强阻尼随机波动方程在有限秩纯跳Lévy强迫下的唯一遍历性

中文摘要 AI 辅助

我们研究三维环面上由从属布朗运动生成的有限秩纯跳Lévy噪声驱动的强阻尼随机波动方程。Lyapunov估计和渐近紧性分解给出不变测度，小噪声事件给出弱不可约性。为证明唯一性，我们将条件Malliavin微积分与终端可观测性以及从高模态出发的全变分流估计相结合。在交替停止块上构造的控制在涉及高模态衰减、谱泄漏和随机时钟的固定秩平衡条件下给出e-性质。因此，不变测度是唯一且遍历的，并且从每个初始状态出发的转移律的Cesàro平均值弱收敛于该测度。我们在每个固定且秩足够大的强迫算子处验证了足够小的正三次系数满足此条件。在另一个显式小性条件下，同步耦合证明了e-性质和转移律的弱收敛，无需对强迫秩施加下界。

英文摘要

We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.

发表机构

• Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
• College of Science, National University of Defense Technology（国防科技大学理学院）

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