Editing Stochastic differential equation (section)

===Terminology===
The most common form of SDEs in the literature is an [[ordinary differential equation]] with the right hand side perturbed by a term dependent on a [[white noise]] variable. In most cases, SDEs are understood as continuous time limit of the corresponding [[stochastic difference equation]]s. This understanding of SDEs is ambiguous and must be complemented by a proper mathematical definition of the corresponding integral.<ref name="rogerswilliams"/><ref name="oksendal"/> Such a mathematical definition was first proposed by [[Kiyosi Itô]] in the 1940s, leading to what is known today as the [[Itô calculus]].
Another construction was later proposed by Russian physicist [[Ruslan L. Stratonovich|Stratonovich]], 
leading to what is known as the [[Stratonovich integral]].
The [[Itô integral]] and [[Stratonovich integral]] are related, but different, objects and the choice between them depends on the application considered. The [[Itô calculus]] is based on the concept of non-anticipativeness or causality, which is natural in applications where the variable is time.
The Stratonovich calculus, on the other hand, has rules which resemble ordinary calculus and has intrinsic geometric properties which render it more natural when dealing with geometric problems such as random motion on [[manifolds]], although it is possible and in some cases preferable to model random motion on manifolds through Itô SDEs,<ref name="Emery"/> for example when trying to optimally approximate SDEs on submanifolds.<ref name="armstrongprojection">Armstrong, J., Brigo, D. and Rossi Ferrucci, E. (2019), Optimal approximation of SDEs on submanifolds: the Itô-vector and Itô-jet projections. Proc. London Math. Soc., 119: 176-213. https://doi.org/10.1112/plms.12226.</ref> 

An alternative view on SDEs is the stochastic flow of diffeomorphisms. This understanding is unambiguous and corresponds to the Stratonovich version of the continuous time limit of stochastic difference equations. Associated with SDEs is the [[Smoluchowski equation]] or the [[Fokker–Planck equation]], an equation describing the time evolution of [[Probability density function|probability distribution function]]s. The generalization of the Fokker–Planck evolution to temporal evolution of differential forms is provided by the concept of [[Supersymmetric theory of stochastic dynamics#Stochastic evolution operator|stochastic evolution operator]].

In physical science, there is an ambiguity in the usage of the term [[Langevin equation|"Langevin SDEs"]]. While Langevin SDEs can be of a [[Langevin equation#Generic Langevin equation|more general form]], this term typically refers to a narrow class of SDEs with gradient flow vector fields. This class of SDEs is particularly popular because it is a starting point of the Parisi–Sourlas stochastic quantization procedure,<ref>{{Cite journal|last1=Parisi|first1=G.|last2=Sourlas|first2=N.|date=1979|title=Random Magnetic Fields, Supersymmetry, and Negative Dimensions |journal=Physical Review Letters|volume=43|issue=11|pages=744–745|doi=10.1103/PhysRevLett.43.744|bibcode=1979PhRvL..43..744P}}</ref> leading to a N=2 supersymmetric model closely related to [[supersymmetric quantum mechanics]]. From the physical point of view, however, this class of SDEs is not very interesting because it never exhibits spontaneous breakdown of topological supersymmetry, i.e., [[Supersymmetric theory of stochastic dynamics#Spontaneous supersymmetry breaking and chaos|(overdamped) Langevin SDEs are never chaotic]].