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

Abstract

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 $\{X_t\ne 0\}$ if and only if $\beta \le {\beta }^{\ast }(\alpha )=\frac{\alpha }{d+\alpha }$. 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 $\{X_t\ne 0\}$. 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 $\{X_t\ne 0\}$. Our characterization is based on the size of $supp(\mu )$, in the sense of Hausdorff measure and dimension. For $s\in [0,d]$, if $\beta \le {\beta }^{\ast }(\alpha ,s)=\frac{\alpha }{d-s+\alpha }$ and $supp(\mu )$ has positive $x^s$-Hausdorff measure, then $\mu (X_t)>0$ a.s. on $\{X_t\ne 0\}$; and when $\beta >{\beta }^{\ast }(\alpha ,s)$, if $\mu$ satisfies a uniform lower density condition which implies $dim(supp(\mu ))<s$, then $P(\mu (X_t)=0|X_t\ne 0)>0$. We also give new result for the fractional PDE ${\partial }_{t}u(t,x)=-(-\Delta {)}^{\alpha /2}u(t,x)-u(t,x{)}^{1+\beta }$ with domain $(t,x)\in (0,\infty )\times R^d$. The initial trace of a solution $u_t(x)$ is a pair $(S,\nu )$, where the singular set $S$ 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 $S\ne R^d$ 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.

45 pages

References in corpus (1)

• Ahlfors-David regular sets and bilipschitz maps