We develop a practical framework for distinguishing diffusive stochastic processes from deterministic signals using only a single discrete time series. Our approach is based on classical excursion and crossing theorems for continuous semimartingales, which correlates number $N_\varepsilon$ of excursions of magnitude at least $\varepsilon$ with the quadratic variation $[X]_T$ of the process. The scaling law holds universally for all continuous semimartingales with finite quadratic variation, including general Ito diffusions with nonlinear or state-dependent volatility, but fails sharply for deterministic systems -- thereby providing a theoretically-certfied method of distinguishing between these dynamics, as opposed to the subjective entropy or recurrence based state of the art methods. We construct a robust data-driven diffusion test. The method compares the empirical excursion counts against the theoretical expectation. The resulting ratio $K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$ is then summarized by a log-log slope deviation measuring the $\varepsilon^{-2}$ law that provides a classification into diffusion-like or not. We demonstrate the method on canonical stochastic systems, some periodic and chaotic maps and systems with additive white noise, as well as the stochastic Duffing system. The approach is nonparametric, model-free, and relies only on the universal small-scale structure of continuous semimartingales.

翻译：我们提出了一种实用框架，仅通过单个离散时间序列即可区分扩散型随机过程与确定性信号。该方法基于连续半鞅的经典游程与穿越定理，将幅度至少为$\varepsilon$的游程数量$N_\varepsilon$与过程的二次变分$[X]_T$相关联。该标度律对所有具有有限二次变分的连续半鞅普遍成立，包括具有非线性或状态依赖波动率的一般伊藤扩散过程，但对确定性系统则明显失效——从而提供了一种理论可验证的动态区分方法，相较于当前基于主观熵或递归的先进方法具有显著优势。我们构建了一种鲁棒的数据驱动扩散检验方法，通过比较经验游程计数与理论期望值，将所得比值$K(\varepsilon)=N_{\varepsilon}^{\mathrm{emp}}/N_{\varepsilon}^{\mathrm{theory}}$通过衡量$\varepsilon^{-2}$律的对数-对数斜率偏差进行汇总，从而实现扩散类与非扩散类系统的分类。我们在典型随机系统、若干周期与混沌映射、加性白噪声系统以及随机杜芬系统上验证了该方法。该框架具有非参数、模型无关的特性，仅依赖于连续半鞅的普适小尺度结构。