A semigroup approach to nonlinear L\'evy processes

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 $(A_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}$ 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.