information induced by the processes. In this talk, I will focus on the adapted Wasserstein distance between continuous Gaussian processes. An explicit formula in terms of their canonical representations will be given. These results cover rough processes such as fractional Brownian motions and fractional Ornstein--Uhlenbeck processes. If time permits, I will also show that the optimal coupling between two 1D additive fractional SDE is driven by the synchronous coupling of the noise.

We introduce a 'causal factorization' as an infinite dimensional Cholesky decomposition on Hilbert spaces. This naturally bridges the probabilistic notion 'causal transport' and the algebraic object 'nest algebra'.  Such a factorization is closely related to the (non)canonical representation of Gaussian processes which is of independent interest. This talk is based on a work-in-progress with Fang Rui Lim.