Martin Huesmann

Optimal transport from Lebesgue to Poisson

This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2. The cost will be defined in terms of an arbitrary increasing function of the distance. The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.

From random walks to rough paths

Donskers invariance principle is shown to hold for random walks in rough path topology. As an application, we obtain Donsker-type weak limit theorems for stochastic integrals and differential equations.

Root to Kellerer

We revisit Kellerer’s Theorem, that is, we show that for a family of real probability distributions (μ t )t ∈ [0, 1] which increases in convex order there exists a Markov martingale (S t )t ∈ [0, 1] s.t. S t ∼ μ t .

Optimal transport between random measures

We study couplings

Self-intersection of optimal geodesics

q^\bullet

Monotonicity preserving transformations of MOT and SEP

of two equivariant random measures

Curvature bounds for configuration spaces

\lambda^\bullet

Transport cost estimates for random measures in dimension one

and

Optimal transport and Skorokhod embedding

\mu^\bullet

Pathwise superreplication via Vovk’s outer measure

on a Riemannian manifold