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 Russia n 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, +∞) × Ω -> Rd be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation dXt=b(Xt)dt+dBt 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 = −∇Φ, μ has Radon–Nikodym derivative dμ(x)dx=exp⁡(−2Φ(x)) and ∫𝐑dexp⁡(−2Φ(x))dx=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

* Voß, Jochen (2004). Some large deviation results for diffusion processes (Thesis). Universität Kaiserslautern: PhD thesis. (See theorem 1.4)

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