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

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 measure of hitting a Borel set A ⊂ 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 δ₀ 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 ⊂ E is a domain such that (0, 0) ∈ δQ, we give a necessary and sufficient condition for the existence on Q of a positive solution of ∂_tu + 1/2Δu = 2u², 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 ≥ 2, the support of super-Brownian motion is intersection-equivalent to the range of Brownian motion.