论文标题
在粗糙道路的精神上对非线性SPDE的一阶描述
A first order description of a nonlinear SPDE in the spirit of rough paths
论文作者
论文摘要
我们考虑以差异形式的非线性随机部分微分方程（SPDE），其中强制项是高斯噪声，在时间上是白色的，在空间上是颜色的，使得溶液的梯度是hölder连续的，但没有可区分的。然后，我们证明了对SPDE的溶液与固定基台围绕其线性化的溶液之间差异的广义taylor膨胀。结果是让人联想到（受控的）粗糙路径的理论，并与一般观察结果一致，即在用粗糙驱动器的设置中，将解决方案减去线性化方程会产生更正常的对象。
We consider a nonlinear stochastic partial differential equation (SPDE) in divergence form where the forcing term is a Gaussian noise, that is white in time and colored in space such that the gradient of the solution is Hölder-continuous, but not differentiable. Then, we prove a generalized Taylor expansion of the difference between the solution to the SPDE and the solution to its linearization around a fixed basepoint. The result is reminiscent of the theory of (controlled) rough paths and agrees with the general observation, that, in settings with a rough driver, subtracting the solution to the linearized equation yields a more regular object.
