  Feynman's path integral approach is studied in the framework of the
Wigner-Dunkl deformation of quantum mechanics. We start with reviewing some
basics from Dunkl theory and investigate the time evolution of a Gaussian wave
packet, which exhibits the same dispersion relation as observed in standard
quantum mechanics. Feynman's path integral approach is then extended to
Wigner-Dunkl quantum mechanics. The harmonic oscillator problem is solved
explicitly. We then look at the Euclidean time evolution and the related Dunkl
process. This process, which exhibit jumps, can be represented by two
continuous Bessel processes, one with reflection and one with absorbtion at the
origin. The Feynman-Kac path integral for the harmonic oscillator problem is
explicitly calculated.
