Extremes of Gaussian processes with a smooth random trend

Let ?(t), t ? R, be a Gaussian zero mean stationary process, and ?(t) another random process, smooth enough, being independent of ?(t). We will consider the process ?(t) + ?(t) such that conditioned on ?(t) it is a Gaussian process. We want to establish an asymptotic exact result for P (t?[o,T] sup (?(t) + ?(t)) > u), as u ? ?, where T > 0.