Gaussian Processes in particular can be very useful in regression problems when you don't want to make strong assumptions about the functional form of the relationship between variables. (You can still introduce more general assumptions, like that the relationship is "smooth" to some degree or is periodic, by your choice of GP kernel function.)
In academia people study stochastic versions of PDEs in order to try to answer regularity and existence questions. Think for example about the famous millennium problem of Navier-Stokes. Sometimes the stochastic viewpoint can even give more results about the non-stochastic setting.
Unless you assume that everything/anything is a constant, stochastic processes are everywhere. They happen to be used a lot in finance and physics, since real world phenomena can be intuitively modeled as a random process, in contrast with deterministic models.
If you're interested try googling "stochastic processes in [insert field]".
We use GPs for our own Bayesian optimization of search engine parameters for relevance. I suspect it could also be a backbone of doing active learning to learn about the search or recos click data’s blind spots. (Basically also Bayesian optimization, but a stronger explore bias)