Multilateral research on infinite dimensional differential operators via stochastic analysis

我主要通过随机分析研究无穷维空间上微分算子的唯一性问题和相应的随机动力学。特别地，我证明了量子场论中出现的Gibbs测度的Dirichlet算子的强唯一性，并利用Dirichlet形式理论构造了相应的随机PDE的唯一解。此外，我将粗糙路径理论与Malliavin微积分相结合，研究了一些振荡泛函积分的精确渐近行为（即定相方法）。在此研究的激励下，我还获得了非对称性对一类三角形晶格上的非对称随机漫步的长时间渐近行为的显式影响。

I mainly studied uniqueness problems of differential operators and corresponding stochastic dynamics on infinite dimensional spaces via stochastic analysis. In particular, I proved strong uniqueness of Dirichlet operators for Gibbs measures which appear in quantum field theory and constructed a unique solution to the corresponding stochastic PDE by using Dirichlet form theory. Besides, I studied precise asymptotic behavior (i.e., the method of stationary phase) of some oscillatory functional integrals by combining Rough Path theory with Malliavin calculus. Motivated by this study of the stationary phase method, I also obtained an explicit effect of non-symmetry on the long time asymptotic behavior for a class of non-symmetric random walks on the triangular lattice.