The Parisi formula is a Hamilton–Jacobi equation in Wasserstein space
نویسندگان
چکیده
Abstract The Parisi formula is a self-contained description of the infinite-volume limit free energy mean-field spin glass models. We showthat this quantity can be recast as solution Hamilton–Jacobi equation in Wasserstein space probability measures on positive half-line.
منابع مشابه
On Differentiability of the Parisi Formula
It was proved by Michel Talagrand in [10] that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application of differentiability to prove non self-averaging of the overlap outside of the...
متن کاملBarycenters in the Wasserstein Space
In this paper, we introduce a notion of barycenter in the Wasserstein space which generalizes McCann’s interpolation to the case of more than two measures. We provide existence, uniqueness, characterizations and regularity of the barycenter, and relate it to the multimarginal optimal transport problem considered by Gangbo and Świȩch in [8]. We also consider some examples and in particular rigor...
متن کاملThe Exponential Formula for the Wasserstein Metric
Many evolutionary partial differential equations may be rewritten as the gradient flow of an energy functional, a perspective which provides useful estimates on the behavior of solutions. The notion of gradient flow requires both the specification of an energy functional and a metric with respect to which the gradient is taken. In recent years, there has been significant interest in gradient fl...
متن کاملLearning in Wasserstein Space
Learning from empirical probability measures is an emerging problem that has potential benefiting multiple domains. My research focuses on developing scalable and effective learning algorithms that handle large-scale data in form of measures. In particular, the Wasserstein space provides a powerful geometry that houses the compositions of data, which can be of great interest to domain experts t...
متن کاملHamilton-jacobi Equations in the Wasserstein Space
Abstract. We introduce a concept of viscosity solutions for Hamilton-Jacobi equations (HJE) in the Wasserstein space. We prove existence of solutions for the Cauchy problem for certain Hamiltonians defined on the Wasserstein space over the real line. In order to illustrate the link between HJE in the Wasserstein space and Fluid Mechanics, in the last part of the paper we focus on a special Hami...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2021
ISSN: ['1496-4279', '0008-414X']
DOI: https://doi.org/10.4153/s0008414x21000031