High extrema of Gaussian chaos processes

Let $\xi(t)$ be a Gaussian stationary vector process. Let $g : R^d \rightarrow R$ be a homogeneous function of positive index. We study probabilities of large extrema of the Gaussian chaos process $g(\xi(t))$. Important examples include products of Gaussian processes and quadratic forms of them. We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process' trajectories.