Freidlin–Wentzell theorem

Let B be a standard Brownian motion on R^d starting at the origin, 0 ∈ R^d, and let X^ε be an R^d-valued Itō diffusion solving an Itō stochastic differential equation of the form

${\displaystyle {\begin{cases}dX_{t}^{\varepsilon }=b(X_{t}^{\varepsilon })\,dt+{\sqrt {\varepsilon }}\,dB_{t},\\X_{0}^{\varepsilon }=0,\end{cases}}}$

where the drift vector field b : R^d → R^d is uniformly Lipschitz continuous. Then, on the Banach space C₀ = C₀([0, T]; R^d) equipped with the supremum norm ||·||_∞, the family of processes (X^ε)_ε>0 satisfies the large deviations principle with good rate function I : C₀ → R ∪ {+∞} given by

${\displaystyle I(\omega )={\frac {1}{2}}\int _{0}^{T}|{\dot {\omega }}_{t}-b(\omega _{t})|^{2}\,dt}$

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

${\displaystyle \limsup _{\varepsilon \downarrow 0}{\big (}\varepsilon \log \mathbf {P} {\big [}X^{\varepsilon }\in F{\big ]}{\big )}\leq -\inf _{\omega \in F}I(\omega )}$

and

${\displaystyle \liminf _{\varepsilon \downarrow 0}{\big (}\varepsilon \log \mathbf {P} {\big [}X^{\varepsilon }\in G{\big ]}{\big )}\geq -\inf _{\omega \in G}I(\omega ).}$

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