Large deviations of infinite intersections of events in Gaussian processes

The large deviations principle for Gaussian measures in Banach space is given by the generalized Schilder's theorem. After assigning a norm ||f|| to paths f in the reproducing kernel Hilbert space of the underlying Gaussian process, the probability of an event A can be studied by minimizing the norm over all paths in A. The minimizing path f*, if it exists, is called the most probable path and it determines the corresponding exponential decay rate. The main objective of our paper is to identify the most probable path for the class of sets A that are such that the minimization is over a closed convex set in an infinite-dimensional Hilbert space. The `smoothness' (i.e., mean-square differentiability) of the Gaussian process involved has a crucial impact on the structure of the solution. Notably, as an example of a non-smooth process, we analyze the special case of fractional Brownian motion, and the set A consisting of paths f at or above the line t in [0,1]. For H>1/2, we prove that there is an s such that 0<s<1/2 and that the optimum path is at the "diagonal" on [0,s] and at t=1, whereas it is strictly above the diagonal for on (s,1); for H<1/2 an analogous result is derived. For smooth processes, such as integrated Ornstein-Uhlenbeck, f* has an essentially different nature, and is found by imposing conditions also on the derivatives of the path. 2000 Mathematics Subject Classification: 60G15, 60K25, 60F10