论文标题
相对于$γ$ -Convergence，凸单单位元素半群及其发电机
Convex monotone semigroups and their generators with respect to $Γ$-convergence
论文作者
论文摘要
我们在连续功能的空间及其在$γ$ - convergence方面的行为上研究了凸单酮操作员的半群。与线性理论相反，发电机的域通常不是在半群下不变。为了克服这个问题，我们考虑了不同版本的不变Lipschitz套件，这些套件被证明是发电机较弱概念的合适域。所谓的$γ$生成剂被定义为上半连续函数空间中$γ$ - convergence的时间导数。在合适的假设下，我们表明$γ$的生成剂唯一地表征了半群，并取决于其在平滑函数上的评估。此外，我们为Chernoff单调半群提供了Chernoff近似结果，并表明基于相同的无限行为的近似方案导致相同的半群。我们的结果应用于与随机最佳控制问题有关的半群，在有限和无限维度以及过渡半群的Wasserstein扰动中。
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.
