Stochastic partial differential equation

Partial differential equations with random force terms and coefficients 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. Examples One of the most studied SPDEs is the stochastic heat equation, which may formally be written as ∂ t u = Δ u + ξ , {\displaystyle \partial _{t}u=\Delta u+\xi \;,} where Δ {\displaystyle \Delta } is the Laplacian and ξ {\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. Discussion 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 soluti