Paper detail

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

We consider the density $X_t(x)$ of the critical $(α,β)$-superprocess in $R^d$ with $α\in (0,2)$ and $β<\frac αd$. A recent result from PDE implies a dichotomy for the density: for fixed $x$, $X_t(x)>0$ a.s. on $\{X_t\neq 0\}$ if and only if $β\leq β^*(α) = \fracα{d+α}$. We strengthen this and show that in the continuous density regime, $β< β^*(α)$ implies that the density function is strictly positive a.s. on $\{X_t\neq 0\}$. We then give close to sharp conditions on a measure $μ$ such that $μ(X_t):=\int X_t(x)μ(dx)>0$ a.s. on $\{X_t\neq 0 \}$. Our characterization is based on the size of $supp(μ)$, in the sense of Hausdorff measure and dimension. For $s \in [0,d]$, if $β\leq β^*(α,s)=\fracα{d-s+α}$ and $supp(μ)$ has positive $x^s$-Hausdorff measure, then $μ(X_t)>0$ a.s. on $\{X_t\neq 0\}$; and when $β> β^*(α,s)$, if $μ$ satisfies a uniform lower density condition which implies $dim(supp(μ)) < s$, then $P(μ(X_t)=0|X_t\neq 0)>0$. We also give new result for the fractional PDE $\partial_t u(t,x) = -(-Δ)^{α/2}u(t,x)-u(t,x)^{1+β}$ with domain $(t,x)\in (0,\infty)\times R^d$. The initial trace of a solution $u_t(x)$ is a pair $(S,ν)$, where the singular set $S$ is a closed set around which local integrals of $u_t(x)$ diverge as $t \to 0$, and $ν$ is a Radon measure which gives the limiting behaviour of $u_t(x)$ on $S^c$ as $t\to 0$. We characterize the existence of solutions with initial trace $(S,0)$ in terms of a parameter called the saturation dimension, $d_{sat}=d+α(1-β^{-1})$. For $S\neq R^d$ with $dim(S)> d_{sat}$ (and in some cases $dim(S)=d_{sat}$) we prove that no such solution exists. When $dim(S)<d_{sat}$ and $S$ is the compact support of a measure satisfying a uniform lower density condition, we prove that a solution exists.