发表机构
Yamagata University; Keio University; Kyushu University（山形大学; 庆应义塾大学; 九州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过图离散化方法，严格推导了加权黎曼流形上带杀灭排斥过程的流体动力学极限，证明了其经验密度场收敛到带杀灭势的热方程的唯一有界弱解，且无需全局曲率或权函数有界性假设。
AI 中文摘要
本文研究在由测地完备加权黎曼流形的分割所构造的近邻图上的带杀灭的排斥过程。我们严格推导了其流体动力学方程，该方程由流形上带杀灭势项的热方程所支配。在我们的图离散化框架内，我们定义了排斥过程的经验密度场，并证明了在适当的时空缩放下，当加权流形是随机完备时，该密度场收敛到流体动力学方程的唯一有界弱解。结合抛物方程的局部正则性与流形上的随机分析技术，我们证明了有界弱解的唯一性，并给出了解关于最小薛定谔核的显式表示。我们的结果既不要求全局下 Ricci 曲率有界，也不要求权函数的全局有界性。
英文摘要
In the present paper, we consider an exclusion process with killing on a proximity graph constructed from a partition of a geodesically complete weighted Riemannian manifold. We rigorously derive its hydrodynamic equation, which is governed by a heat equation with a killing potential term on the manifold. Within our graph discretization framework, we define an empirical density field for the exclusion process and prove its convergence, under an appropriate space-time scaling, to the unique bounded weak solution of the hydrodynamic equation, provided that the weighted manifold is stochastically complete. Combining local regularity for the parabolic equation with techniques from stochastic analysis on manifolds, we prove the uniqueness of the bounded weak solution and give the explicit representation of the solution in terms of the minimal Schrödinger kernel. Our result requires neither a global lower Ricci curvature bound nor global boundedness of the weight function.
Comments40 pages, 2 figures
