Gradient flow approach to local mean-field spin systems

It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems; see e.g. Maas (2011), Fathi and Simon (2016), Er...

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Bibliographic Details
Published in:Stochastic processes and their applications 2020-03, Vol.130 (3), p.1461-1514
Main Authors: Bashiri, K., Bovier, A.
Format: Article
Language:English
Subjects:
Gradient flow, Large deviation principle, Hydrodynamic limit, Gamma-convergence
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Summary:It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems; see e.g. Maas (2011), Fathi and Simon (2016), Erbar (2016). In this paper we establish such a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use this gradient flow representation to prove a large deviation principle for the empirical process associated to this system. This is done by using the criterion established in Fathi (2016). Finally, the corresponding hydrodynamic limit is shown by using the approach initiated in Sandier and Serfaty (2004) and Serfaty (2011).
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2019.05.006