Modulated logarithmic Sobolev inequalities and generation of chaos

Matthew Rosenzweig; Sylvia Serfaty

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arXiv · 2307.07587

Modulated logarithmic Sobolev inequalities and generation of chaos

Matthew Rosenzweig·Sylvia Serfaty

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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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Matthew Rosenzweig, Sylvia Serfaty. 2023-07-14. Modulated logarithmic Sobolev inequalities and generation of chaos. https://arxiv.org/abs/2307.07587

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