具有奇异势的流体动力学导波理论中的指数混合
Exponential mixing in a hydrodynamic pilot--wave theory with singular potentials
AI总结：
本文通过Lyapunov技术和渐近耦合方法，证明了受奇异势和加性高斯噪声驱动的随机导波动力学指数收敛到唯一不变测度，并给出了二维库仑势中不变测度的数值示例。
摘要：
我们对一个随机流体动力学导波理论进行了分析，该理论是一个具有记忆核的Langevin方程，描述了受排斥奇异势和通过加性高斯噪声随机扰动的行走液滴（或“walker”）的动力学。在奇异性的适当假设下，我们证明了walker动力学指数吸引到唯一的不变概率测度。证明依赖于Lyapunov技术和针对我们设置特别定制的渐近耦合的组合。我们还给出了从二维库仑势中walker的数值模拟获得的不变测度示例。我们的结果扩展了先前关于光滑限制势中随机导波动力学遍历性的工作。
英文摘要：
We conduct an analysis of a stochastic hydrodynamic pilot-wave theory, which is a Langevin equation with a memory kernel that describes the dynamics of a walking droplet (or "walker") subjected to a repulsive singular potential and random perturbations through additive Gaussian noise. Under suitable assumptions on the singularities, we show that the walker dynamics is exponentially attracted toward the unique invariant probability measure. The proof relies on a combination of the Lyapunov technique and an asymptotic coupling specifically tailored to our setting. We also present examples of invariant measures, as obtained from numerical simulations of the walker in two-dimensional Coulomb potentials. Our results extend previous work on the ergodicity of stochastic pilot-wave dynamics established for smooth confining potentials.
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