Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces

Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).