带噪声的平均场耦合映射的一致时间混沌传播
Uniform in time propagation of chaos for noisy mean-field coupled maps
浏览论文内容
中文总结 AI 辅助
本文针对环面上带加性噪声的离散时间平均场系统，在噪声密度下界足够大的条件下，证明了传递算子的压缩性、唯一不变测度及均匀时间混沌传播。
中文摘要 AI 辅助
我们研究环面上受加性噪声影响的离散时间$N$维平均场系统，其概率密度有正下界。给定一个Lipschitz单粒子映射和相互作用项，我们证明，如果噪声密度的下界足够大，则$N$维传递算子$\mathcal P_N$保持一类与维度无关的次高斯概率测度。此外，$\mathcal P_N$在Dobrushin-Wasserstein距离下是一步压缩的，因此具有唯一的不变测度$\rho_N$，该测度本身是次高斯的。在相同的噪声强度条件下，我们证明相关的自洽传递算子在Wasserstein距离下是一步压缩的，因此具有唯一的不动点$\rho$。我们进一步利用这些压缩估计证明，$\mathcal P_N$在Dobrushin-Wasserstein度量下保持乘积测度$\rho^{\otimes N}$的$O(N^{-1/2})$邻域，特别地，$\rho_N$属于该邻域。最后，我们在Dobrushin-Wasserstein度量下建立了均匀时间混沌传播，并在噪声密度具有有界变差的额外假设下，对每个固定维边际建立了全变差意义下的均匀时间混沌传播。
英文摘要
We study discrete-time $N$-dimensional mean-field systems on the torus subject to additive noise whose probability density is bounded away from zero. Given a Lipschitz one-particle map and interaction term, we prove that, if the lower bound on the noise density is sufficiently large, the $N$-dimensional transfer operator $\mathcal P_N$ preserves a dimension-independent class of sub-Gaussian probability measures. Moreover, $\mathcal P_N$ is a one-step contraction in the Dobrushin-Wasserstein distance and therefore admits a unique invariant measure $ρ_N$, which is itself sub-Gaussian. Under the same condition on the noise strength, we show that the associated self-consistent transfer operator is a one-step contraction in Wasserstein distance and hence admits a unique fixed point $ρ$. We further prove, using these contraction estimates, that $\mathcal P_N$ preserves an $O(N^{-1/2})$ neighbourhood of the product measure $ρ^{\otimes N}$ in the Dobrushin-Wasserstein metric, and in particular that $ρ_N$ belongs to this neighbourhood. Finally, we establish uniform-in-time propagation of chaos in the Dobrushin-Wasserstein metric and, under the additional assumption that the noise density is of bounded variation, in total variation for every fixed-dimensional marginal.
发表机构
- Imperial College London（帝国理工学院）
- King’s College London（伦敦国王学院）
机构由 AI 辅助整理，请以论文原文为准。
