Small Deviations for Some Gaussian Chaos

The small ball probability or small deviation studies the behavior of log (x: jjxjj ") as " tend to zero for a given measure and a norm k k on a Banach space. In the literature, small deviation probabilities of various types are studied and applied to many problems of interest under dierent names such as small value probabilities, lower tail behavior, two sided boundary crossing probabilities, the rst exit probabilities, the asymptotes of Laplace transforms, etc. We will present recent developments for certain family of Gaussian chaos dened by stochastic integrals. Connections and interplays between Gaussian chaos and Markov processes are viewed through simple examples