Differentiability of a convex semigroup on Rn

It is proved that each continuous semigroup {P(t)}_t≥0 of convex operators P(t):Rⁿ→Rⁿ is continuously differentiable with respect to t. This note represents a first step towards a better understanding of semigroups formed by convex operators. We establish the differentiability of a convex semigroup in the finite dimensional case, generalizing a basic result from linear semigroup theory. Our motivation for the study of semigroups of convex operators comes from the theory of Markov decision processes. In 1] and in 2] it was shown that the maximum reward of these processes can be described by a certain nonlinear semigroup. The nonlinear operators are defined as suprema of linear operators (plus a constant), hence they are convex operators. It seems that the convexity assumption keeps its smoothing influence even in the infinite dimensional situation. We hope to discuss this in a future paper.