含小幅度层状稳定噪声的非线性朗之万系统在全变差意义下的截断现象
The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise
AI总结：
本文研究小幅度层状稳定噪声扰动下的单稳态非线性朗之万系统，在全变差距离下证明其指数遍历性与截断现象，推广了相关结果并给出可验证的充分条件，证明技巧异于高斯情形。
摘要：
本文针对一类原型非线性朗之万系统开展了截断现象的拓展案例研究，该系统具有单一稳定态，受小幅度ε>0的加性纯跳 Lévy 噪声扰动，驱动噪声过程为层状稳定类型。在漂移强制性条件下，相关过程族X^ε被证明在全变差距离下具有指数遍历性，其平衡分布为μ^ε，这将Peng和Zhang（2018）的结果推广到了任意多项式矩的情形。本文的主要结果在Blumenthal-Getoor指数α>3/2的充分光滑条件下，确立了全变差意义下的截断现象。也就是说，在该设定下，我们确定了一个确定性时间尺度t_ε^cut，满足当ε→0时t_ε^cut→∞，以及对应的时间窗口t_ε^cut ± o(t_ε^cut)，在此窗口内，当前状态与其平衡分布μ^ε之间的全变差距离会随着ε趋于零而本质上骤降。此外，我们将描述该现象的动力学特征——即该距离收敛到唯一的轮廓函数，最初由Barrera和Jara（2020）针对线性漂移情形确立——推广到了非线性漂移的Lévy情形。这推导出了可通过实例验证的充分条件，例如受小幅度对称α-稳定噪声作用且α>3/2的梯度系统。由于缺乏能在短时间内渐近耦合非线性方程与线性近似的相应Girsanov变换，本文的证明技巧与高斯情形完全不同。
英文摘要：
This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude $\varepsilon>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^\varepsilon$ turns out to be exponentially ergodic with equilibrium distribution $μ^{\varepsilon}$ in total variation distance which extends a result from Peng and Zhang (2018) to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index $α>3/2$. That is to say, in this setting we identify a deterministic time scale $\mathfrak{t}_{\varepsilon}^{\mathrm{cut}}$ satisfying $\mathfrak{t}_ \varepsilon^{\mathrm{cut}} \rightarrow \infty$, as $\varepsilon \rightarrow 0$, and a respective time window, $\mathfrak{t}_\varepsilon^{\mathrm{cut}} \pm o(\mathfrak{t}_\varepsilon^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium $μ^{\varepsilon}$ essentially collapses as $\varepsilon$ tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in Barrera and Jara (2020) to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric $α$-stable noise for $α>3/2$. The proof techniques differ completely from the Gaussian case due to the absence of respective Girsanov transforms which couple the nonlinear equation and the linear approximation asymptotically even for short times.
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