Editing Stochastic differential equation (section)

==Use in physics==
{{See also|Langevin equation}}
In physics, SDEs have wide applicability ranging from molecular dynamics to neurodynamics and to the dynamics of astrophysical objects. More specifically, SDEs describe all dynamical systems, in which quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the [[dynamical systems theory]] to models with noise. This is an important generalization because real systems cannot be completely isolated from their environments and for this reason always experience external stochastic influence.

There are standard techniques for transforming higher-order equations into several coupled first-order equations by introducing new unknowns. Therefore, the following is the most general class of SDEs:

:<math id="genSDE">\frac{\mathrm{d}x(t)}{\mathrm{d}t} = F(x(t)) + \sum_{\alpha=1}^ng_\alpha(x(t))\xi^\alpha(t),\,</math>

where <math>x\in X </math> is the position in the system in its [[phase space|phase (or state) space]], <math>X</math>, assumed to be a differentiable manifold, the <math>F\in TX</math> is a flow vector field representing deterministic law of evolution, and <math>g_\alpha\in TX </math> is a set of vector fields that define the coupling of the system to Gaussian white noise, <math>\xi^\alpha</math>. If <math> X </math> is a linear space and <math>g</math> are constants, the system is said to be subject to additive noise, otherwise it is said to be subject to multiplicative noise. For additive noise, the Itô and Stratonovich forms of the SDE generate the same solution, and it is not important which definition is used to solve the SDE. For multiplicative noise SDEs the Itô and Stratonovich forms of the SDE are different, and care should be used in mapping between them.<ref>{{Cite book| edition = 4th| publisher = Springer-Verlag| isbn = 978-3-540-70712-7| last = Gardiner| first = C W| title = Handbook of Stochastic Methods| location = Berlin| year = 2009}}</ref>

For a fixed configuration of noise, SDE has a unique solution differentiable with respect to the initial condition.<ref>{{Cite journal|last=Slavík|first=A.|title=Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters|journal=Journal of Mathematical Analysis and Applications|language=en|volume=402|issue=1|pages=261–274|doi=10.1016/j.jmaa.2013.01.027|year=2013|doi-access=free}}</ref> Nontriviality of stochastic case shows up when one tries to average various objects of interest over noise configurations. In this sense, an SDE is not a uniquely defined entity when noise is multiplicative and when the SDE is understood as a continuous time limit of a [[stochastic difference equation]]. In this case, SDE must be complemented by what is known as "interpretations of SDE" such as Itô or a Stratonovich interpretations of SDEs. Nevertheless, when SDE is viewed as a continuous-time stochastic flow of diffeomorphisms, it is a [[Supersymmetric theory of stochastic dynamics#Resolution of Ito–Stratonovich dilemma and operator ordering conventions|uniquely defined mathematical object]] that corresponds to Stratonovich approach to a continuous time limit of a stochastic difference equation.

In physics, the main method of solution is to find the [[probability distribution]] function as a function of time using the equivalent [[Fokker–Planck equation]] (FPE). The Fokker–Planck equation is a deterministic [[partial differential equation]]. It tells how the probability distribution function evolves in time similarly to how the  [[Schrödinger equation]] gives the time evolution of the quantum wave function or the [[diffusion equation]] gives the time evolution of chemical concentration. Alternatively, numerical solutions can be obtained by [[Monte Carlo Method|Monte Carlo]] simulation. Other techniques include the [[path integral formulation|path integration]] that draws on the analogy between statistical physics and [[quantum mechanics]] (for example, the Fokker-Planck equation can be transformed into the [[Schrödinger equation]] by rescaling a few variables) or by writing down [[ordinary differential equations]] for the statistical [[moment (mathematics)|moments]] of the probability distribution function. {{Citation needed|date=August 2011}}