Stochastic Analysis on nonlinear sets

在包含一般维纳空间的高斯空间中，我们给出了凸集的散度公式。我们发现，在高斯测度下，过程的极大值的某种规律性起着重要的作用，而不是认为的马尔可夫性质。我们还成功地利用马尔可夫性质恢复了已知的公式，这是我们散度公式的具体表达。我们也得到了分形集上的一个临界概率。我们已经成功地可视化了相互作用粒子的运动，以及由Levy型噪声驱动的随机偏微分方程的解。

We have formulated a divergence formula on convex sets in the Gaussian space which include usual Wiener spaces. We found that a certain regularity of the maximal of the processes under the Gaussian measure plays an essential role rather than Markov properties which was believed. We also succeeded in recovering known formulae using Markov properties then, which is a concrete expression of our divergence formulae. We have also obtained a critical probability for a percolation on a fractal set. We have succeeded in visualizing a motion of interacting particles, solutions to stochastic partial differential equations driven by Levy type noises.