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.

中文翻译：

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分段扩散马尔可夫过程的数值近似和收敛分析及其在胶质瘤细胞迁移中的应用


在本文中，我们专注于分段扩散马尔可夫过程 （PDifMPs） 的数值近似，特别是当显式流图不可用时。我们的方法基于对跳跃机制进行建模的细化方法，并结合了 Euler-Maruyama 方案来近似底层流动动力学。对于提出的近似方案，我们研究了均方收敛和弱收敛。算法的弱收敛是通过马丁格尔问题公式建立的。此外，我们利用这些结果来模拟在微观水平上移动神经胶质瘤细胞所表现出的迁移模式。此外，我们为这个 PDifMP 模型开发并实现了一种分裂方法，并在我们的模拟示例中采用了 Thinned Euler-Maruyama 和分裂方案，使我们能够比较这两种方法。