The density of the (α,d,β)-superprocess and singular solutions to a fractional non-linear PDE

Abstract

Abstract: We consider the density Xt(x) of the critical (α, β)-superprocess in R d with α ∈ (0, 2) and β < α d . Our starting point is a recent result from PDE [2] which implies the following dichotomy: if x ∈ R is fixed and β ≤ β∗(α) := α d+α , then Xt(x) > 0 a.s. on {Xt 6= 0}; otherwise, the probability that Xt(x) is positive when conditioned on {Xt 6= 0} has power law decay. We strengthen this and prove probabilistically that if β < β∗(α) and the density is continuous, which holds if and only if d = 1 and α > 1+ β, then Xt(x) > 0 for all x ∈ R a.s. on {Xt 6= 0}. The above complements a classical superprocess result that if Xt is non-zero, then it charges every open set almost surely. We unify and extend these results by giving close to sharp conditions on a measure μ such that μ(Xt) := ∫ Xt(x)μ(dx) > 0 a.s. on {Xt 6= 0}. Our characterization is based on the size of supp(μ), in the sense of Hausdorff measure and dimension. For s ∈ [0, d], if β ≤ β∗(α, s) = α d−s+α and supp(μ) has positive x-Hausdorff measure, then μ(Xt) > 0 a.s. on {Xt 6= 0}; and when β > β ∗(α, s), if μ satisfies a uniform lower density condition which implies dim(supp(μ)) < s, then P (μ(Xt) = 0 |Xt 6= 0) > 0. Our methods also give new results for the fractional PDE which is dual to the (α, β)superprocess, i.e. ∂tu(t, x) = ∆αu(t, x)− u(t, x) 1+β with domain (t, x) ∈ (0,∞) × R, where ∆α = −(−∆) α