Convergence of Infinitesimal Generators and Stability of Convex Monotone Semigroups

Based on the convergence of their infinitesimal generators in the mixed topology, we provide a stability result for strongly continuous convex monotone semigroups on spaces of continuous functions. In contrast to previous results, we do not rely on the theory of viscosity solutions but use a recently established comparison principle which uniquely determines semigroups via their upper [Formula: see text]-generator defined on their upper Lipschitz sets and therefore resembles the classical analogue from the linear case. The framework also allows for discretizations in both time and space and covers a variety of applications. This includes Euler schemes and Yosida-type approximations for upper envelopes of families of linear semigroups, stability results and finite-difference schemes for convex HJB equations, Freidlin–Wentzell-type results, and Markov chain approximations for a class of stochastic optimal control problems.