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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions
Probability, Uncertainty and Quantitative Risk ( IF 1.0 ) Pub Date : 2020-11-03 , DOI: 10.1186/s41546-020-00049-8
Rainer Buckdahn , Christian Keller , Jin Ma , Jianfeng Zhang

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:

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。
更新日期:2020-11-04
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