text: 隨機偏微分方程類似於一般的隨機微分方程。其本質上是帶有隨機項和隨機係數的偏微分方程。隨機微分方程在量子場論、統計力學、金融數學中有著廣泛的應用。Examples.One of the most studied SPDEs is the stochastic heat equation, which may formally be written asformula_1where formula_2 is the Laplacian and formula_3 denotes space-time white noise.Discussion.One difficulty is their lack of regularity. In one space dimension, solutions to the stochastic heat equation are only almost 1/2-Hölder continuous in space and 1/4-Hölder continuous in time. For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributions.
metadata.dataset: erhwenkuo/wikipedia-zhtw
metadata.title: 隨機偏微分方程
metadata.url: https://zh.wikipedia.org/wiki?curid=4180306