  This third part extends the theory of Generalized Poisson-Kac (GPK) processes
to nonlinear stochastic models and to a continuum of states. Nonlinearity is
treated in two ways: (i) as a dependence of the parameters (intensity of the
stochastic velocity, transition rates) of the stochastic perturbation on the
state variable, similarly to the case of nonlinear Langevin equations, and (ii)
as the dependence of the stochastic microdynamic equations of motion on the
statistical description of the process itself (nonlinear Fokker-Planck-Kac
models). Several numerical and physical examples illustrate the theory.
Gathering nonlinearity and a continuum of states, GPK theory provides a
stochastic derivation of the nonlinear Boltzmann equation, furnishing a
positive answer to the Kac's program in kinetic theory. The transition from
stochastic microdynamics to transport theory within the framework of the GPK
paradigm is also addressed.
