Upper Bound Estimates of the Cramér Functionals for Markov Processes

We prove that the Cramér functional of a Markov process can be controlled by a function of an integral functional if the transition semigroup is uniformly integrable in L ^p . As an application of this result, a general large deviation upper bound is obtained. Then, the notation of F-Sobolev inequality is extended to general Markov processes by replacing the Dirichlet form with the Donsker–Varadhan entropy. As the other application, it is proved that the uniform integrability of a transition semigroup implies a F-Sobolev inequality.