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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 → R be a Lipschitz continuous vector field. Let X : [0, +∞) × Ω → R be an Itō diffusion defined on a probability space (Ω, Σ, P) and solving the Itō stochastic differential equation $${\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\mathrm {d} B_{t}}$$ with square-integrable initial condition, i.e. X₀ ∈ L(Ω, Σ, P; R). Then the following are equivalent:

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

(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(−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)

• Stochastic differential equations
• Probability theorems