Reversible diffusion explained

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$$\backslash mathrm\; X\_\; =\; b(X\_)\; \backslash ,\; \backslash mathrm\; t\; +\; \backslash mathrm\; B\_$$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 $$\backslash frac\; =\; \backslash exp\; \backslash left(-\; 2\; \backslash Phi\; (x)\; \backslash right)$$ and $$\backslash int\_\; \backslash exp\; \backslash left(-\; 2\; \backslash Phi\; (x)\; \backslash right)\; \backslash ,\; \backslash mathrm\; x\; =\; 1.$$

(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

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