Extremes of Gaussian chaos processes with Trend

Let $\mathit{X}(t)=(X_1(t),\dots ,X_d(t)),t\in [0,S]$ be a Gaussian vector process and let $g(\mathit{x}),\mathit{x}\in {\mathbb{R}}^d$ be a continuous homogeneous function. In this paper we are concerned with the exact tail asymptotics of the chaos process $g(\mathit{X}(t))+h(t),t\in [0,S]$ with trend function $h$. Both scenarios $\mathit{X}(t)$ is locally-stationary and $\mathit{X}(t)$ is non-stationary are considered. Important examples include the product of Gaussian processes and chi-processes.