Spectral and stochastic properties of the f-Laplacian, solutions of PDE's at infinity and geometric applications

The aim of this paper is to suggest a new viewpoint to study qualitative properties
of solutions of semilinear elliptic PDE's defined outside a compact set. The relevant tools
come from spectral theory and from a combination of stochastic properties of the relevant
differential operators. Possible links between spectral and stochastic properties are
analyzed in detail.