We show existence of an infinitesimally invariant measure m for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in \(L^s({\mathbb {R}}^d,m)\), \(s\in [1,\infty ]\), including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of m, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.

我们显示了存在一类具有局部Sobolev正则扩散系数和某些局部可积阶漂移的椭圆型二阶偏微分算子的一类无穷大的无穷度量m的存在。随后，我们导出\（L ^ s（{\ mathbb {R}} ^ d，m）\），\（s [in，[infty] \）中定义的相应半群的正则性质，包括古典的强Feller性质和古典的不可约性。这导致了Hunt进程的转换功能，该功能明确标识为SDE的解决方案。此亨特过程的其他属性，例如非爆炸，矩不等式，递归和瞬变以及遍历，包括对象的不变性和唯一性米，并在法律上的独特性，然后可以使用派生分析工具和工具，从广义狄氏型理论进行研究。