Large deviations for functionals of Gaussian processes

Xiaoming Song

Published 2018-02-12Version 1

We prove large deviation principles for $\int_0^t \gamma(X_s)ds$, where $X$ is a $d$-dimensional Gaussian process and $\gamma(x)$ takes the form of the Dirac delta function $\delta(x)$, $|x|^{-\beta}$ with $\beta\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-\beta_i}$ with $\beta_i>0$ and $\sum_{i=1}^d\beta_i\in(0,d)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, bi-fractional Brownian motion, and sub-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.