Stochastic boundary values and boundary singularities for solutions of the equation Lu=ua

We investigate positive solutions of a nonlinear equation Lu = u(alpha) where L is a second order elliptic differential operator in a Riemannian manifold E and 1 < alpha less than or equal to 2. The restriction alpha less than or equal to 2 is imposed because our main tool is (L, alpha)-superdiffusion X which is not defined for alpha > 2. We establish a 1-1 correspondence between the set u of positive solutions and a class 3 of functionals of X which we call linear boundary functionals (they depend only on the behavior of X near the Martin boundary E'). The class 3 is a closed convex cone and u is an element of u is a subadditive function of Z is an element of 3. Special roles belong to moderate solutions corresponding to Z with finite mathematical expectations and to a family of solutions determined by the range of X. A new formula is deduced connecting u, Z and L-diffusions conditioned to hit the boundary E' at a given point y. A concept of a singular boundary point for u is introduced in terms of the conditioned diffusion. (C) 1998 Academic Press.