Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\{X_t\neq 0\}$ if and only if $\beta \leq
\beta^*(\alpha) = \frac{\alpha}{d+\alpha}$. We strengthen this and show that in
the continuous density regime, $\beta < \beta^*(\alpha)$ 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 $\mu$ such that
$\mu(X_t):=\int X_t(x)\mu(dx)>0$ a.s. on $\{X_t\neq 0 \}$. 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 $supp(\mu)$ has positive
$x^s$-Hausdorff measure, then $\mu(X_t)>0$ a.s. on $\{X_t\neq 0\}$; and when
$\beta > \beta^*(\alpha,s)$, if $\mu$ satisfies a uniform lower density
condition which implies $dim(supp(\mu)) < s$, then $P(\mu(X_t)=0|X_t\neq 0)>0$.
  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 $S\neq
R^d$ 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.

26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas