Stochastic Boundary Values and Boundary Singularities for Solutions of the EquationLu=uα

We investigate positive solutions of a nonlinear equationLu=uαwhereLis a second order elliptic differential operator in a Riemannian manifoldEand 1<α⩽2. The restrictionα⩽2 is imposed because our main tool is (L, α)-superdiffusionXwhich is not defined forα>2. We establish a 1-1 correspondence between the set U of positive solutions and a class Z of functionals ofXwhich we call linear boundary functionals (they depend only on the behavior ofXnear the Martin boundaryE′). The class Z is a closed convex cone andu∈U is a subadditive function ofZ∈Z. Special roles belong to moderate solutions corresponding toZwith finite mathematical expectations and to a family of solutions determined by the range ofX. A new formula is deduced connectingu,ZandL-diffusions conditioned to hit the boundaryE′ at a given pointy. A concept of a singular boundary point foruis introduced in terms of the conditioned diffusion.