Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome.

## Motivation/Background

I was investigating whether it would be good idea to use Gaussian Process Regression in my application. In the usual use case one conditions a Gaussian process using sample values and that results in a predictive distribution that can be used for a test input. This is a nice approach for making predictions for a single point, however in my application I need to work with intervals. So here comes the question:

Let $K \in \mathbb{R}$ and $f(t)$ be a Gaussian random process with the known mean function $m(t)$ and the known covariance function $c(t_1, t_2)$. What is $\textrm{Pr}\left\{\exists t \in \left[t_1, t_2\right] \text{ s.t. } f(t) \geq K\right\}$?

With words, I am trying to find the probability of the event that the process will exceed a threshold at least at one point during the interval.

I am interested in both analytical and numerical approaches. You are welcome to assume specific forms for $m(t)$ and $c(t_1, t_2)$ (e.g. $m(t) = 0$ and $c(t_1, t_2) = \text{exp}\left(-\frac{1}{2}\left|t_1 - t_2\right|\right)$) to demonstrate an approach.

Note: Edited to fix typos.