The density of the $(\alpha ,\beta )$-superprocess and singular solutions to a fractional non-linear PDE

We consider the density $X_t(x)$ of the critical $(\alpha ,\beta )$-superprocess in $R^d$ with $\alpha \in (0,2)$ and $\beta<\frac \alpha d$. A recent result from PDE implies a dichotomy for the density: for fixed $x$, $X_t(x)>0$ a.s. on $\{X_t\mathrm{\ne }0\}$ if and only if $\beta \leq \beta^(\alpha) = \frac{\alpha}{d+\alpha}$. We strengthen this and show that in the continuous density regime, $\beta <{\beta }^{\ast }(\alpha )$ implies that the density function is strictly positive a.s. on $\{X_t\mathrm{\ne }0\}$. We then give close to sharp conditions on a measure $\mu$ such that $\mu (X_t):=\int X_t(x)\mu (dx)>0$ a.s. on $\{X_t\mathrm{\ne }0\}$. Our characterization is based on the size of $supp(\mu )$, in the sense of Hausdorff measure and dimension. For $s\in [0,d]$, if $\beta \leq \beta^(\alpha,s)=\frac{\alpha}{d-s+\alpha}$and$supp(\mu)$haspositive$x^s$-Hausdorffmeasure,then$\mu(X_t)>0$a\mathrm{.}s\mathrm{.}on${X_t\neq 0}$;andwhen$\beta > \beta^*(\alpha,s)$,if$\mu$ satisfies a uniform lower density condition which implies $dim(supp(\mu ))<s$, then $P(\mu (X_t)=0\mathrm{\mid }{X}_t\mathrm{\ne }0)>0$. We also give new result for the fractional PDE $\partial_t u(t,x) = -(-\Delta)^{\alpha/2}u(t,x)-u(t,x)^{1+\beta}$withdomain$(t,x)\in (0,\infty)\times R^d$\mathrm{.}Theinitialtraceofasolution$u_t(x)$isapair$(S,\nu)$,wherethesingularset$S$ is a closed set around which local integrals of $u_t(x)$ diverge as $t\to 0$, and $\nu$ 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, $d_{sat}=d+\alpha (1-{\beta }^{-1})$. For $S\neq R^d$with$dim(S)> d_{sat}$(andinsomecases$dim(S)=d_{sat}$) 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.