Let $X=(X_t)_{t \in S}$ be a real-valued Gaussian random process indexed by S with mean zero. General upper and lower estimates are given for the lower tail probability $\mathbb{P}(\sup_{t \in S} (X_t-X_{t_0}) \leq x )$ as $x \to 0$, with $t_0\in S$ fixed. In particular, sharp rates are given for fractional Brownian sheet. Furthermore, connections between lower tail probabilities for Gaussian processes with stationary increments and level crossing probabilities for stationary Gaussian processes
are studied. Our methods also provide useful information on a random pursuit problem for fractional Brownian particles.