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.

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. X 0  ∈ L 2 (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 \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(−2Φ(·)) is a probability density function with integral 1.) ==References== (See theorem 1.4)

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