  The standard Levy walk is performed by a particle that moves ballistically
between randomly occurring collisions, when the intercollision time is a random
variable governed by a power-law distribution. During instantaneous collision
events the particle randomly changes the direction of motion but maintains the
same constant speed. We generalize the standard model to incorporate velocity
fluctuations into the process. Two types of models are considered, namely, (i)
with a walker changing the direction and absolute value of its velocity during
collisions only, and (ii) with a walker whose velocity continuously fluctuates.
We present full analytic evaluation of both models and emphasize the importance
of initial conditions. We show that the type of the underlying Levy walk
process can be identified by looking at the ballistic regions of the diffusion
profiles. Our analytical results are corroborated by numerical simulations.
