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。

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

where ${\displaystyle \mathrm{\Delta <!--\; \Delta \; -->}}$ is the Laplacian and ${\displaystyle \mathrm{\xi <!--\; \xi \; -->}}$ denotes space-time white noise.

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.