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Propagation of chaos, Wasserstein gradient flows and toric Kahler-Einstein metrics | Robert J. Berman
; Magnus Onnheim
; | Date: |
30 Jan 2015 | Abstract: | Motivated by a probabilistic approach to Kahler-Einstein metrics we consider
a general non-equlibrium statistical mechanics model in Euclidean space
consisting of the stochastic gradient flow of a given quasi-convex N particle
interaction energy. We show that a deterministic macroscopic evolution equation
emerges in the large N-limit of many particles. The proof uses the theory of
weak gradient flows on the Wasserstein space and in particular De Georgi’s
notion of minimizing movements. Applied to the setting of permanental point
processes at negative temperature the corresponding limiting evolution equation
yields a new drift-diffusion equation, coupled to the Monge-Ampere operator,
whose static solutions correspond to toric Kahler-Einstein metrics. This
drift-diffusion equation is the gradient flow on the Wasserstein space of
probability measures of the K-energy functional in Kahler geometry and it can
be seen as a fully non-linear version of various extensively studied
dissipative evolution equations and conservations laws, including the
Keller-Segal equation and Burger’s equation. We also obtain a real
probabilistic analog of the complex geometric Yau-Tian-Donaldson conjecture in
this setting. In another direction applications to singular pair interactions
in 1D are given. Complex geometric aspects of these results will be discussed
elsewhere. | Source: | arXiv, 1501.7820 | Services: | Forum | Review | PDF | Favorites |
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