论文标题
关于几乎肯定$ c^0 $随机过程的持续同源性
On the persistent homology of almost surely $C^0$ stochastic processes
论文作者
论文摘要
本文研究了几乎肯定是$ [0，t] $的持续随机过程所引起的持续图。我们将研究重点放在两个变量上，这些变量共同表征了条形码：矩形$] \！ - ！ - \！\ infty，x] \ times [x+\ varepsilon，\ varepsilon，\ infty [$，$ n $ n^\ varepsilon $。对于具有强大马尔可夫属性的过程，我们显示了这两个变量都承认了一个瞬间生成功能，特别是每个顺序的时刻。我们将注意力转移到半木星上，我们显示了$ n^\ varepsilon $和$ n^{x，x+\ varepsilon} $的渐近行为为$ \ varepsilon \ to 0 $ to $ n^\ varepsilon $ as $ \ varepsilon $ as $ \ varepsilon \。最后，我们研究了条形码的经典稳定性定理的影响，并通过一些例子说明了我们的结果，其中最著名的是布朗运动和经验功能融合到布朗桥。
This paper investigates the propreties of the persistence diagrams stemming from almost surely continuous random processes on $[0,t]$. We focus our study on two variables which together characterize the barcode : the number of points of the persistence diagram inside a rectangle $]\!-\!\infty,x]\times [x+\varepsilon,\infty[$, $N^{x,x+\varepsilon}$ and the number of bars of length $\geq \varepsilon$, $N^\varepsilon$. For processes with the strong Markov property, we show both of these variables admit a moment generating function and in particular moments of every order. Switching our attention to semimartingales, we show the asymptotic behaviour of $N^\varepsilon$ and $N^{x,x+\varepsilon}$ as $\varepsilon \to 0$ and of $N^\varepsilon$ as $\varepsilon \to \infty$. Finally, we study the repercussions of the classical stability theorem of barcodes and illustrate our results with some examples, most notably Brownian motion and empirical functions converging to the Brownian bridge.
