Extinction of Superdiffusions and Semilinear Partial Differential Equations

✍ Scribed by E.B. Dynkin; E. Kuznetsov

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 231 KB
• Volume: 162
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1998.3355

✦ Synopsis

A superdiffusion is a measure-valued branching process associated with a nonlinear operator Lu& (u) where L is a second order elliptic differential operator and is a function from R d _R + to R + . In the case L1 0 (the so-called subcritical case), the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1. Most results on connections between the superdiffusion and differential equations involving Lu& (u) were obtained for the subcritical case. In the present paper, we assume only that L 1 is a bounded function. In this more general setting, the probability of extinction can be smaller than 1, and we show that his happens if and only if there exists a strictly positive solution of the equation Lu& (u)=0 in R d . We also establish a relation between the probability of extinction in a domain D and strictly positive solutions of equation Lu& (u)=0 in D that are equal to 0 on the boundary of D. We call such solutions special. For the equation (2+c) u=u : , where 2 is the Beltrami Laplace operator on a complete Riemannian manifold E, strictly positive solutions in E were studied in connection with a geometrical problem: Which two functions represent scalar curvatures of two Riemannian metrics related by a conformal mapping (see [1] and references there)?

1999 Academic Press