Stochastic partial differential equation
Differential equations  

Navier–Stokes differential equations used to simulate airflow around an obstruction.  
Classification  
Types


Relation to processes 

Solution  
General topics 

Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
They have relevance to quantum field theory and statistical mechanics.
Examples
One of the most studied SPDEs is the stochastic heat equation, which may formally be written as
where is the Laplacian and denotes spacetime white noise.
Discussion
One difficulty is their lack of regularity. In one space dimension, solutions to the stochastic heat equation are only almost 1/2Hölder continuous in space and 1/4Hölder continuous in time. For dimensions two and higher, solutions are not even functionvalued, but can be made sense of as random distributions.