989 research outputs found
On the convexity of the entropy along entropic interpolations
Convexity properties of the entropy along displacement interpolations are
crucial in the Lott-Sturm-Villani theory of lower bounded curvature of geodesic
measure spaces. As discrete spaces fail to be geodesic, an alternate analogous
theory is necessary in the discrete setting. Replacing displacement
interpolations by entropic ones allows for developing a rigorous calculus, in
contrast with Otto's informal calculus. When the underlying state space is a
Riemannian manifold, we show that the first and second derivatives of the
entropy as a function of time along entropic interpolations are expressed in
terms of the standard Bakry-\'Emery operators and . On the
other hand, in the discrete setting new operators appear. Our approach is
probabilistic; it relies on the Markov property and time reversal. We
illustrate these calculations by means of Brownian diffusions on manifolds and
random walks on graphs. We also give a new unified proof, covering both the
manifold and graph cases, of a logarithmic Sobolev inequality in connection
with convergence to equilibrium
Girsanov theory under a finite entropy condition
This paper is about Girsanov's theory. It (almost) doesn't contain new
results but it is based on a simplified new approach which takes advantage of
the (weak) extra requirement that some relative entropy is finite. Under this
assumption, we present and prove all the standard results pertaining to the
absolute continuity of two continuous-time processes with or without jumps. We
have tried to give as much as possible a self-contained presentation. The main
advantage of the finite entropy strategy is that it allows us to replace
martingale representation results by the simpler Riesz representations of the
dual of a Hilbert space (in the continuous case) or of an Orlicz function space
(in the jump case)
A large deviation approach to optimal transport
A probabilistic method for solving the Monge-Kantorovich mass transport
problem on is introduced. A system of empirical measures of independent
particles is built in such a way that it obeys a doubly indexed large deviation
principle with an optimal transport cost as its rate function. As a
consequence, new approximation results for the optimal cost function and the
optimal transport plans are derived. They follow from the Gamma-convergence of
a sequence of normalized relative entropies toward the optimal transport cost.
A wide class of cost functions including the standard power cost functions
enter this framework
A survey of the Schr\"odinger problem and some of its connections with optimal transport
This article is aimed at presenting the Schr\"odinger problem and some of its
connections with optimal transport. We hope that it can be used as a basic
user's guide to Schr\"odinger problem. We also give a survey of the related
literature. In addition, some new results are proved.Comment: To appear in Discrete \& Continuous Dynamical Systems - Series A.
Special issue on optimal transpor
Characterization of the optimal plans for the Monge-Kantorovich transport problem
We present a general method, based on conjugate duality, for solving a convex
minimization problem without assuming unnecessary topological restrictions on
the constraint set. It leads to dual equalities and characterizations of the
minimizers without constraint qualification. As an example of application, the
Monge-Kantorovich optimal transport problem is solved in great detail. In
particular, the optimal transport plans are characterized without restriction.
This characterization improves the already existing literature on the subject.Comment: 39 page
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