Matrix displacement convexity along density flows. (English) Zbl 07922985
Summary: A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schrödinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.
MSC:
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 49Qxx | Manifolds and measure-geometric topics |
| 60Jxx | Markov processes |
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