On infinitesimal generators of sublinear Markov semigroups

We establish a Dynkin formula and a Courr`ege-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: $$Af(x)=\underset{\alpha \in I}{\sup }(-q_{\alpha }(x,D)f)(x)\mathrm{.}$$ As an immediate consequence, we obtain a representation formula for infinitesimal generators of sublinear Markov semigroups with a sufficiently rich domain. We give applications in the theory of non-linear Hamilton–Jacobi–Bellman equations and L'evy processes for sublinear expectations.