Stochastic partial differential equations (SPDEs) generalize
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
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
Spatial may refer to:
* Dimension
* Space
* Three-dimensional space
See also
*
*
{{disambig ...
.
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. Other examples also include stochastic versions of famous linear equations, such as
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
and
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
.
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 Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
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
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'' SPDE's.
See also
*
Brownian surface
*
Kardar–Parisi–Zhang equation
*
Kushner equation
*
Malliavin calculus
*
Wick product
*
Zakai equation
References
Further reading
*
External links
*
* {{cite arXiv , title=An Introduction to Stochastic PDEs , first=Martin , last=Hairer , author-link=Martin Hairer , year=2009 , class=math.PR , eprint=0907.4178
Stochastic differential equations
Partial differential equations