We consider a particular class of multidimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant over cones. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the n-dimensional space and can be extremely resistant to a more detailed understanding. One can see that the structure of the invariant probability measure is intertwined with the geometry of the cones where the drift function is a constant. In this article we pursue results that make this connection precise. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. As broad classes of examples we analyze interactions defined by Coxeter groups actions and weighted graphs by this method.