Mathematische Zeitschrift

, Volume 270, Issue 1, pp 165–177

Reverse Khas’minskii condition


DOI: 10.1007/s00209-010-0790-6

Cite this article as:
Valtorta, D. Math. Z. (2012) 270: 165. doi:10.1007/s00209-010-0790-6


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.


p-ParabolicitySuperharmonic functionsKhas’minskii conditionEvans potentials

Mathematics Subject Classification (2000)


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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di Milano (in collaborazione con l’Università dell’Insubria di Como)MilanoItaly