Langevin dynamic for the 2D Yang–Mills measure
Langevin dynamic for the 2D Yang–Mills measure
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DOI：
10.1007/s10240-022-00132-0
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发表时间：
2020-06
期刊：
影响因子：
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通讯作者：
A. Chandra;I. Chevyrev;Martin Hairer;Hao Shen
中科院分区：
文献类型：
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作者：
A. Chandra;I. Chevyrev;Martin Hairer;Hao Shen
我们定义了一个自然状态空间和与二维随机Yang-Mills热流相关联的Markov过程，为了实现这一点，我们首先引入了一个分布连接空间，在这个空间中，沿着充分正则曲线的完整性（Wilson圈观测量）和一组相关的规范变换的作用都是定义良好的，并且满足良好的连续性。所需的状态空间被获得为在此群作用下的轨道的对应空间，并且当配备有自然Hausdorff度量时被示出为波兰空间。为了构造马尔可夫过程，我们示出随机杨-米尔斯热流在我们的连接空间中取值，并且使用引入依赖于时间的规范变换的“DeTurck技巧”来示出不变性，我们求解杨-米尔斯热流的主要工具是正则性结构理论，沿着我们还开发了一个“无基”框架，用于在向量值噪声的背景下应用正则性结构理论。这提供了一个概念框架来解释几个以前的结构，我们希望这个框架是独立的利益。
We define a natural state space and Markov process associated to the stochastic Yang–Mills heat flow in two dimensions.To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both well-defined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric.To construct the Markov process we show that the stochastic Yang–Mills heat flow takes values in our space of connections and use the “DeTurck trick” of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations.Our main tool for solving for the Yang–Mills heat flow is the theory of regularity structures and along the way we also develop a “basis-free” framework for applying the theory of regularity structures in the context of vector-valued noise – this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.