论文标题
临界维度中非线性随机热方程系统的几乎所有点的极性
Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension
论文作者
论文摘要
我们将矢量值解决方案$ u（t，x）\ in \ mathbb {r}^d $ in到具有乘法噪声的非线性随机加热方程的系统：\ begin {equation*} \ frac {\ partial} {\ partial} {\ partial t} {\ partial t} u（t，x）+σ（u（t，x））\ dot {w}（t，x）。 \ end {equation*}在这里$ t \ geq 0 $，$ x \ in \ mathbb {r} $和$ \ dot {w}（t，x）$是$ \ mathbb {r}^d $ - valuew $ valued-valued时空白噪声。我们说，如果\ begin {equation*} p \ {u（t，x）= z \ text {对于某些$ t> 0 $和$ x \ in \ mathbb {r} $} $} $} = 0。 \ end {equation*}我们表明，在关键尺寸$ d = 6 $中，几乎所有点$ \ m athbb {r}^d $ is polar is polar。
We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+σ(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar.
