1,753 research outputs found
The Monge problem for supercritical Mane potentials on compact manifolds
We prove the existence of an optimal map for the Monge problem when the cost
is a supercritical Mane potential on a compact manifold. Supercritical Mane
potentials form a class of costs which generalize the Riemannian distances. We
describe new links between this transportation problem and viscosity
subsolutions of the Hamilton-Jacobi equation
Entropic and displacement interpolation: a computational approach using the Hilbert metric
Monge-Kantorovich optimal mass transport (OMT) provides a blueprint for
geometries in the space of positive densities -- it quantifies the cost of
transporting a mass distribution into another. In particular, it provides
natural options for interpolation of distributions (displacement interpolation)
and for modeling flows. As such it has been the cornerstone of recent
developments in physics, probability theory, image processing, time-series
analysis, and several other fields. In spite of extensive work and theoretical
developments, the computation of OMT for large scale problems has remained a
challenging task. An alternative framework for interpolating distributions,
rooted in statistical mechanics and large deviations, is that of Schroedinger
bridges (entropic interpolation). This may be seen as a stochastic
regularization of OMT and can be cast as the stochastic control problem of
steering the probability density of the state-vector of a dynamical system
between two marginals. In this approach, however, the actual computation of
flows had hardly received any attention. In recent work on Schroedinger bridges
for Markov chains and quantum evolutions, we noted that the solution can be
efficiently obtained from the fixed-point of a map which is contractive in the
Hilbert metric. Thus, the purpose of this paper is to show that a similar
approach can be taken in the context of diffusion processes which i) leads to a
new proof of a classical result on Schroedinger bridges and ii) provides an
efficient computational scheme for both, Schroedinger bridges and OMT. We
illustrate this new computational approach by obtaining interpolation of
densities in representative examples such as interpolation of images.Comment: 20 pages, 7 figure
General equilibrium analysis in ordered topological vector spaces
The second welfare theorem and the core-equivalence theorem have been proved to be fundamental tools for obtaining equilibrium existence theorems, especially in an infinite dimensional setting. For well-behaved exchange economies that we call proper economies, this paper gives (minimal) conditions for supporting with prices Pareto optimal allocations and decentralizing Edgeworth equilibrium allocations as non-trivial equilibria. As we assume neither transitivity nor monotonicity on the preferences of consumers, most of the existing equilibrium existence results are a consequence of our results. A natural application is in Finance, where our conditions lead to new equilibrium existence results, and also explain why some financial economies fail to have equilibrium.Equilibrium; Valuation equilibrium; Pareto-optimum; Edgeworth equilibrium; Properness; ordered topological vector spaces; Riesz-Kantorovich formula; sup-convolution
Optimal transportation for a quadratic cost with convex constraints and applications
We prove existence of an optimal transport map in the Monge-Kantorovich
problem associated to a cost which is not finite everywhere, but
coincides with if the displacement belongs to a given convex
set and it is otherwise. The result is proven for satisfying
some technical assumptions allowing any convex body in and any convex
polyhedron in , . The tools are inspired by the recent
Champion-DePascale-Juutinen technique. Their idea, based on density points and
avoiding disintegrations and dual formulations, allowed to deal with
problems and, later on, with the Monge problem for arbitrary norms
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