Reversible diffusion

In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov.

Kolmogorov's characterization of reversible diffusions

Let "B" denote a "d"-dimensional standard Brownian motion; let "b" : R^"d" → R^"d" be a Lipschitz continuous vector field. Let "X" : [0, +∞) × Ω → R^"d" be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation

:$mathrm\{d\}\; X\_\{t\}\; =\; b(X\_\{t\})\; ,\; mathrm\{d\}\; t\; +\; mathrm\{d\}\; B\_\{t\}$

with square-integrable initial condition, i.e. "X"₀ ∈ "L"²(Ω, Σ, P; R^"d"). Then the following are equivalent:

* The process "X" is reversible with stationary distribution "μ" on R^"d".

* There exists a scalar potential Φ : R^"d" → R such that "b" = −∇Φ, "μ" has Radon-Nikodym derivative

::$frac\{mathrm\{d\}\; mu\; (x)\}\{mathrm\{d\}\; x\}\; =\; exp\; left(\; -\; 2\; Phi\; (x)\; ight)$

:and

::$int\_\{mathbf\{R\}^\{d\; exp\; left(\; -\; 2\; Phi\; (x)\; ight)\; ,\; mathrm\{d\}\; x\; =\; 1.$

(Of course, the condition that "b" be the negative of the gradient of Φ only determines Φ up to an additive constant; this constant may be chosen so that exp(−2Φ(·)) is a probability density function with integral 1.)

References

* cite book
last = Voß
first = Jochen
title = Some large deviation results for diffusion processes
year = 2004
publisher = PhD thesis
location = Universität Kaiserslautern
language = English (See theorem 1.4)