([0, 1]) given by X = {f 1 0

Next define a new process on the same canonical space: Xt = T G(s, t) dBs, where {Bs, s T} is the standard BM on the same space. This being a linear transformation, {Xt, t T} is also a centered Gaussian process having the same covariance function r as the Xt-process. Since a Gaussian process is determined by its mean and covariance functions, these two processes can be identified, Xt = Xt, a.e., t T. Now f = R½h, h L2 (T,dt) and R-½ exists, and so by Theorem 5, one has the following exact form of the conjugate function of :

(f) = 12 R –½

f 2 = 1 2

(R-1 f, f) L2(T, dt)), f M = R½ L2(T, dt)), (31) and (f) = , otherwise. This may be summerized as:

Theorem 6. Let {X t . Xt,t T, > 0} be a centered Gaussian process

with a continuous covariance function. Then X t -process obeys the large de

viation principle with the action or rate functional given by (20).