On infinitesimal generators of sublinear Markov semigroups

Abstract

We establish a Dynkin formula and a Courrege-von Waldenfels theorem for sublinear Markov semigroups. In particular, we show that any sublinear operator $A$ on $C_c^{\infty}(\mathds{R}^d)$ satisfying the positive maximum principle can be represented as supremum of a family of pseudo-differential operators: \begin{equation*} Af(x) = \sup_{\alpha \in I} (-q_{\alpha}(x,D) f)(x). \end{equation*} 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 Levy processes for sublinear expectations.