Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit $S(t):=\lim _{n\to \infty }(I(t 2^{-n}))^{2^{n}}$ S ( t ) : = lim n → ∞ ( I ( t 2 − n ) ) 2 n exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by $\lim _{h\downarrow 0}\tfrac {I(h)x-x}{h}$ lim h ↓ 0 I ( h ) x − x h , whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.

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We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to Γ-convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under the semigroup. To overcome this issue, we consider different versions of invariant Lipschitz sets which turn out to be suitable domains for weaker notions of the generator. The so-called Γ-generator is defined as the time derivative with respect to Γ-convergence in the space of upper semicontinuous functions. Under suitable assumptions, we show that the Γ-generator uniquely characterizes the semigroup and is determined by its evaluation at smooth functions. Furthermore, we provide Chernoff approximation results for convex monotone semigroups and show that approximation schemes based on the same infinitesimal behaviour lead to the same semigroup. Our results are applied to semigroups related to stochastic optimal control problems in finite and infinite-dimensional settings as well as Wasserstein perturbations of transition semigroups.

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