On infinitesimal generators of sublinear Markov semigroups

Abstract

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

References in corpus (2)

• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear Lévy(-type) processes