Editing Langevin equation (section)

== Mathematical aspects ==
A strictly <math>\delta</math>-correlated fluctuating force <math>\boldsymbol{\eta}\left( t\right)</math> is not a function in the usual mathematical sense and even the derivative <math>\mathrm d\mathbf{v}/\mathrm{d}t</math> is not defined in this limit. This problem disappears when the Langevin equation is written in integral form 
<math display="block">m\mathbf{v} = \int^t \left( -\lambda \mathbf{v} + \boldsymbol{\eta}\left( t\right)\right)\mathrm{d}t.</math> 

Therefore, the differential form is only an abbreviation for its time integral. The general mathematical term for equations of this type is "[[stochastic differential equation]]".<ref>{{cite journal | last1 = Huang | first1 = Yunfei. | display-authors = etal | year = 2022 | title = Sparse inference and active learning of stochastic differential equations from data | journal = Scientific Reports | volume = 12 | number = 1| page = 21691 | doi = 10.1038/s41598-022-25638-9  | pmid = 36522347 | doi-access = free| pmc = 9755218 }}</ref>

Another mathematical ambiguity occurs for Langevin equations with multiplicative noise, which refers to noise terms that are multiplied by a non-constant function of the dependent variables, e.g., <math>\left|\boldsymbol{v}(t)\right| \boldsymbol{\eta}(t)</math>. If a multiplicative noise is intrinsic to the system, its definition is ambiguous, as it is equally valid to interpret it according to Stratonovich- or Ito- scheme (see [[Itô calculus]]). Nevertheless, physical observables are independent of the interpretation, provided the latter is applied consistently when manipulating the equation. This is necessary because the symbolic rules of calculus differ depending on the interpretation scheme. If the noise is external to the system, the appropriate interpretation is the Stratonovich one.<ref name="vanKampen1981">{{cite journal | last=van Kampen | first=N. G. | title=Itô versus Stratonovich | journal=Journal of Statistical Physics | publisher=Springer Science and Business Media LLC | volume=24 | issue=1 | year=1981 | issn=0022-4715 | doi=10.1007/bf01007642 | pages=175–187| bibcode=1981JSP....24..175V | s2cid=122277474 }}</ref><ref name="Stochastic Processes in Physics and Chemistry 2007 p.">{{cite book | last=van Kampen | first=N. G. | title=Stochastic Processes in Physics and Chemistry | publisher=Elsevier | year=2007 | isbn=978-0-444-52965-7 | doi=10.1016/b978-0-444-52965-7.x5000-4 }}</ref>