Extremes DOI 10.1007/s10687-016-0239-3 High extrema of Gaussian chaos processes Vladimir I. Piterbarg 1 Received: 12 September 2015 / Revised: 13 January 2016 / Accepted: 19 January 2016 © Springer Science+Business Media New York 2016 Abstract Let ξ (t) = (ξ 1 (t),...,ξ d (t)) be a Gaussian stationary vector process. Let g : R d → R be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(ξ (t)). Important examples include g(ξ (t)) =  d i =1 ξ i (t) and g(ξ (t)) = ∑ d i =1 a i ξ 2 i (t). 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. Keywords Gaussian chaos · Wiener chaos · Gaussian vector processes · Large excursions · Asymptotic methods · Double sum method AMS 2000 Subject Classification Primary 60G15; Secondary 60K30 · 60K40 · 60G70 1 Introduction Let g(x), x ∈R d ,d ≥ 2, be a homogeneous function of a positive order β> 0, meaning that g(ax) = a β g(x) for any a> 0 and all x. Assume that g(x)> 0 for some x. Furthermore, let ξ (t), t ∈ R, be an R d -valued Gaussian vector process. We study the behavior of the probability P  max t ∈[0,p] g(ξ (t)) > u  ,p> 0, (1)  Vladimir I. Piterbarg piter@mech.math.msu.su 1 Lomonosov Moscow State University, Moscow, Russia