Stochastic processes with Student-t marginals. Much as Student-\(t\) distributions generalise Gaussian distributions, \(t\)-processes generalise Gaussian processes.
There are a couple of classic cases in ML where \(t\)-processes arise, e.g. in Bayes NNs (Neal 1996) or GP literature (9.9 Rasmussen and Williams 2006). Recently there has been an uptick in actual applications of these processes in regression (Chen, Wang, and Gorban 2020; Shah, Wilson, and Ghahramani 2014; Tang et al. 2017; Tracey and Wolpert 2018). See Wilson and Ghahramani (2011) for a Generalized Wishart Process construction that may be helpful? This prior is available in GPyTorch. Recent papers (Shah, Wilson, and Ghahramani 2014; Tracey and Wolpert 2018) make it seem fairly straightforward.
I am interested in seeing if these can be pressed into service as a model for mis-specification in Gaussian process regression.
Some papers discuss this in term of inference using Inverse Wishart
Process with t-distributed increments is in fact a Lévy process, which follows from the fact that the Student-\(t\) distribution is divisible. As far as I can see here Grigelionis (2013) is the definitive collation of results on that observation.