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首页> 外文期刊>Infinite dimensional analysis, quantum probability, and related topics >AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL
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AN INVARIANCE PRINCIPLE FOR THE TAGGED PARTICLE PROCESS IN CONTINUUM WITH SINGULAR INTERACTION POTENTIAL

机译：具有奇异相互作用势的连续体中带标记的粒子过程的不变性原理

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摘要

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential mdissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials.
机译：我们考虑了根据随机梯度动力学在无限粒子环境中移动的带标记粒子的动力学。对于奇异的相互作用势，该标记的粒子动力学首先在参考文献中建立。 7，使用参考文献中已经提出的Dirichlet之前形式的封闭形式。图13和24.分别构建了环境动力学以及标记粒子与环境的耦合动力学。在这里，我们继续对这些过程进行分析：为Ref耦合动力学生成器证明必不可少的磁化结果。从图7中可以看出，这种动力学不仅包含环境动力学（作为一个组件），而且在给定后者的情况下，它是耦合过程的唯一可能选择。此外，我们将环境的均匀运动确定为标记粒子的反向运动。 （由于动力学是在配置空间上作为mar解决方案构建的，因此这不是立即的。）此外，只要底层参考度量是系统的纯阶段，我们就证明了环境动力学的遍历性。最后，我们证明了这些考虑因素足以应用参考文献。图4用于证明标记粒子过程的不变性原理。我们指出，参考文献之前曾研究过这种不变性原理。参考第13章中的平滑电位，并由Ref。1中的抽象Dirichlet形式方法显示。 24为奇异电位。

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