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Author: Laschos, Vaios ×

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    Gradient flow structure for McKean-Vlasov equations on discrete spaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Erbar, Matthias; Fathi, Max; Laschos, Vaios; Schlichting, André
    In this work, we show that a family of non-linear mean-field equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity.
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    Geometric properties of cones with applications on the Hellinger-Kantorovich space, and a new distance on the space of probability measures
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Laschos, Vaios; Mielke, Alexander
    By studying general geometric properties of cone spaces, we prove the existence of a distance on the space of Probability measures that turns the Hellinger--Kantorovich space into a cone space over the space of probabilities measures. Here we exploit a natural two-parameter scaling property of the Hellinger-Kantorovich distance. For the new space, we obtain a full characterization of the geodesics. We also provide new geometric properties for the original space, including a two-parameter rescaling and reparametrization of the geodesics, local-angle condition and some partial K-semiconcavity of the squared distance, that it will be used in a future paper to prove existence of gradient flows.
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    Large deviations for empirical measures generated by Gibbs measures with singular energy functionals
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dupuis, Paul; Laschos, Vaios; Ramanan, Kavita
    We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on n-particle configurations, each of which is defined in terms of an inverse temperature bn and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (bn)/(n) ® ¥, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed bn =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in ``A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in ``First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.