AI 中文总结
研究通过Monna映射构造[0,1]上的p进扩散相关强马尔可夫过程，计算对应扩散算子的谱，用有限集随机游走近似求解热方程柯西问题，还提出了p进扩散的可视化方法。
AI 中文摘要
通过Monna映射将p进单位圆盘上的积分算子迁移至实单位区间，构造出[0, 1]上的强马尔可夫过程，其路径右连续，仅存在跳跃型间断。计算了对应扩散算子的谱，这类算子的核函数依赖于p进距离。此外，利用由p进距离诱导的单位区间分层划分得到的有限集上的连续时间随机游走，近似求解其热方程的柯西问题。通过Monna映射将p进扩散迁移至实域，可得到一种简单的可视化方法，文末给出了具体示例的图示。
英文摘要
Integral operators on the real unit interval are constructed as transported from ones on the $p$-adic unit disc via the Monna map. This gives rise to strong Markov processes on $[0, 1]$, whose paths are right-continuous and have no discontinuities other than jumps. The spectra of the corresponding diffusion operators, whose kernel functions depend on a $p$-adic distance, are calculated. Further, the solution to the Cauchy problem for their heat equations are approximated via continuous- time random walks on finite sets coming from a hierarchical partition of the unit interval induced by the $p$-adic distance. The transport of $p$-adic diffusion to the real domain via the Monna map gives rise to a simple visualisation method. Illustrations of concrete examples are undertaken in the end.
Comments23 pages, 4 figures; v2: some clarification as well as references added in introduction section
