Elements of the Stochastic Calculus for a Class of Boltzmann type processes and Application to Regularity of Densities

For a class of piecewise deterministic Markov processes we introduce a stochastic calculus which is a certain non-Gaussian counterpart to the classical Malliavin calculus. As an application we investigate the regularity of densities of $N$-particle Boltzmann type processes at time $t>0$.