The Koopmanization of controlled nonlinear It\^o stochastic differential systems and its comparison with the Carleman embedding: new results

The Koopmanization embeds the bilinearization via the action of the infinitesimal stochastic Koopman operator on the observables associated with the controlled nonlinear It\^o stochastic differential system without explicit linearizations. The stochastic evolutions of controlled Markov processes assume the structure of controlled nonlinear It\^o stochastic differential equations. This paper sketches a Koopman operator framework for the filtering of the controlled nonlinear It\^o stochastic differential system. The major ingredients of this paper are the construction of the eigenfunctions, action of the infinitesimal stochastic Koopman operator, multi-dimensional It\^o differential rule and filtering concerning the controlled nonlinear It\^o stochastic differential system. In this paper, we illustrate the filtering in the Koopman setting for a polynomial system and compare with the filtering in the Carleman setting.