一种关于自发随机性的测度论方法
A Measure-Theoretic Approach to Spontaneous Stochasticity
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中文总结 AI 辅助
研究自发随机性问题，引入测度论形式体系，将其作为测度选择原则。通过正则化流研究推送测度，建立有限维系统相关结构结果，分析与初始数据敏感性的关系，发展重整化群观点并举例说明各因素在自发随机性出现中的相互作用。
中文摘要 AI 辅助
自发随机性（SpSt）源于理查森的湍流扩散图景和洛伦兹关于有限时间可预测性丧失的欧拉观点，后由加韦茨基及其合作者以该名称提出，并由迈利鲍耶夫及其合作者在壳模型中发展。它是否出现在充分发展的湍流中仍是一个主要的开放性问题。除了少数特定类型的系统外，SpSt缺乏一般的数学定义。我们引入一种测度论形式体系，将其理解为一种测度选择原则。给定一个无粘问题、一个适定的正则化以及一个环境测度，我们研究正则化流对该测度的推送。当这些推送测度收敛到一个非狄拉克概率律时，强SpSt发生，用统计选择取代经典的确定性选择。对于有限维系统，我们建立了几个结构结果。我们的中心可达性定理表明，只要无粘问题不唯一，任何支撑在无粘状态集上的概率测度都可以被选为合适正则化的极限律。我们还通过迪尼型方向增长识别出无粘动力学中的奇异集，将其作为非唯一性的必要障碍。我们分析了SpSt与对初始数据的敏感性之间的关系，阐明了受湍流启发的有限时间分离标准的范围和局限性。最后，我们发展了一种重整化（半）群观点，其中极限统计作为统计吸引子出现。明确的例子说明了环境测度、无粘奇点、正则化尺度和初始数据敏感性在SpSt出现过程中是如何相互作用的。
英文摘要
Spontaneous stochasticity (SpSt), originating in Richardson's picture of turbulent dispersion and Lorenz's Eulerian view of finite-time loss of predictability, was later formulated under this name by Gawędzki and collaborators and developed in shell models by Mailybaev and collaborators. Whether it occurs in fully developed turbulence remains a major open question. Beyond a few specific classes of systems, however, SpSt has lacked a general mathematical definition. We introduce a measure-theoretic formalism in which it is understood as a measure-selection principle. Given an inviscid problem, a well-posed regularization, and an ambient measure, we study the pushforward of that measure by the regularized flow. Strong SpSt occurs when these pushforward measures converge to a non-Dirac probability law, replacing classical deterministic selection by statistical selection. For finite-dimensional systems, we establish several structural results. Our central attainability theorem shows that, whenever the inviscid problem is nonunique, any probability measure supported on the set of inviscid states can be selected as the limiting law of a suitable regularization. We also identify singular sets in the inviscid dynamics, detected through Dini-type directional growth, as necessary obstructions underlying nonuniqueness. We analyze the relation between SpSt and sensitivity to initial data, clarifying the scope and limitations of turbulence-inspired finite-time separation criteria. Finally, we develop a renormalization-(semi)group viewpoint in which limiting statistics arise as statistical attractors. Explicit examples illustrate how ambient measures, inviscid singularities, regularization scales, and initial-data sensitivity interact in the emergence of SpSt.
