Freidlin–Wentzell theorem

In mathematics, the Freidlin–Wentzell theorem (due to Mark Freidlin and Alexander D. Wentzell) is a result in the large deviations theory of stochastic processes.

Roughly speaking, the Freidlin–Wentzell theorem gives an estimate for the probability that a (scaled-down) sample path of an Itō diffusion will stray far from the mean path.

This statement is made precise using rate functions.

Let B be a standard Brownian motion on Rd starting at the origin, 0 ∈ Rd, and let Xε be an Rd-valued Itō diffusion solving an Itō stochastic differential equation of the form where the drift vector field b : Rd → Rd is uniformly Lipschitz continuous.

Then, on the Banach space C0 = C0([0, T]; Rd) equipped with the supremum norm ||⋅||∞, the family of processes (Xε)ε>0 satisfies the large deviations principle with good rate function I : C0 → R ∪ {+∞} given by if ω lies in the Sobolev space H1([0, T]; Rd), and I(ω) = +∞ otherwise.