SUPERDIFFUSIONS AND REMOVABLE SINGULARITIES FOR QUASI-LINEAR PARTIAL-DIFFERENTIAL EQUATIONS

To every second-order elliptic differential operator L and to every nu mber alpha is an element of (1,2] there is a corresponding measure-val ued Markov process X called the (L, alpha)-superdiffusion. Suppose tha t Gamma is a closed set in R(d). It is known that the following three statements are equivalent: (alpha) the range of X does not hit Gamma; (beta) if u greater than or equal to 0 and Lu = u(alpha) in R(d)\Gamma , then u = 0 (in other words, Gamma is a removable singularity for all solutions of equation Lu = u(alpha)); (gamma) Cap(2,alpha')(Gamma) = 0 where 1/alpha + 1/alpha' = 1 and Cap(gamma,q) is the so-called Besse l capacity. The equivalence of (beta) and (gamma) was established by B aras and Pierre in 1984 and the equivalence of (alpha) and (beta) was proved by Dynkin in 1991. In this paper, we consider sets Gamma on the boundary partial derivative D of a bounded domain D and we establish (assuming that partial derivative D is smooth) the equivalence of the following three properties: (a) the range of X in D does not hit Gamma ; (b) if u greater than or equal to 0 and Lu = u(a)lpha in D, and if u --> 0 as a is an element of partial derivative D\Gamma, then u = 0; ( C) Cap(2/alpha,alpha')(partial derivative)(Gamma) = 0 where Cap(gamma, q)(partial derivative) is the Bessel capacity on partial derivative D. This implies positive answers to two conjectures posed by Dynkin a fe w years ago. (The conjectures have already been confirmed for alpha = 2 and L = Delta in a recent paper of Le Gall.) By using a combination of probabilistic and analytic arguments we not only prove the equivale nce of (a)-(c) but also give a new, simplified proof of the equivalenc e of (alpha)-(gamma). The paper consists of an Introduction (Section 1 ) and two parts, probabilistic (Sections 2 and 3) and analytic (Sectio ns 4 and 5), that can be read independently. An important probabilisti c lemma, stated in the Introduction, is proved in the Appendix. (C) 19 96 John Wiley & Sons, Inc.