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和随机系数偏微分方程。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

 

where   is the Laplacian and   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

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