Schilder's theorem

In mathematics, Schilder's theorem is a generalization of the Laplace method from integrals on

The theorem is used in the large deviations theory of stochastic processes.

Roughly speaking, out of Schilder's theorem one gets an estimate for the probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0).

This statement is made precise using rate functions.

Let C0 = C0([0, T]; Rd) be the Banach space of continuous functions

be the subspace of absolutely continuous functions whose derivative is in

Define the rate function on

(the "action") has a unique minimum

Then under some differentiability and growth assumptions on

denotes expectation with respect to the Wiener measure

Let B be a standard Brownian motion in d-dimensional Euclidean space Rd starting at the origin, 0 ∈ Rd; let W denote the law of B, i.e. classical Wiener measure.

For ε > 0, let Wε denote the law of the rescaled process √εB.

Then, on the Banach space C0 = C0([0, T]; Rd) of continuous functions

, equipped with the supremum norm ||⋅||∞, the probability measures Wε satisfy the large deviations principle with good rate function I : C0 → R ∪ {+∞} given by if ω is absolutely continuous, and I(ω) = +∞ otherwise.

In other words, for every open set G ⊆ C0 and every closed set F ⊆ C0, and Taking ε = 1/c2, one can use Schilder's theorem to obtain estimates for the probability that a standard Brownian motion B strays further than c from its starting point over the time interval [0, T], i.e. the probability as c tends to infinity.

Here Bc(0; ||⋅||∞) denotes the open ball of radius c about the zero function in C0, taken with respect to the supremum norm.

First note that Since the rate function is continuous on A, Schilder's theorem yields making use of the fact that the infimum over paths in the collection A is attained for ω(t) = ⁠t/T⁠ .

This result can be heuristically interpreted as saying that, for large c and/or large T In fact, the above probability can be estimated more precisely: for B a standard Brownian motion in Rn, and any T, c and ε > 0, we have: