  We study the relation between L\'evy processes under nonlinear expectations,
nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one
relation between nonlinear L\'evy processes and nonlinear Markovian convolution
semigroups. Second, we provide a condition on a family of infinitesimal
generators $(A_\lambda)_{\lambda\in\Lambda}$ of linear L\'evy processes which
guarantees the existence of a nonlinear L\'evy processes such that the
corresponding nonlinear Markovian convolution semigroup is a viscosity solution
of the fully nonlinear PDE $\partial_t u=\sup_{\lambda\in \Lambda} A_\lambda
u$. The results are illustrated with several examples.
