标题： 收缩和集中度量及其在理论神经科学中的应用 显示英文标题

标题： Contraction and concentration of measures with applications to theoretical neuroscience

摘要： 我们研究了与具有全局收缩或$B_{r}$-收缩漂移项的随机动力系统相关的概率测度的渐近行为。虽然经典结果通常假设扩散系数恒定且基于梯度的漂移，我们将分析扩展到空间非均匀扩散和非可积向量场。我们建立了在全局收缩下平稳测度存在性和唯一性的充分条件，表明当收缩率主导扩散非均匀性时，收敛性得以保持。对于仅在紧集外收缩且扩散系数恒定的系统，我们证明了在与非凸势能相关的最小值附近出现质量集中现象，类似于多稳定态情况。理论发现通过Hopfield网络得到说明，突显了其对噪声环境中记忆检索动态的影响。 显示英文摘要

摘要： We investigate the asymptotic behavior of probability measures associated with stochastic dynamical systems featuring either globally contracting or $B_{r}$-contracting drift terms. While classical results often assume constant diffusion and gradient-based drifts, we extend the analysis to spatially inhomogeneous diffusion and non-integrable vector fields. We establish sufficient conditions for the existence and uniqueness of stationary measures under global contraction, showing that convergence is preserved when the contraction rate dominates diffusion inhomogeneity. For systems contracting only outside of a compact set and with constant diffusion, we demonstrate mass concentration near the minima of an associated non-convex potential, like in multistable regimes. The theoretical findings are illustrated through Hopfield networks, highlighting implications for memory retrieval dynamics in noisy environments.