  We consider a nonlinear random walk which, in each time step, is free to
choose its own transition probability within a neighborhood (w.r.t. Wasserstein
distance) of the transition probability of a fixed L\'evy process. In analogy
to the classical framework we show that, when passing from discrete to
continuous time via a scaling limit, this nonlinear random walk gives rise to a
nonlinear semigroup. We explicitly compute the generator of this semigroup and
corresponding PDE as a perturbation of the generator of the initial L\'evy
process.
