Stochastic partial differential equation

Differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction.
Scope Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Delay
Solution
General topics Picard–Lindelöf theorem Wronskian Phase portrait Phase space Lyapunov / Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Finite difference (Crank–Nicolson) Finite element Finite volume Galerkin Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients
People Isaac Newton Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Wilhelm Kutta

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 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.

See also

• Kardar–Parisi–Zhang equation
• Zakai equation
• Kushner equation