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 Ã¢Â†Â’ R^d be a Lipschitz continuous vector field. Let X : [0, +∞) × ÃŽÂ© Ã¢Â†Â’ R^d be an Itō diffusion defined on a probability space (Ω, ÃŽÂ£, P) and solving the Itō stochastic differential equation

with square-integrable initial condition, i.e. X₀ Ã¢ÂˆÂˆ L²(Ω, ÃŽÂ£, 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 Ã¢Â†Â’ R such that b = −∇Φ, μ has Radon–Nikodym derivative and

(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

• (See theorem 1.4)