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We study semigroups of convex monotone operators on spaces of continuousfunctions and their behaviour with respect to $\Gamma$-convergence. In contrastto the linear theory, the domain of the generator is, in general, not invariantunder the semigroup. To overcome this issue, we consider different versions ofinvariant Lipschitz sets which turn out to be suitable domains for weakernotions of the generator. The so-called $\Gamma$-generator is defined as thetime derivative with respect to $\Gamma$-convergence in the space of uppersemicontinuous functions. Under suitable assumptions, we show that the$\Gamma$-generator uniquely characterizes the semigroup and is determined byits evaluation at smooth functions. Furthermore, we provide Chernoffapproximation results for convex monotone semigroups and show thatapproximation schemes based on the same infinitesimal behaviour lead to thesame semigroup. Our results are applied to semigroups related to stochasticoptimal control problems in finite and infinite-dimensional settings as well asWasserstein perturbations of transition semigroups.

Convex monotone semigroups and their generators with respect to $\Gamma$-convergence