Gaussian Processes - Wiener process - The drunk particle


I recently read about Gaussian Processes ( for non linear regression. I am not an expert but I found it quite interesting how to see for example a wiener process from the Gaussian processes perspective.

An interesting example came to my mind.

Imaging that we have a particle moving in a 1D space Wiener process. So, each incremental random step (stationary increments) follow a Gaussian distribution N(u,sigma^2)..

Let's say, there is a relationship between the stationary increments and the probability of the particle of being drunk p(drunk) = 1-p(no_drunk). Let say, if the increment are within sigma (68%) the probability of being drunk is less than if they are bigger and bigger. Let's assume that when the increments are closer to 3-sigma the particle is more drunk that in 2-sigma and almost no-drunk within the sigma region.

Consider that the drunk effect is a dynamic process, so we cannot observe the particle at one single instant time, we need to observe it in a window (e.g.: 10 steps) in order to evaluate how much drunk the particle is. Because the fact of being drunk evolves like e.g.: normal->drunk->verydrunk->lessdrunk->normal

The first question I have is: How do we map the probability of being drunk p(drunk) w.r.t the Wiener process? Is the Wiener process the correct model for it?

The second one is: Can we learn hyperparameters using Gaussian processes? In order to predict if the particle has a certain probability of being drunk in the future (prediction). If this is possible, which hyperparameters to learn, because a Wiener process is mainly driven by sigma (std deviation)

Thanks in advance,