Regularity of semigroups generated by Lévy type operators via coupling (Q988677)

Let \(X_t\) be the \(d\)-dimensional Markov process generated by the Lévy type operator \(Lf(x)=\frac{1}{2}\sum_{i,j}a_{ij}(x)\partial_{ij}f(x)+\sum_ib_i(x)\partial_if(x)+\int(f(x+z)-f(x)-\nabla f(x)\cdot z1_{\{|z|\leq 1\}})\nu(x,dz),\) where \((a,b,\nu)\) are Lévy characteristics. Let \(P_t\) denote the semigroup of operators generated by \(L.\) The process \(X_t\) has \(C_b\)-Feller continuous property if for all \(t>0,\) \(P_t:C_b(\mathbb R^d)\to C_b(\mathbb R^d);\) \(X_t\) has the strong Feller continuous property if \(P_t:C_b(\mathbb R^d)\to C_b(\mathbb R^d);\) finally, \(X_t\) has Lipschitz continuous property if for all \(t>0,\) \(P_t:\mathrm{Lip}_b(\mathbb R^d)\to\mathrm{Lip}_b(\mathbb R^d),\) where \(\mathrm{Lip}_b\) stands for Lipschitz bounded. The main results of the paper give the verifiable sufficient conditions which guarantee that \(X_t\) has the above mention properties. As a corollary the criterion for the equi-continuity of \(P_t\) on \(\mathrm{Lip}_b(\mathbb R^d)\) (the so-called \(e\)-property) is also given. It is shown that the genuine Lévy processes as well as the Ornstein-Uhlenbeck type processes have \(e\)-property.