On the existence of an invariant measure for Markov-Feller operators
Let \((X,\rho )\) be a metric space, \({\mathcal B}(X)\) the \(\sigma\)-algebra of Borel subsets of \(X\), \({\mathcal M}(X)\) the family of all finite measures on \(X\), \({\mathcal M}_s (X)\) the space of all signed measures on \(X\), \({\mathcal M}_1 (X)=\{ \mu : \mu\in{\mathcal M}(X), \mu (X)=1\}\), \(C(X)\) the space of continuous functions on \(X\), \(\langle f,\mu\rangle =\int_X f(x)\mu (dx)\). An operator \(P:{\mathcal M}(X)\rightarrow {\mathcal M}(X)\) is called a Markov operator if \(P(\lambda_1 \mu_1 +\lambda_2 \mu_2 )=\lambda_1 P\mu_1 +\lambda_2 P\mu_2\), \(\lambda_1 ,\lambda_2 \in{\mathbb R}_+\), \(\mu_1 , \mu_2 \in {\mathcal M}(X)\) and \(P\mu (X)=\mu (X)\), \(\mu\in {\mathcal M}(X)\). The operator \(P\) is called Feller operator if there is an operator \(U:{\mathcal B}(X)\rightarrow {\mathcal B}(X)\) such that \(\langle Uf,\mu \rangle =\langle f,P\mu\rangle\), \(f\in {\mathcal B}(X)\), \(\mu\in {\mathcal M}(X)\), and if \(Uf\in C(X)\) for \(f\in C(X)\). A measure \(\mu\) is called invariant with respect to \(P\) if \(P\mu =\mu\). The main result of the paper is as follows. Let \(P:{\mathcal M}(X)\rightarrow {\mathcal M}(X)\) be a Feller operator. Assume that \(\exists z\in X\) such that \(\forall \sigma >0\) \(\forall {\overline \mu}\in{\mathcal M}_1 (X)\) such that \(\liminf_{n\rightarrow\infty} P^n {\overline \mu}(B(z,\sigma ))>0\) (\(B(z,\sigma )\) is the ball with center \(z\) and radius \(\sigma \)). Moreover, assume that the family of functions \(\{ X\ni x\mapsto P^n \delta_x \in{\mathcal M}_1 (X):n\in {\mathbb N}\}\) is equicontinuous at the point \(z\) (\({\mathcal M}_1 (X)\) is endowed with the Prokhorov metric, \(\delta_x\) is Dirac measure supported at \(x\)). Then \(P\) admits an invariant measure. A counterexample, which shows that the equicontinuity condition cannot be omitted, is constructed.
scientific article; zbMATH DE number 2085922