\(\alpha\)-transience and \(\alpha\)-recurrence of right processes (Q2505389)

Let \(X=(X_t)_{t\geq 0}\) be a right process on \((E,{\mathcal F})\) with transition function \(p_t\) and resolvent \(U^\alpha\). The \(\alpha\)-excessive function is defined as a non-negative function \(f\) such that \(e^{-\alpha t}p_tf \uparrow f\) as \(t\downarrow 0\). The typical \(\alpha\)-excessive function is given by \(\alpha\)-potential of the form \(f=U^\alpha h\) for \(h\geq 0\). In these definitions, \(\alpha\) is usually considered non-negative because, if otherwise, the convergence is not clear. In this paper, the potential theoretic notions such as \(\alpha\)-excessive functions, \(\alpha\)-potentials, \(\alpha\)-order hitting probabilities are are considered for \(\alpha<0\) and the similar results to the case of \(\alpha\geq 0\) are given. In particular, \(\alpha\)-tansience of \(X_t\) is defined as an existence of finely continuous function \(0<h_n\uparrow \infty\) such that \(U^\alpha h_n <\infty\). If \(U^\alpha(x,B)=0\) or \(U^\alpha(x,B)=\infty\) for any \(B\), then \(X_t\) is called \(\alpha\)-recurrent. The main results of this paper are characterizations of the \(\alpha\)-transience and \(\alpha\)-recurrence by means of \(\alpha\)-excessive functions and \(\alpha\)-potentials. Furthermore, the numbers \(0\geq \alpha_1\geq \alpha_2\) such that \(X_t\) is \(\alpha\)-transient for \(\alpha>\alpha_1\) and \(\alpha\)-recurrent for \(\alpha<\alpha_2\) are characterized.