分段扩散马尔可夫过程的数值逼近与收敛性分析，及其在神经胶质瘤细胞迁移中的应用
Numerical Approximations and Convergence Analysis of Piecewise Diffusion Markov Processes, with Application to Glioma Cell Migration
AI总结：
本文提出基于thin方法与Euler-Maruyama格式的PDifMP数值逼近方案，证明其均方与弱收敛性，并应用于神经胶质瘤细胞迁移模拟。
摘要：
本文研究分段扩散马尔可夫过程（PDifMPs）的数值逼近，特别是在显式流映射不可用时。该方法基于thin方法对跳跃机制建模，并结合Euler-Maruyama格式逼近底层流动力学。针对所提逼近格式，研究均方收敛和弱收敛。通过鞅问题表述建立算法的弱收敛性。此外，应用这些结果模拟微观层面神经胶质瘤细胞表现的迁移模式。进一步，为该PDifMP模型开发并实现分裂方法，在模拟示例中采用Thinned Euler-Maruyama和分裂格式，以比较两种方法。
英文摘要：
In this paper, we focus on numerical approximations of Piecewise Diffusion Markov Processes (PDifMPs), particularly when the explicit flow maps are unavailable. Our approach is based on the thinning method for modelling the jump mechanism and combines the Euler-Maruyama scheme to approximate the underlying flow dynamics. For the proposed approximation schemes, we study both the mean-square and weak convergence. Weak convergence of the algorithms is established by a martingale problem formulation. Moreover, we employ these results to simulate the migration patterns exhibited by moving glioma cells at the microscopic level. Further, we develop and implement a splitting method for this PDifMP model and employ both the Thinned Euler-Maruyama and the splitting scheme in our simulation example, allowing us to compare both methods.
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