Reverse Khas’minskii condition

Abstract

The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity for complete Riemannian manifolds in terms of existence of special exhaustion functions. In particular, Khas’minskii in Ergodic properties of recurrent diffusion prossesses and stabilization of solution to the Cauchy problem for parabolic equations (Theor Prob Appl 5(2), 1960) proved that if there exists a 2-superharmonic function ${\mathcal{K}}$ defined outside a compact set on a complete Riemannian manifold R such that ${\lim_{x\to \infty} \mathcal{K}(x)=\infty}$, then R is 2-parabolic, and Sario and Nakai in Classification theory of Riemann surfaces (Springer, Berlin, 1970) were able to improve this result by showing that R is 2-parabolic if and only if there exists an Evans potential, i.e. a 2-harmonic function ${E:R{\setminus} K \to \mathbb{R}^+}$ with ${\lim_{x\to \infty}\mathcal{E}(x)=\infty}$. In this paper, we will prove a reverse Khas’minskii condition valid for any p > 1 and discuss the existence of Evans potentials in the nonlinear case.