Answer by Yoav Kallus for Infimum of Gaussian process

Googling "Stationary Gaussian process" and "level crossing" or "first passage" brings up a lot of relevant-looking material. This one seems to address your question most directly out of the ones I peeked at: Zero Crossing Probabilities for Gaussian Stationary Processes by G. F. Newell and M. Rosenblatt. The authors look at the quantity $H_X(T) = P[X(t)>0\text{ for all } 0\le t \le T]$ and establish some upper and lower bounds.