89 research outputs found
A Benamou-Brenier formulation of martingale optimal transport
We introduce a Benamou-Brenier formulation for the continuous-time martingale
optimal transport problem as a weak length relaxation of its discrete-time
counterpart. By the correspondence between classical martingale problems and
Fokker-Planck equations, we obtain an equivalent PDE formulation for which
basic properties such as existence, duality and geodesic equations can be
analytically studied, yielding corresponding results for the stochastic
formulation. In the one dimensional case, sufficient conditions for finiteness
of the cost are also given and a link between geodesics and porous medium
equations is partially investigated
Curvature bounds for configuration spaces
We show that the configuration space over a manifold M inherits many
curvature properties of the manifold. For instance, we show that a lower Ricci
curvature bound on M implies for the configuration space a lower Ricci
curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality,
gradient estimates and Wasserstein contraction. Moreover, we show that the heat
flow on the configuration space, or the infinite independent particle process,
can be identified as the gradient flow of the entropy.Comment: 34 page
Optimal Transport and Skorokhod Embedding
The Skorokhod embedding problem is to represent a given probability as the
distribution of Brownian motion at a chosen stopping time. Over the last 50
years this has become one of the important classical problems in probability
theory and a number of authors have constructed solutions with particular
optimality properties. These constructions employ a variety of techniques
ranging from excursion theory to potential and PDE theory and have been used in
many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts
from optimal mass transport. In analogy to the celebrated article of Gangbo and
McCann on the geometry of optimal transport, we establish a geometric
characterization of Skorokhod embeddings with desired optimality properties.
This leads to a systematic method to construct optimal embeddings. It allows
us, for the first time, to derive all known optimal Skorokhod embeddings as
special cases of one unified construction and leads to a variety of new
embeddings. While previous constructions typically used particular properties
of Brownian motion, our approach applies to all sufficiently regular Markov
processes.Comment: Substantial revision to improve the readability of the pape
- …