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

✍ Scribed by E.B. Dynkin

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 435 KB
• Volume: 153
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3176

✦ Synopsis

We investigate positive solutions of a nonlinear equation Lu=u : where L is a second order elliptic differential operator in a Riemannian manifold E and 1<: 2. The restriction : 2 is imposed because our main tool is (L, :)-superdiffusion X which is not defined for :>2. We establish a 1-1 correspondence between the set U of positive solutions and a class Z of functionals of X which we call linear boundary functionals (they depend only on the behavior of X near the Martin boundary E$). The class Z is a closed convex cone and u # U is a subadditive function of Z # Z. Special roles belong to moderate solutions corresponding to Z with finite mathematical expectations and to a family of solutions determined by the range of X. A new formula is deduced connecting u, Z and L-diffusions conditioned to hit the boundary E$ at a given point y. A concept of a singular boundary point for u is introduced in terms of the conditioned diffusion. 1998 Academic Press g(x, y)= | 0 p t (x, y) dt (1.1) article no. FU973176