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A basic and natural coupling between two probabilities on $\mathbb R^N$ isgiven by the Knothe-Rosenblatt coupling. It represents a multiperiod extensionof the quantile coupling and is simple to calculate numerically. We considerthe distance on $\mathcal P (\mathbb R^N)$ that is induced by considering thetransport costs associated to the Knothe-Rosenblatt coupling. We show that thisKnothe-Rosenblatt distance metrizes the adapted weak topology which is astochastic process version of the usual weak topology and plays an importantrole, e.g. concerning questions on stability of stochastic control andprobabilistic operations. We also establish that the Knothe-Rosenblatt distanceis a geodesic distance, give a Skorokhod representation theorem for the adaptedweak topology, and provide multi-dimensional versions of our results.

The Knothe-Rosenblatt distance and its induced topology