Free energy Wasserstein gradient flow and their particle counterparts: toy model, (degenerate) PL inequalities and exit times
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912659264241664 |
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| author | Monmarché, Pierre |
| author_facet | Monmarché, Pierre |
| contents | In finite dimension, the long-time and metastable behavior of a gradient flow perturbated by a small Brownian noise is well understood. A similar situation arises when a Wasserstein gradient flow over a space of probability measure is approximated by a system of mean-field interacting particles, but classical results do not apply in these infinite-dimensional settings. This work is concerned with the situation where the objective function of the optimization problem contains an entropic penalization, so that the particle system is a Langevin diffusion process. We consider a very simple class of models, for which the infinite-dimensional behavior is fully characterized by a finite-dimensional process. The goal is to have a flexible class of benchmarks to fix some objectives, conjectures and (counter-)examples for the general situation. Inspired by the systematic study of these toy models, one application is presented on the continuous Curie-Weiss model in a symmetric double-well potential. We show that, at the critical temperature, although the $N$-particle Gibbs measure does not satisfy a uniform-in-$N$ standard log-Sobolev inequality (the optimal constant growing like $\sqrt{N}$), it does satisfy a more general Lojasiewicz inequality uniformly in $N$, inducing uniform polynomial long-time convergence rates, propagation of chaos at stationarity and uniformly in time, and creation of chaos. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16506 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free energy Wasserstein gradient flow and their particle counterparts: toy model, (degenerate) PL inequalities and exit times Monmarché, Pierre Probability Analysis of PDEs Optimization and Control In finite dimension, the long-time and metastable behavior of a gradient flow perturbated by a small Brownian noise is well understood. A similar situation arises when a Wasserstein gradient flow over a space of probability measure is approximated by a system of mean-field interacting particles, but classical results do not apply in these infinite-dimensional settings. This work is concerned with the situation where the objective function of the optimization problem contains an entropic penalization, so that the particle system is a Langevin diffusion process. We consider a very simple class of models, for which the infinite-dimensional behavior is fully characterized by a finite-dimensional process. The goal is to have a flexible class of benchmarks to fix some objectives, conjectures and (counter-)examples for the general situation. Inspired by the systematic study of these toy models, one application is presented on the continuous Curie-Weiss model in a symmetric double-well potential. We show that, at the critical temperature, although the $N$-particle Gibbs measure does not satisfy a uniform-in-$N$ standard log-Sobolev inequality (the optimal constant growing like $\sqrt{N}$), it does satisfy a more general Lojasiewicz inequality uniformly in $N$, inducing uniform polynomial long-time convergence rates, propagation of chaos at stationarity and uniformly in time, and creation of chaos. |
| title | Free energy Wasserstein gradient flow and their particle counterparts: toy model, (degenerate) PL inequalities and exit times |
| topic | Probability Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2510.16506 |