Some Non-Uniform Large Deviation Results

Abstract

Thus far we have a quite satisfactory theory of large deviations for Markov processes under the assumption (6.1), in case (D), or (6.8), in case (C). Unfortunately, when E is not compact, such an assumption will not be satisfied, except in very special situations (e.g. the Sanov theorem). For example, let E = R¹ and P(t,x,dy) = g(1-e^-2t,y-e^-tx)dy, where g(s, ξ) = (2πs)^-1/2 exp(-ξ²/2s). Clearly, the associated process (i.e. the Ornstein-Uhlenbeck process) has strong ergodic properties and one ought to be able to study the large deviation theory. At the same time, it is equally clear that P(t,x,•) fails to satisfy (6.8), and therefore the theory developed in section 6) is not applicable. Of course, the reason why section 6) cannot handle this process is clear; namely: when section 6) applies, the resulting large deviation principle is uniform, whereas one should not expect a uniform principle to hold for the Ornstein-Uhlenbeck process. Indeed, one can hope for uniform large deviation principles only in the presence of uniform ergodicity.