  The index Whittaker convolution operator, recently introduced by the authors,
gives rise to a convolution measure algebra having the property that the
convolution of probability measures is a probability measure. In this paper, we
introduce the class of L\'evy processes with respect to the index Whittaker
convolution and study their basic properties. We prove that the square root of
the Shiryaev process belongs to our family of L\'evy process, and this is shown
to yield a martingale characterization of the Shiryaev process analogous to
L\'evy's characterization of Brownian motion.
  Our results demonstrate that a nice theory of L\'evy processes with respect
to generalized convolutions can be developed even if the usual compactness
assumption on the support of the convolution fails, shedding light into the
connection between the properties of the convolution algebra and the nature of
the singularities of the associated differential operator.
