Modulated logarithmic Sobolev inequalities and generation of chaos

2023/07/14 by Rosenzweig, Matthew, Serfaty, Sylvia · 2 citations
#35Q70 #35Q82 #39B62 #82B40 #82C22 #82C40 #94A17 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2307.07587

Abstract

We consider mean-field limits for overdamped Langevin dynamics of N particles with possibly singular interactions. It has been shown that a modulated free energy method can be used to prove the mean-field convergence or propagation of chaos for a certain class of interactions, including Riesz kernels. We show here that generation of chaos, i.e. exponential-in-time convergence to a tensorized (or iid) state starting from a nontensorized one, can be deduced from the modulated free energy method provided a uniform-in-N "modulated logarithmic Sobolev inequality" holds. Proving such an inequality is a question of independent interest, which is generally difficult. As an illustration, we show that uniform modulated logarithmic Sobolev inequalities can be proven for a class of situations in one dimension.

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• A sharp commutator estimate for all Riesz modulated energies
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