Freidlin-Wentzell theorem

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.

tatement of the 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

:

where the drift vector field "b" : R^"d" → R^"d" is uniformly Lipschitz continuous. Then, on the Banach space "C"₀ = "C"₀( [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" : "C"₀ → 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 "H"¹( [0, "T"] ; R^"d"), and "I"("ω") = +∞ otherwise. In other words, for every open set "G" ⊆ "C"₀ and every closed set "F" ⊆ "C"₀,

:

and

:

References

* cite book
last = Freidlin
first = Mark I.
coauthors = Wentzell, Alexander D.
title = Random perturbations of dynamical systems
series = Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 260
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 1998
pages = pp. xii+430
isbn = 0-387-98362-7 MathSciNet|id=1652127
* cite book
last= Dembo
first = Amir
coauthors = Zeitouni, Ofer
title = Large deviations techniques and applications
series = Applications of Mathematics (New York) 38
edition = Second edition
publisher = Springer-Verlag
location = New York
year = 1998
pages = pp. xvi+396
isbn = 0-387-98406-2 MathSciNet|id=1619036 (See chapter 5.6)