Abstract.We study the Fokker-Planck equation as the many-particle limit of a stochastic particle system on one hand and as a Wasserstein gradient flow on the other. We write the path-space rate functional, which characterises the large deviations from the expected trajectories, in such a way that the free energy appears explicitly. Next we use this formulation via the contraction principle to prove that the discrete time rate functional is asymptotically equivalent in the Gamma-convergence sense to the functional derived from the Wasserstein gradient discretization scheme.Mathematics Subject Classification. 35A15, 5Q84.
We prove that, for the case of Gaussians on the real line, the functional derived by a time discretization of the diffusion equation as entropic gradient flow is asymptotically equivalent to the rate functional derived from the underlying microscopic process. This result strengthens a conjecture that the same statement is actually true for all measures with second finite moment. C 2012 American Institute of Physics.
We study general geometric properties of cone spaces, and we apply them on the Hellinger-Kantorovich space (M(X), HK α,β ). We exploit a two-parameter scaling property of the Hellinger-Kantorovich metric HK α,β , and we prove the existence of a distance SHK α,β on the space of Probability measures that turns the Hellinger-Kantorovich space (M(X), HK α,β ) into a cone space over the space of probabilities measures (P(X), SHK α,β ). We provide a two parameter rescaling of geodesics in (M(X), HK α,β ), and for (P(X), SHK α,β ) we obtain a full characterization of the geodesics. We finally prove finer geometric properties, including local-angle condition and partial K-semiconcavity of the squared distances, that will be used in a future paper to prove existence of gradient flows on both spaces.
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