无限维Dyson布朗运动
Infinite-dimensional Dyson Brownian motion(s)
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中文总结 AI 辅助
本研究研究无限维Dyson布朗运动，构造行列式过程并证明收敛性，推广了Katori-Tanemura和Tsai-Osada的结果，并推导了Burgers型随机偏微分方程。
中文摘要 AI 辅助
我们研究了无限维Dyson布朗运动，这些运动是有限系统在不重新缩放实际随机动力学的情况下的极限。对于β=2，我们在初始数据的扩展空间上构造了行列式过程，并证明了在本质上最优条件下其有限维分布的收敛性。这推广了Katori和Tanemura的开创性结果。额外的参数记录了无穷远处的信息，并通过关联的Laguerre-Pólya整函数进入。此外，对于显式配置类，我们建立了路径空间上的收敛性和马尔可夫性质。我们证明了从具有幂律计数指数q∈(0,2)的任意对称初始配置出发，在有限维分布上向平稳扩展Sine过程的重新缩放长期收敛。这推广了Katori和Tanemura的整数格松弛结果，该结果是显式确定性初始条件的唯一此类结果。对于β≥1，我们证明了从规则初始数据出发的有限粒子系统收敛到某个刚性路径正则性类中无限维随机微分方程的独特强解。这推广了Tsai和Osada的开创性工作。最后，我们为动力学的Stieltjes变换推导了一个Burgers型随机偏微分方程。
英文摘要
We study infinite-dimensional Dyson Brownian motions obtained as limits of finite systems without rescaling the actual stochastic dynamics. For $β=2$, we construct determinantal processes on an extended space of initial data and prove convergence of their finite-dimensional distributions under essentially optimal conditions. This extends the seminal results of Katori and Tanemura. The additional parameters record information at infinity and enter through an associated Laguerre-Pólya entire function. Moreover, for explicit classes of configurations, we establish convergence on path space and the Markov property. We prove rescaled long-time convergence, in finite-dimensional distributions, to the stationary extended $\mathsf{Sine}$ process from arbitrary symmetric initial configurations with power-law counting exponent $q\in(0,2)$. This extends the integer lattice relaxation result of Katori and Tanemura which was the only such result for explicit deterministic initial conditions. For $β\geq1$, we prove convergence of finite particle systems from regular initial data to the unique strong solution of an infinite-dimensional stochastic differential equation in a certain rigid-path-regularity class. This extends seminal works of Tsai and Osada. We finally derive a stochastic partial differential equation of Burgers-type for the Stieltjes transform of the dynamics.
