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. X₀ ∈ L²(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 $$\frac{\mathrm{d}\mu (x)}{\mathrm{d}x}=\exp (-2\Phi (x))$$ and $$\underset{{𝐑}^d}{\int }\exp (-2\Phi (x))\mathrm{d}x=1.$$

(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

• (See theorem 1.4)