  L\'evy walk process is one of the most effective models to describe
superdiffusion, which underlies some important movement patterns and has been
widely observed in the micro and macro dynamics. From the perspective of random
walk theory, here we investigate the dynamics of L\'evy walks under the
influences of the constant force field and the one combined with harmonic
potential. Utilizing Hermite polynomial approximation to deal with the
spatiotemporally coupled analysis challenges, some striking features are
detected, including non Gaussian stationary distribution, faster diffusion, and
still strongly anomalous diffusion, etc.
