A study was conducted to show that the composition of the generalized Levy Laplacian of order p and a certain linear transformation of the domain of the function to which it was applied was proportional to the generalized Lévy Laplacian of smaller or larger order. The construction was based on an expression of generalized Cesàro means in terms of ordinary ones. The study considered a relationship between the generalized Lévy Laplacian and quantum random processes. The study also defined generalized Cesàro means and demonstrated how to reduce calculating them to calculating usual Cesàro means. It was observed that the non-classical Lévy Laplacians coincided with the compositions of the classical Levy Laplacian and operators of the form Nq.