19 - Support theorem and large deviations

Summary

We now discuss some classical results of diffusion theory: the Stroock–Varadhan support theorem and Freidlin–Wentzell large deviation estimates. Everything relies on the fact that the Stratonovich SDE

can be solved as an RDE solution which depends continuously on enhanced Brownian motion in rough path topology, subject to the suitable Lipregularity assumptions on the vector fields. (A summary of the relevant continuity results was given in Section 17.1.)

Support theorem for SDEs driven by Brownian motion

Theorem 19.1 (Stroock–Varadhan support theorem)Assume that V = (V₁, …, V_d) is a collection of Lip²-vector fields on ℝ^e, and V₀is a Lip¹-vector field on ℝ^e. Let B be a d-dimensional Brownian motion and consider the unique (up to indistinguishability) Stratonovich SDE solution Y on [0, T] to

Let us write y^h = π(V, V₀) (0, y₀; (h, t)) for the ODE solution to

started at y₀ ∈ ℝ^ewhere h is a Cameron–Martin path, i.e. h. Then, for any α ∈ [0, 1/2) and any δ > 0,

(where the Euclidean norm is used for conditioning |B − h|_∞;[0, T] < ∈).

Proof. Without loss of generality, α ∈ (1/3, 1/2). Let us write, for a fixed Cameron–Martin path h,

From Theorem 17.3, there is a unique solution π(V, V₀) (0, y₀; (B, t)) to the RDE with drift

which then solves the Stratonovich equation (19.1).