超线性力作用下欠阻尼朗之万扩散的反射耦合与定量收缩

中文摘要 AI 辅助

欠阻尼朗之万扩散广泛应用于采样和动力学模型，但当力仅为局部利普希茨且增长速度快于线性时，定量收敛难以实现。噪声仅作用于速度，而标准反射耦合估计依赖全局利普希茨界。针对该情形，我们提出一种局部化反射-同步耦合方法。该构造结合了亚椭圆李雅普诺夫函数与加权凹距离，使得局部耦合误差被全局漂移吸收。由于主动反射区域可与反射方向的奇异集相交，我们对耦合进行正则化并取弱极限，同时保持正确的边际分布。这在适配的坎托罗维奇距离下，对一类受限的超线性力，得到了马尔可夫半群的显式指数收缩。因此，吉布斯测度在对应的有限代价类内是唯一的，且动力学指数收敛至平衡态。该结果将定量路径收缩扩展至全局利普希茨力之外的情形，并为非线性动力学采样器提供可计算的稳定性估计。

英文摘要

Underdamped Langevin diffusions model kinetic sampling and thermally driven inertial dynamics, but quantitative convergence is difficult when degenerate noise is coupled with a nonconvex superlinear force. We prove unit-prefactor exponential contraction in an explicit weighted Kantorovich cost under radial confinement, a hypocoercive Lyapunov condition, and \(|\nabla U(x)-\nabla U(y)|\leq L_1(1+|x|^\ell+|y|^\ell)|x-y|\), \(0\leq\ell\leq2\). No smallness condition is imposed on \(L_1\). At the critical exponent \(\ell=2\), a quartic lower bound on \(U\) is used together with an anisotropic position--velocity localization to obtain a nonempty endpoint regime. The proof combines regularized reflection--synchronous coupling, an exponential Lyapunov weight, tightness, and an occupation-time argument at the singular coupling direction. The result yields exponential convergence and uniqueness of the invariant Gibbs measure within the finite-cost class. It covers power-law confining potentials below quartic growth and nonconvex quartic Duffing potentials with cubic force.

发表机构

• School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）
• School of Mathematics and Statistics, Northeast Normal University（东北师范大学数学与统计学院）
• School of Mathematical Sciences, Tiangong University（天津工业大学数学科学学院）

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