临界Ornstein--Uhlenbeck过程的平稳测度
On the stationary measures of the critical Ornstein--Uhlenbeck process
- Technische Universität Berlin（柏林工业大学）
- Technische Universität Braunschweig（布伦瑞克工业大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究临界Ornstein--Uhlenbeck过程的平稳测度，给出电导的尖锐准则保证可逆性，并揭示非可逆、时间周期等丰富行为，兼论随机电导情形。
AI中文摘要：
我们研究$\mathbb{R}^{\mathbb{Z}^d}$上的线性对称扩散过程，这些过程可视为给定电导下（非齐次）调和晶格的Langevin动力学，并特别关注其平稳分布。尽管相互作用是线性的，我们展示了由电导所编码的无序性所导致的丰富多样的行为。我们的主要结果提供了关于电导的基本上尖锐的准则，确保每个平稳测度都是可逆的。我们还提供了这些准则失效的例子，其中要么不存在平稳测度，要么可逆与非可逆平稳测度共存。此外，我们表明该线性系统甚至可以表现出非平凡的时间周期行为，并提供了此类振荡发生的谱刻画。确定性电导的情形由在相当一般的矩假设下对随机电导情形的研究所补充。
英文摘要：
We study linear and symmetric diffusion processes on $\mathbb{R}^{\mathbb{Z}^d}$ that can be seen as the Langevin dynamics of the (inhomogeneous) harmonic crystal for given conductances, with a particular focus on their stationary distributions. Despite the linearity of the interaction, we exhibit a rich variety of behaviours, depending on the disorder encoded by the conductances. Our main results provide essentially sharp criteria on the conductances ensuring that every stationary measure is reversible. We also provide examples for which these criteria fail and where either no stationary measures exist or reversible and non-reversible stationary measures coexist. Additionally, we show that the linear system can even exhibit non-trivial time-periodic behaviour and provide a spectral characterisation of the occurrence of such oscillations. The case of deterministic conductances is complemented by a study of the case of random conductances under quite general moment assumptions.
