论文标题
$（α，β）$ - 超级进程和分数非线性PDE的奇异解决方案的密度
The density of the $(α,β)$-superprocess and singular solutions to a fractional non-linear PDE
论文作者
论文摘要
我们考虑关键$（α，β）$的密度$ x_t（x）$ - $ r^d $ in $ r^d $，$α\ in（0,2）$和$β<\fracαd$。 PDE的最新结果暗示着密度的二分法：对于固定的$ x $，$ x_t（x）> 0 $ a.s.在$ \ {x_t \ neq 0 \} $上，仅当$β\ leqβ^*（α）= \fracα{d+α} $时，才。我们加强了这一点，并表明在连续密度状态下，$β<β^*（α）$意味着密度函数严格呈阳性A.S.在$ \ {x_t \ neq 0 \} $上。 然后，我们在$μ$上给出接近尖锐的条件，以使$μ（x_t）：= \ int x_t（x）μ（dx）> 0 $ a.s.在$ \ {x_t \ neq 0 \} $上。我们的表征是基于$ supp（μ）$的大小，从Hausdorff度量和尺寸的意义上讲。对于$ s \ in [0，d] $，如果$β\ leqβ^*（α，s）= \fracα{d-s+α} $和$ supp（μ）$具有正$ x^s $ -hausdorff量，则是$μ（x_t）> 0 $ A.S.在$ \ {x_t \ neq 0 \} $上;当$β>β^*（α，s）$时，如果$μ$满足均匀的较低密度条件，则暗示$ dim（supp（μ））<s $，则$ p（μ（x_t）= 0 | x_t \ neq 0）> 0 $。 我们还为分数PDE $ \ partial_t u（t，x）= - （ - δ）^{α/2} u（t，x）-u（t，x）-u（t，x）^{1+β} $带有域$（t，x）解决方案$ u_t（x）$的初始痕迹是一对$（s，ν）$，其中单数集$ s $是封闭式套装，围绕$ u_t（x）$ diverge的本地积分为$ t \ to 0 $，而$ν$是radon量度，它具有$ u_t（x）$ $ u_t（x）$ s $ s $ s^c $ as $ as us s^c $ as as as as as as as as as as as as as as as as as as a f t t t的极限行为。我们在称为饱和度的参数中表征了具有初始跟踪$（s，0）$的解决方案的存在，$ d_ {sat} = d+α（1-β^{ - 1}）$。对于$ s \ neq r^d $，带有$ dim（s）> d_ {sat} $（在某些情况下$ dim（s）= d_ {sat} $），我们证明不存在此类解决方案。当$ dim（s）<d_ {sat} $和$ s $是满足均匀密度条件的度量的紧凑支持时，我们证明存在解决方案。
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.
