Systems Driven by an Infinite Dimensional Brownian Noise

Abstract

In Chap. 8 we gave a representation for positive functionals of a Hilbert space valued Brownian motion. This chapter will apply the representation to study the large deviation properties of infinite dimensional small noise stochastic dynamical systems. In the application, the driving noise is given by a Brownian sheet, and so in this chapter we will present a sufficient condition analogous to Condition 9.1 (but there will be no Poisson noise in this chapter) that covers the setting of such noise processes (see Condition 11.15). Another formulation of an infinite dimensional Brownian motion that will be needed in Chap. 12 is as a sequence of independent Brownian motions regarded as a \(\mathscr {C}([0,T]:\mathbb {R}^{\infty })\)-valued random variable. We also present the analogous sufficient condition (Condition 11.12) for an LDP to hold for this type of driving noise.