A Semigroup Approach to Nonlinear Lévy Processes

We study the relation between Levy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear Levy processes and nonlinear Markovian convolution semigroups. Second, we provide a condition on a family of infinitesimal generators (A(lambda))(lambda is an element of Lambda) of linear Levy processes which guarantees the existence of a nonlinear Levy process such that the corresponding nonlinear Markovian convolution semigroup is a viscosity solution of the fully nonlinear PDE partial derivative(t)u = sup(lambda is an element of Lambda) A(lambda)u. The results are illustrated with several examples. (C) 2019 Published by Elsevier B.V.