发表机构
University of California, Los Angeles; Weizmann Institute of Science; University of Leicester（加州大学洛杉矶分校; 魏茨曼科学研究所; 莱斯特大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究基于次椭圆Itô扩散的Kolmogorov生成元的Ruelle--Pollicott谱，构建了适用于随机多稳亚稳系统的tipping转变统一理论，明确早期预警信号的属性，区分分岔与噪声诱导的tipping，并关联Doob漂移与最优Girsanov采样。
AI 中文摘要
临界点是系统统计状态发生的突然、可能不可逆的重组，通常可通过临界慢化来预判：恢复速度减慢、自相关和方差升高、频谱红移。该范式在简单平衡分岔附近很有效，但并非适用于随机、多稳或亚稳系统的通用理论。我们从次椭圆Itô扩散的Kolmogorov生成元的Ruelle--Pollicott（RP）谱出发，构建了这样的理论。解析不变统计量对RP谱块的敏感性表明，每个贡献可分解为一个谱分母和一个将谱块与可观测量及扰动方向耦合的残差；小分母会引发大响应，残差则产生这类响应。因此，RP谱隙闭合不足以作为早期预警，而残差增长可在无谱隙闭合时产生经典预警信号。故早期预警信号是RP谱块、可观测量和扰动方向三者的属性。我们在一个RP谱完全冻结的随机非正规系统上验证了这一点，此时经典指标比真正谱隙闭合时更具警示性。RP分解将此归因于残差增长，并得到一个指标$\u2115(f)$，对于可逆动力学满足$\u2115(f)\u22641$，因此$\u2115(f)>1$可证明残差驱动的放大效应。对于亚稳系统，一个被“杀死”的问题会产生两个互补的谱对象：Doob Q-过程隔离了阱内恢复，而逃逸时钟和提交子加权的目标概率则控制阱间转变。在一维折叠中，这些分别按$(\u03b5_c-\u03b5)^{1/2}$和$(\u03b5_c-\u03b5)^{3/2}$缩放，区分了分岔诱导的 tipping 与噪声诱导的 tipping。Doob漂移还与最优Girsanov采样相关联。
英文摘要
Tipping points---abrupt, potentially irreversible reorganizations of a system's statistical state---are commonly anticipated through critical slowing down: recovery slows, autocorrelation and variance rise, and spectra redden. This paradigm is powerful near simple equilibrium bifurcations but is not a general theory for stochastic, multistable, or metastable systems. We develop such a theory from the Ruelle--Pollicott (RP) spectrum of Kolmogorov generators of hypoelliptic Itô diffusions. Resolving the sensitivity of invariant statistics on RP spectral blocks shows that each contribution factorizes into a spectral denominator and a residue coupling the block to both the observable and perturbation direction. Small denominators permit large responses; residues produce them. Thus a closing RP gap is not sufficient for an early warning, while growing residues can generate the classical signature with no gap closure. An early-warning signal is therefore a property of a triple: RP block, observable, and perturbation direction. We demonstrate this on a stochastic non-normal system whose RP spectrum is exactly frozen, yet classical indicators become more alarming than during genuine gap closure. The RP decomposition attributes this to residue growth and yields an index $\mathcal{N}(f)$ satisfying $\mathcal{N}(f)\leq 1$ for reversible dynamics; hence $\mathcal{N}(f)>1$ certifies residue-driven amplification. For metastable systems, one killed problem yields two complementary spectral objects: the Doob $Q$-process isolates in-well recovery, while escape clocks and committor-weighted destination probabilities govern interwell transitions. In a one-dimensional fold, these scale as $(ε_c-ε)^{1/2}$ and $(ε_c-ε)^{3/2}$, separating bifurcation-induced from noise-induced tipping. Doob drift also connects to optimal Girsanov sampling.
Comments37 pages; 2 figures
