  This paper introduces Switching Processes, called SP. Their constructions are
inspired by the PDMP's ones (PDMP stands for Piecewise Deterministic Markov
Process). A Markov process, called the intrinsic process, replaces the PDMP's
flow. Jumps are added ; they occur randomly as their locations ; their
distributions depend on the process's trajectory between them. When the
intrinsic process is a Levy process, thanks to its L{\'e}vy-It{\^o}
decomposition as a semi-martingale, we obtain the expected Kolmogorov equations
for the SP. The results are extended to It{\^o}-L{\'e}vy processes, in
particular to diffusion processes.
