# Stochastic partial differential equation

Differential equations :
Scope :
Fields
Natural sciencesEngineering
Astronomy
Physics
Chemistry
Biology
Geology
Applied mathematics
Continuum mechanics
Chaos theory
Dynamical systems
Social sciences
Economics
Population dynamics
List of named differential equations :
Classification :
Types
Ordinary
Partial
Differential-algebraic
Integro-differential
Fractional
Linear
Non-linear
By variable type
Dependent and independent variables
Autonomous
Coupled / Decoupled
Exact
Homogeneous / Nonhomogeneous
Features
Order
Operator
Notation :
Relation to processes
Difference (discrete analogue)
Stochastic
Stochastic partial
Delay :
Solution :
Existence and uniqueness
Picard–Lindelöf theorem
Peano existence theorem
Carathéodory's existence theorem
Cauchy–Kowalevski theorem :
General topics
Initial conditions
Boundary values
Dirichlet
Neumann
Robin
Cauchy problem
Wronskian
Phase portrait
Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions
Numerical integration
Dirac delta function :
Solution methods
Inspection
Method of characteristics
Euler
Exponential response formula
Finite difference (Crank–Nicolson)
Finite element
Infinite element
Finite volume
Galerkin
Petrov–Galerkin
Green's function
Integrating factor
Integral transforms
Perturbation theory
Runge–Kutta
Separation of variables
Undetermined coefficients
Variation of parameters :
People :
List
Isaac Newton
Gottfried Leibniz
Jacob Bernoulli
Leonhard Euler
Joseph-Louis Lagrange
Józef Maria Hoene-Wroński
Joseph Fourier
Augustin-Louis Cauchy
George Green
Carl David Tolmé Runge
Martin Kutta
Rudolf Lipschitz
Ernst Lindelöf
Émile Picard
Phyllis Nicolson
John Crank :
vte :
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
    $\partial _{t}u=\Delta u+\xi \;,$
where $\Delta$ is the Laplacian and $\xi$ 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 solution via semigroup techniques.
However, problems start to appear when considering non-linear equations. For example
    $\partial _{t}u=\Delta u+P(u)+\xi ,$
where $P$ 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.
An early attempt to circumvent such problems for some specific equations was the so called da Prato–Debussche trick which involved studying such non-linear equations as perturbations of linear ones. However, this can only be used in very restrictive settings, as it depends on both the non-linear factor and on the regularity of the driving noise term. In recent years, the field has drastically expanded, and now there exists a large machinery to guarantee local existence for a variety of sub-critical SPDEs.