Freidlin-Wentzell theorem

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In mathematics, the Freidlin-Wentzell theorem 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. The Freidlin-Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.

[edit] Statement of the theorem

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

\begin{cases} \mathrm{d} X_{t}^{\varepsilon} = b(X_{t}^{\varepsilon}) \, \mathrm{d} t + \sqrt{\varepsilon} \, \mathrm{d} B_{t}; \\ X_{0}^{\varepsilon} = 0; \end{cases}

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

I(\omega) = \frac{1}{2} \int_{0}^{T} | \dot{\omega}_{t} - b(\omega_{t}) |^{2} \, \mathrm{d} t

if ω lies in the Sobolev space H1([0, T]; Rd), and I(ω) = +∞ otherwise. In other words, for every open set G ⊆ C0 and every closed set F ⊆ C0,

\limsup_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P} \big[ X^{\varepsilon} \in F \big] \leq - \inf_{\omega \in F} I(\omega)

and

\liminf_{\varepsilon \downarrow 0} \varepsilon \log \mathbf{P} \big[ X^{\varepsilon} \in G \big] \geq - \inf_{\omega \in G} I(\omega).

[edit] References

  • Freidlin, Mark I.; Wentzell, Alexander D. (1998). Random perturbations of dynamical systems, Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260, New York: Springer-Verlag, pp. xii+430. ISBN 0-387-98362-7.  MR1652127
  • Dembo, Amir; Zeitouni, Ofer (1998). Large deviations techniques and applications, Second edition, Applications of Mathematics (New York) 38, New York: Springer-Verlag, pp. xvi+396. ISBN 0-387-98406-2.  MR1619036 (See chapter 5.6)