Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions
Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions
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DOI：
10.1007/s11118-020-09872-2
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发表时间：
2017-11
影响因子：
1.1
通讯作者：
Y. Kawamoto;H. Osada;H. Tanemura
中科院分区：
文献类型：
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作者：
Y. Kawamoto;H. Osada;H. Tanemura
研究了与相互作用的布朗运动的各种无穷系统有关的Dirichlet形式。对于给定的随机点场μ，在L2(S，μ)上存在两个描述相互作用的布朗运动的自然无限体积狄利克雷形式，每个形式都具有未标记的平衡态μ。前者是这种有限体积Dirichlet形式格式的下限，后者是另一种有限体积Dirichlet形式格式的递增极限。此外，后者是前者的延伸。我们给出了这两个Dirichlet形式相同的充分条件。在第一个主要定理(定理3.1)中，文[1]给出的马尔可夫半群与自然无限维随机微分方程(ISDE)相联系。在第二个主要定理(定理3.2)中，我们利用ISDE弱解的唯一性证明了这些Dirichlet形式彼此重合。我们将定理3.1应用于随机矩阵理论产生的随机动力学，如Sine，Bessel和Ginibre相互作用的布朗运动和相互作用的布朗运动与Ruelle类相互作用势，以及定理3.2应用于Sine2相互作用的布朗运动和相互作用的布朗运动与Ruelle类相互作用势。
The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point fieldμ, there exist two natural infinite-volume Dirichlet formsandonL2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium stateμ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given byis associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of-class.