This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups in the characteristic case via the Fichera function. Probabilistically, our result may be stated as follows: We construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves continuously in the interior of the state space, without reaching the boundary. We make use of the Hille–Yosida–Ray theorem that is a Feller semigroup version of the classical Hille–Yosida theorem in terms of the positive maximum principle. Our proof is based on a method of elliptic regularizations essentially due to Oleĭnik and Radkevič. The weak convergence of approximate solutions follows from the local sequential weak compactness of Hilbert spaces and Mazur's theorem in normed linear spaces.

中文翻译：

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Feller半群和退化椭圆算子III

本文致力于通过Fichera函数对特征情况下的Feller半群构造问题进行泛函分析。概率上，我们的结果可以说如下：我们构造了一个Feller半群，对应于这样一种扩散现象，即马尔可夫粒子在状态空间内部连续移动而没有到达边界。根据正最大原理，我们使用了Hille-Yosida-Ray定理，它是经典Hille-Yosida定理的Feller半群版本。我们的证明是基于椭圆正则化的方法，该方法主要归因于Oleĭnik和Radkevič。近似解的弱收敛来自希尔伯特空间的局部顺序弱紧性和范线性空间中的Mazur定理。