Reversible diffusion

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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.

[edit] 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

\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; Rd). Then the following are equivalent:

\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.)

[edit] References

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