  We investigate the connections between the mean pathwise regularity of
stochastic processes and their L^r(P)-functional quantization rates as random
variables taking values in some L^p([0,T],dt)-spaces (0 < p <= r). Our main
tool is the Haar basis. We then emphasize that the derived functional
quantization rate may be optimal (e.g., for Brownian motion or symmetric stable
processes) so that the rate is optimal as a universal upper bound. As a first
application, we establish the O((log N)^{-1/2}) upper bound for general It\^o
processes which include multidimensional diffusions. Then, we focus on the
specific family of L\'evy processes for which we derive a general quantization
rate based on the regular variation properties of its L\'evy measure at 0. The
case of compound Poisson processes, which appear as degenerate in the former
approach, is studied specifically: we observe some rates which are between the
finite-dimensional and infinite-dimensional ``usual'' rates
