Reverse Khas'minskii condition

The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity in terms of existence of special exhaustion functions. In particular, Khas'minskii in [K] proved that if there exists a 2-superharmonic function k defined outside a compact set such that $\lim_{x\to \infty} k(x)=\infty$, then R is 2-parabolic, and Sario and Nakai in [SN] 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 $E:R\setminus K \to \R^+$ with $\lim_{x\to \infty} \E(x)=\infty$. 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.