We study the infinitesimal generators of evolutions of linear mappings on the space of polynomials, which correspond to a special class of Markov processes with polynomial regressions called quadratic harnesses. We relate the infinitesimal generator to the unique solution of a certain commutation equation, and we use the commutation equation to find an explicit formula for the infinitesimal generator of free quadratic harnesses.

We consider the stochastic differential equation (1) $du\left(t\right)=a(u\left(t\right),\xi \left(t\right))dt+{\int }_{\Theta }\sigma {(u\left(t\right),\theta )}_p(dt,d\theta )$ for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${P^t}_{t\ge 0}$ corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup ${P^t}_{t\ge 0}$ describing the evolution of measures along trajectories and vice versa.