Diffusion process: Difference between revisions

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In [[probability theory]] and [[statistics]], a '''diffusion process''' is a solution to a [[stochastic differential equation]]. It is a continuous-time [[Markov process]] with [[almost surely]] [[continuous function|continuous]] sample paths. [[Brownian motion]], [[reflected Brownian motion]] and [[Ornstein–Uhlenbeck processes]] are examples of diffusion processes.

A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called [[Brownian motion]]. The position of the particle is then random; its [[probability density function]] as a [[function of space and time]] is governed by an [[advection equation|advection]]-–[[diffusion equation]].

== Mathematical definition ==

A ''diffusion process'' is a [[Markov process]] with [[Sample-continuous_process|continuous sample paths]] for which the [[Kolmogorov_equations|Kolmogorov forward equation]] is the [[Fokker-PlanckFokker–Planck equation]].<ref>{{cite web|title=9. Diffusion processes|url=http://math.nyu.edu/faculty/varadhan/stochastic.fall08/sec10.pdf|format=pdf|accessdate=October 10, 2011}}</ref>

== See also ==