On the equivalence of stochastic completeness and Liouville and Khas’minskii conditions in linear and nonlinear settings

Abstract

Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled after the p-Laplacian with potential. In particular, we discuss the equivalence between the Lioville property and the Khas'minskii condition, i.e. the existence of an exhaustion functions which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors and answers to a question in [26].Date: September 10, 2018. 2010 Mathematics Subject Classification. Primary 31C12 (potential theory on Riemannian manifolds, Secondary: 35B53 (Liouville theorems), 58J65 (stochastic equations and processes on manifolds), 58J05 (elliptic equations on manifolds).

Maximum Principles at Infinity and the Ahlfors-Khas’minskii Duality: An Overview

“…Even for q = 2, the equivalence of q-parabolicity with the existence of q-harmonic Evans potentials is still an open problem, although results for more general operators on rotationally symmetric manifolds (cf. the last section in [54]) indicate that it is likely to hold.…”

“…It is therefore natural to ask whether this is specific to the operators ∆u and ∆u − λu or if it is a more general fact, and, in the latter case, how one can take advantage from such an equivalence. This is the starting point of the papers [54,50].…”

“…with A a Caratheódory map that locally behaves like a q-Laplacian, and B non-decreasing in u with uB(x, u) 0, the AK-duality in a slightly less general version was first established in [54]. The use of weak instead of viscosity solutions allows to work with very general A , B, since a comparison theorem is easy to show and the obstacle problem is solvable by classical results.…”

“…Differently from the Hessian principle, in view of elliptic estimates for semilinear equations, the viscosity weak Laplacian principle is equivalent to its corresponding classical C 2 principle, that is, to the stochastic completeness of X. In this case, the AK-duality improves on the original results in [42,54]. Regarding the viscosity, strong Laplacian principle, that is, the Ahlfors property for the subequation {Tr(A) 1} ∪ {|p| 1} = F ∪ E, its equivalence with the classical, C 2 one (that is, Yau's principle) seems quite delicate and is currently unknown.…”

On the role of gradient terms in coercive quasilinear differential inequalities on Carnot groups

“…Our choice here is to emphasize the next aspect: in the case l(0) > 0, if u * is attained and assuming that everything be smooth enough, evaluating (1.34) at a maximum point gives f (u * ) ≤ 0. Therefore, property f (u * ) ≤ 0 can be thought as the validity of a weak maximum principle at infinity, according to the point of view adopted in [32,27], see also [22].…”