Editing Brownian motion (section)

===Riemannian manifolds===
[[File:BMonSphere.jpg|thumb|upright=1.8|Brownian motion on a sphere]]
Brownian motion is usually considered to take place in [[Euclidean space]]. It is natural to consider how such motion generalizes to more complex shapes, such as [[surface]]s or higher dimensional [[manifold]]s. The formalization requires the space to possess some form of a [[derivative]], as well as a [[metric space|metric]], so that a [[Laplacian]] can be defined. Both of these are available on [[Riemannian manifold]]s.

Riemannian manifolds have the property that [[geodesic]]s can be described in [[polar coordinates]]; that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space.

The [[Infinitesimal generator (stochastic processes)|infinitesimal generator]] (and hence [[characteristic operator]]) of Brownian motion on Euclidean {{math|'''R'''<sup>''n''</sup>}} is {{math|{{sfrac|1|2}}Δ}}, where {{math|Δ}} denotes the [[Laplace operator]]. Brownian motion on an {{mvar|m}}-dimensional [[Riemannian manifold]] {{math|(''M'', ''g'')}} can be defined as diffusion on {{mvar|M}} with the characteristic operator given by {{math|{{sfrac|1|2}}Δ<sub>LB</sub>}}, half the [[Laplace–Beltrami operator]] {{math|Δ<sub>LB</sub>}}.

One of the topics of study is a characterization of the [[Poincaré recurrence theorem|Poincaré recurrence time]] for such systems.<ref name=Grigoryan-1999/>