随机偏微分方程

随机偏微分方程（英文：Stochastic partial differential equations (SPDEs)）类似于一般的随机微分方程。其本质上是带有随机项和随机系数的偏微分方程。随机微分方程在量子场论、统计力学、金融数学中有着广泛的应用。

Examples

One of the most studied SPDEs is the stochastic heat equation, which may formally be written as

${\displaystyle \partial _{t}u=\Delta u+\xi \;,}$ 

where ${\displaystyle \Delta }$  is the Laplacian and ${\displaystyle \xi }$  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.

参见

• Brownian surface
• Kardar–Parisi–Zhang equation
• Kushner equation
• Wick product
• Zakai equation

延伸阅读

• Holden, H., Øksendal, B., Ubøe, J, Zhang, T., 2010. Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach （页面存档备份，存于互联网档案馆）. Universitext, Springer, New York, NY, ed. 2.