凸单调半群及其关于 $\Gamma$-收敛的生成元
Convex monotone semigroups and their generators with respect to $Γ$-convergence
AI总结：
本文研究连续函数空间上凸单调算子半群关于$\Gamma$-收敛的生成元理论，引入$\Gamma$-生成元并证明其唯一表征半群，提供Chernoff逼近结果，应用于随机最优控制与Wasserstein扰动。
摘要：
我们研究了连续函数空间上的凸单调算子半群及其关于 $\Gamma$-收敛的行为。与线性理论不同，生成元的定义域一般不在半群作用下保持不变。为克服这一问题，我们考虑了不变Lipschitz集的不同版本，这些集合被证明是较弱生成元概念的合适定义域。所谓的 $\Gamma$-生成元被定义为上半连续函数空间中关于 $\Gamma$-收敛的时间导数。在适当的假设下，我们证明了 $\Gamma$-生成元唯一地表征了半群，并且由其在一类光滑函数上的取值所决定。此外，我们提供了凸单调半群的Chernoff逼近结果，并表明基于相同无穷小行为的逼近格式会导致相同的半群。我们的结果应用于有限维和无限维设置中与随机最优控制问题相关的半群，以及转移半群的Wasserstein扰动。
英文摘要：
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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