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Infinite-dimensional stochastic differential equations and tail $$\sigma $$-fields
Probability Theory and Related Fields ( IF 2 ) Pub Date : 2020-07-04 , DOI: 10.1007/s00440-020-00981-y
Hirofumi Osada , Hideki Tanemura

We present general theorems solving the long-standing problem of the existence and pathwise uniqueness of strong solutions of infinite-dimensional stochastic differential equations (ISDEs) called interacting Brownian motions. These ISDEs describe the dynamics of infinite-many Brownian particles moving in $ \mathbb{R}^d $ with free potential $ \Phi $ and mutual interaction potential $ \Psi $. We apply the theorems to essentially all interaction potentials of Ruelle's class such as the Lennard-Jones 6-12 potential and Riesz potentials, and to logarithmic potentials appearing in random matrix theory. We solve ISDEs of the Ginibre interacting Brownian motion and the sine$_{\beta}$ interacting Brownian motion with $ \beta = 1,2,4$. We also use the theorems in separate papers for the Airy and Bessel interacting Brownian motions. One of the critical points for proving the general theorems is to establish a new formulation of solutions of ISDEs in terms of tail $ \sigma $-fields of labeled path spaces consisting of trajectories of infinitely many particles. These formulations are equivalent to the original notions of solutions of ISDEs, and more feasible to treat in infinite dimensions.

中文翻译:

无限维随机微分方程和尾部$$\sigma $$-fields

我们提出了一般定理来解决称为相互作用布朗运动的无限维随机微分方程 (ISDE) 的强解的存在性和路径唯一性这一长期存在的问题。这些 ISDE 描述了无限多布朗粒子在 $\mathbb{R}^d $ 中运动的动力学,具有自由势 $\Phi $ 和相互作用势 $\Psi $。我们将定理应用于基本上所有 Ruelle 类的相互作用势,例如 Lennard-Jones 6-12 势和 Riesz 势,以及随机矩阵理论中出现的对数势。我们解决了 Ginibre 相互作用布朗运动和正弦 $_{\beta}$ 相互作用布朗运动的 ISDE,$ \beta = 1,2,4$。我们还在单独的论文中将定理用于艾里和贝塞尔相互作用的布朗运动。证明一般定理的关键点之一是根据由无限多个粒子的轨迹组成的标记路径空间的尾部 $\sigma $-场来建立 ISDE 解的新公式。这些公式等价于 ISDE 解的原始概念,在无限维中处理更可行。
更新日期:2020-07-04
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