Extremes and crossings for differentiable stationary processes with application to Gaussian processes in m and Hilbert space

Let {[omega](t)}t[greater-or-equal, slanted]0 be a stochastically differentiable stationary process in m and let satisfy limu[short up arrow]u2P{[omega](0) [set membership, variant] Au} = 0. We give a method to find the asymptotic behaviour of P{[union operator]0[less-than-or-equals, slant]t[less-than-or-equals, slant]h{[omega](t) [set membership, variant] Au}} as u [short up arrow]u2. We use our method to study hitting probabilities for small sets with application to Gaussian processes and to study suprema of processes in with application to (the norm of) Gaussian processes in Hilbert space.