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Freidlin–Wentzell theorem

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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. The Freidlin–Wentzell theorem generalizes Schilder's theorem for standard Brownian motion.

Statement

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

{ d X t ε = b ( X t ε ) d t + ε d B t , X 0 ε = 0 , {\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 → R is uniformly Lipschitz continuous. Then, on the Banach space C0 = C0(; R) equipped with the supremum norm ||⋅||∞, the family of processes (X)ε>0 satisfies the large deviations principle with good rate function I : C0 → R ∪ {+∞} given by

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

if ω lies in the Sobolev space H(; R), and I(ω) = +∞ otherwise. In other words, for every open set G ⊆ C0 and every closed set F ⊆ C0,

lim sup ε ↓ 0 ( ε log ⁡ P [ X ε ∈ F ] ) ≤ − inf ω ∈ F I ( ω ) {\displaystyle \limsup _{\varepsilon \downarrow 0}{\big (}\varepsilon \log \mathbf {P} {\big }{\big )}\leq -\inf _{\omega \in F}I(\omega )}

and

lim inf ε ↓ 0 ( ε log ⁡ P [ X ε ∈ G ] ) ≥ − inf ω ∈ G I ( ω ) . {\displaystyle \liminf _{\varepsilon \downarrow 0}{\big (}\varepsilon \log \mathbf {P} {\big }{\big )}\geq -\inf _{\omega \in G}I(\omega ).}

References

  • Freidlin, Mark I.; Wentzell, Alexander D. (1998). Random perturbations of dynamical systems. Grundlehren der Mathematischen Wissenschaften 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)
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