Superdiffusions and removable singularities for quasilinear partial differential equations

To every second‐order elliptic differential operator L and to every number α ϵ (1, 2] there is a corresponding measure‐valued Markov process X called the ( L , α)‐superdiffusion. Suppose that Γ is a closed set in R d . It is known that the following three statements are equivalent: (α) the range of X does not hit Γ; (β) if u ≥ 0 and Lu = u α in R d \Γ, then u = 0 (in other words, Γ is a removable singularity for all solutions of equation Lu = u α ); (γ) Cap 2,α′ (Γ) = 0 where 1/α + 1/α′ = 1 and Cap γ, q is the so‐called Bessel capacity. The equivalence of (β) and (γ) was established by Baras and Pierre in 1984 and the equivalence of (α) and (β) was proved by Dynkin in 1991. In this paper, we consider sets Γ on the boundary ∂ D of a bounded domain D and we establish (assuming that ∂ D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Γ (b) if u ≥ 0 and Lu = u α in D , and if u → 0 as x → α ϵ ∂ D \Γ, then u = 0; (c) Cap ∂ 2/α,α′ (Γ) = 0 where Cap ∂ γ‐ q is the Bessel capacity on ∂ D. This implies positive answers to two conjectures posed by Dynkin a few years ago. (The conjectures have already been confirmed for α = 2 and L = Δ in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivalence of (a)‐(c) but also give a new, simplified proof of the equivalence of (α)‐(γ). The paper consists of an Introduction (Section 1) and two parts, probabilistic (Sections 2 and 3) and analytic (Sections 4 and 5), that can be read independently. An important probabilistic lemma, stated in the Introduction, is proved in the Appendix. © 1996 John Wiley & Sons, Inc.