配置空间上扩散的碰撞与非碰撞
Collision and non-collision for diffusions on configuration space
AI总结：
研究实线上可逆无限多相互作用扩散过程的碰撞与非碰撞标准，基于位势理论和容量估计，结果与模型无关，还确定了\(\mathsf{Sine}_\beta\)对称狄氏型扩散的碰撞阈值，给出无限粒子对应物。
AI中文摘要：
我们为实线上可逆的无限多个相互作用扩散过程建立了碰撞和非碰撞标准。该方法基于位势理论，依赖于配置空间上对称狄氏型的容量估计。主要结果与模型无关，无需行列式或普法夫结构、规定的相互作用势或显式标记的随机微分方程。非碰撞标准仅涉及可逆测度的二阶和三阶相关函数，碰撞标准基于可逆测度有限体积条件密度的局部下界。作为应用，我们确定了与\(\mathsf{Sine}_\beta\)对称狄氏型相关扩散的精确碰撞阈值：当且仅当\(\beta\geq1\)时，碰撞集是极集。这为具有逆温度\(\beta\)的有限粒子戴森布朗运动的经典碰撞阈值提供了无限粒子对应物。
英文摘要：
We develop criteria for collision and non-collision of reversible infinitely many interacting diffusion processes in the real line. The approach is potential-theoretic and is based on capacity estimates for symmetric Dirichlet forms on the configuration space. Our main results are model-independent in the sense that no determinantal or Pfaffian structure, prescribed interaction potential, or explicit labelled stochastic differential equation is required. The non-collision criterion involves only the second and third correlation functions of the reversible measure, whereas the collision criterion is based on a local lower bound for a finite-volume conditional density of the reversible measure. As an application, we identify the sharp collision threshold for the diffusion associated with the $\mathsf{Sine}_β$-symmetric Dirichlet form: the collision set is polar if and only if $β\ge 1$. This provides an infinite-particle counterpart of the classical collision threshold for the finite-particle Dyson Brownian motion with inverse temperature $β$.
