We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.

中文翻译：

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完全非线性的随机和粗糙PDE：经典和粘性解决方案

我们研究完全非线性的二阶（正向）随机PDE。它们也可以被视为与前向路径相关的PDE，在统一框架下将被视为粗糙的PDE。对于最一般的完全非线性情况，我们发展了经典解的局部理论，然后通过平滑测试函数定义了粘度解。我们的粘度解决方案概念等同于使用半喷嘴的替代方案。接下来，我们将证明基本属性，例如一致性，稳定性和一般设置中的部分比较原理。如果扩散系数是半线性的（即溶液的梯度是线性的，溶液的非线性是；漂移仍然可以是完全非线性的），则我们建立了一个完整的理论，包括全局存在性和比较原理。