# arg max of a Gaussian process

So suppose I have some random function distributed as a Gaussian process on say, the interval $[0,1]$ with mean function $\mu$ and covariance function $\Lambda$. When is there a nice formula for the probability that the $\arg \max$ of the random function lies in some sub-interval $[a,b]$? Obviously there are trivial cases, like where the mean function is zero the covariance function is translation invariant, but what about other cases? Any help appreciated, even if it just pointing me to resources that might help, thanks.

First note that if we define a random variable $$M = \text{arg max}_t \Lambda(t)$$, then your question is equivalent to asking when does the cumulative distribution function of $$M$$ have a nice formula. One nontrivial case where this happens is when $$\Lambda$$ is a Brownian motion (see the third arcsine law).