AI 中文总结
针对带基尔霍夫与粘性顶点条件的网络扩散过程，提出离散马尔可夫链逼近方法，证明其依分布收敛，进而推导网络上哈密尔顿-雅可比-贝尔曼方程的全离散半拉格朗日格式并建立收敛性。
AI 中文摘要
我们提出了一种针对带有基尔霍夫条件和粘性顶点条件的网络上扩散过程的离散马尔可夫链逼近方法。粘性通过顶点处的概率驻留机制建模，而沿边的运动在扩散尺度上遵循欧拉-丸山型更新。我们利用Ethier-Kurtz框架证明，相关的时间插值链在斯科罗霍德空间中依分布收敛到极限扩散过程。基于该构造，我们推导了网络上哈密尔顿-雅可比-贝尔曼方程的全离散半拉格朗日格式，并利用粘性解技术建立了其收敛性。
英文摘要
We propose a discrete Markov-chain approximation of diffusion processes on networks with both Kirchhoff and sticky vertex conditions. Stickiness is modeled by a probabilistic residence mechanism at the vertex, while the motion along the edges follows an Euler-Maruyama-type update at the diffusive scale. We prove that the associated time-interpolated chain converges in distribution to the limiting diffusion in the Skorokhod space using the Ethier-Kurtz framework. Based on this construction, we derive a fully discrete semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations on networks and establish its convergence using viscosity solution techniques.
