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Center for Mathematical Economics Working Papers No. 662
Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld
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.
Convex monotone semigroup
Lipschitz set
Chernoff approximation
optimal control
Wasserstein perturbation
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