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 : Rd → Rd be a Lipschitz continuous vector field. Let X : [0, +∞) &times; Ω → Rd 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. X0 ∈ L2(Ω, Σ, P; Rd). Then the following are equivalent:


 * The process X is reversible with stationary distribution μ on Rd.
 * There exists a scalar potential Φ : Rd → R such that b = &minus;∇Φ, μ has Radon–Nikodym derivative $$\frac{\mathrm{d} \mu (x)}{\mathrm{d} x} = \exp \left( - 2 \Phi (x) \right)$$ and $$\int_{\mathbf{R}^{d}} \exp \left( - 2 \Phi (x) \right) \, \mathrm{d} x = 1.$$

(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(&minus;2Φ(·)) is a probability density function with integral 1.)