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

Motivation/Background
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I was investigating whether it would be good idea to use [Gaussian Process Regression](http://www.gaussianprocess.org) 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 $\textrm{Pr}\left\{\exists t \in \left[t\_1, t\_2\right] \text{ s.t. } f(t) \geq K\right\}$?
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> 
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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.