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'' :  , +∞) × Ω → R^''d'' be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation
$\mathrm X_ = b(X_) \, \mathrm t + \mathrm B_$
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 $\frac = \exp \left( - 2 \Phi (x) \right)$ and $\int_ \exp \left( - 2 \Phi (x) \right) \, \mathrm x = 1.$

(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 thesis
, last = Voß
, first = Jochen
, title = Some large deviation results for diffusion processes
, year = 2004
, publisher = PhD thesis
, location = Universität Kaiserslautern
, url = https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1559
(See theorem 1.4)

Stochastic differential equations
Probability theorems