In this project, we will expand on recent developments in Koopman operator theory for stochastic control systems. It has been shown that the Koopman approach leads to an elegant, low-dimensional, and low-complexity representation for systems which are affine-linear with respect to control, but may be fully non-linear otherwise. This setting, however, is too simple for many complex systems of interest. A natural generalization is multi-linearity of the control input. This opens up a direct connection to tensor decompositions and low-rank representations, which have been studied extensively in numerical linear algebra, and applied successfully in multiple disciplines. The goal of this project is to explore the theoretical foundations of this setting and to develop efficient algorithms for practical applications. In particular, we also aim to explore known connections to data-driven reduced-order modelling in this context.