SIAM Journal on Mathematical Analysis, Volume 52, Issue 2, Page 1943-1979, January 2020.
We introduce a new definition of viscosity solution to path-dependent partial differential equations, which is a slight modification of the definition introduced in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204--236]. With the new definition, we prove the two important results, until now missing in the literature, namely, a general stability result and a comparison result for semicontinuous sub-/supersolutions. As an application, we prove the existence of viscosity solutions using the Perron method. Moreover, we connect viscosity solutions of path-dependent PDEs with viscosity solutions of partial differential equations on Hilbert spaces.

中文翻译：

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随时间变化的路径相关PDE的粘度解

SIAM数学分析杂志，第52卷，第2期，第1943-1979页，2020年1月。
我们为与路径相关的偏微分方程引入了粘度解的新定义，它是对[I. Ekren等，Ann。Probab。，42（2014），第204--236页]。利用新的定义，我们证明了两个重要的结果，直到现在在文献中都没有，即一般稳定性结果和半连续子/上解的比较结果。作为应用，我们使用Perron方法证明了粘度溶液的存在。此外，我们将路径依赖的PDE的粘度解与希尔伯特空间上偏微分方程的粘度解联系起来。