Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(0).

What can be said on the infimum of $g(t)$? For example, what is the distribution of $\inf_A g(t)$? What is the mean? What is the median?

Related and perhaps simpler version would be to ask what is $p(c)=Pr(\{g(t)>c\ :\ \forall t\in A\})$? Or at least what is $p(0)$, the probability of no zero crossing in $[0,T]$?

Note that unlike [related question][1] I'm not interested in the asymptotic behavior for large $c$ but rather what is the probability of infimum near 0 (or how $p(c)$ behave for small $c$).


  [1]: http://mathoverflow.net/questions/99001/calculating-the-probability-of-an-event-defined-by-a-condition-on-a-gaussian-ran