CHARACTERIZATION OF G-REGULARITY FOR SUPER-BROWNIAN MOTION AND CONSEQUENCES FOR

PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

JEAN-FRANÇOIS DELMAS AND JEAN-STÉPHANE DHERSIN

Abstract. We give a characterization of G-regularity for super-Brownian motion and the Brownian snake. More precisely, we define a capacity on E = (0, ∞) × ℝ^{d}, which is not invariant by translation. We then prove that the hitting probability of a Borel set A ⊂ E for the graph of the Brownian snake 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 δ_{0} hits immediately A (that is (0, 0) is G-regular for A^{c}) if and only if its capacity is infinite. As a direct consequence, if Q ⊂ E is a domain such that (0, 0) ∈ ∂Q, we give a necessary and sufficient condition for the existence on Q of a positive solution of ∂_{t}u+ \frac{1}{2}Δu = 2u^{2} which blows up at (0, 0). We also give an estimation of the hitting probabilities for the support of super-Brownian motion at fixed time. We prove that if d ≥ 2, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.