Characterization of G-regularity for super-Brownian motion and consequences for parabolic partial differential equations

• 成果类型: Article
• 署名作者: Delmas, JF; Dhersin, JS
• 署名单位: Institut Polytechnique de Paris; Ecole des Ponts ParisTech; Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677384
• 发表日期: 1999
• 页码: 731-750
• 关键词: superprocesses

摘要

We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on E = (0, infinity) x R-d, which is not invariant by translation. We then prove that the measure of hitting a Borel set A subset of E for the graph of the Brownian snake excursion starting at (0, 0) is comparable, up to multiplicative constants, to its capacity. This implies that super-Brownian motion started at time 0 at the Dirac mass delta(0) hits immediately A [i.e., (0, 0) is G-regular for A(c)] if and only if its capacity is infinite. As a direct consequence, if Q subset of E is a domain such that (0, 0) is an element of partial derivative Q, we give a necessary and sufficient condition for the existence on Q of a positive solution of partial derivative(t)u + 1/2 Delta u = 2u(2), which blows up at (0, 0). We also give an estimate of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if d greater than or equal to 2, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.