Title: Monge-Ampère gravitating fluids. Least action principles and particle systems Show Chinese title

Title: 蒙日-安培引力流体。 最小作用原理和粒子系统

Abstract: The Monge-Amp\`ere gravitation theory (MAG) was introduced by Brenier to obtain an approximate solution of the early Universe reconstruction problem. It is a modification of Newtonian gravitation which is based on quadratic optimal transport. Later, Brenier, then Ambrosio, Baradat and Brenier discovered a double large deviation principle for Brownian particles whose rate function is precisely MAG's action functional. In the present article, following Brenier we first recap MAG's theory. Then, we slightly extend it from particles to fluid. This allows us to revisit the Ambrosio-Baradat-Brenier particle system and propose another one which is easier to interpret and whose large deviation rate function is MAG's action functional for fluids. This model leads to a Gibbs conditioning principle that is an entropy minimization problem close to the Schr\"odinger problem. While the setting of the Schr\"odinger problem is a system of noninteracting particles, the particle system we work with is subject to some branching mechanism which regulates the thermal fluctuations and some quantum force which balances them. Show Chinese abstract

Abstract: 莫加-阿姆佩尔引力理论（MAG）由布雷尼尔提出，以获得早期宇宙重建问题的近似解。 它是一种基于二次最优传输的牛顿引力的修改版本。 后来，布雷尼尔、安布罗西奥、巴拉达特和布雷尼尔发现了布朗粒子的双重大偏差原理，其速率函数正是MAG的作用泛函。 在本文中，按照布雷尼尔的方法，我们首先回顾MAG的理论。 然后，我们将其从粒子稍微扩展到流体。 这使我们能够重新审视安布罗西奥-巴拉达特-布雷尼尔的粒子系统，并提出另一个更容易解释的系统，其大偏差速率函数是流体的MAG作用泛函。 这个模型导致了一个吉布斯条件原理，这是一个接近薛定谔问题的熵最小化问题。 虽然薛定谔问题的设置是一个非相互作用粒子系统，但我们所研究的粒子系统受到某种分支机制的制约，该机制调节热涨落，并有一种量子力来平衡它们。