Stochastic partial differential equation

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, statistical mechanics, and spatial modeling.

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

where ${\displaystyle \Delta }$ is the Laplacian and ${\displaystyle \xi }$ denotes space-time white noise. Other examples also include stochastic versions of famous linear equations, such as the wave equation and the Schrödinger equation.

One difficulty is their lack of regularity. In one dimensional space, 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.

For linear equations, one can usually find a mild solution via semigroup techniques.

However, problems start to appear when considering non-linear equations. For example

where ${\displaystyle P}$ is a polynomial. In this case it is not even clear how one should make sense of the equation. Such an equation will also not have a function-valued solution in dimension larger than one, and hence no pointwise meaning. It is well known that the space of distributions has no product structure. This is the core problem of such a theory. This leads to the need of some form of renormalization.