Extremes of L^p-norm of vector-valued Gaussian processes with trend

Let (Formula presented.) be a Gaussian vector process and (Formula presented.) be a continuous function. The asymptotics of distribution of (Formula presented.), the (Formula presented.) norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution, we are concerned with the exact tail asymptotics of (Formula presented.) with trend (Formula presented.) over (Formula presented.). Both scenarios that (Formula presented.) is locally stationary and non-stationary are considered. Important examples include (Formula presented.) and chi-square processes with trend, i.e. (Formula presented.). These results are of interest in applications in engineering, insurance, and statistics, etc.