论文标题
Heisenberg Groups的抛物线Anderson模型：ITô设置
Parabolic Anderson model on Heisenberg groups: the Itô setting
论文作者
论文摘要
在本说明中，我们将注意力集中在Heisenberg Group $ \ Mathbf {H}^{n} $ $ n $的随机热方程上。该方程式写为$ \ partial_t u = \ frac {1} {2}ΔU+u \ u \ dot {w}_α$，其中$Δ$是$ \ mathbf {h}^{n}^{n} $和$ \ \ \ \ \ { α> 0 \} $是一个高斯时空噪音的家族，它们是白色的，并且在空间中具有$（ - δ）^{ - α} $产生的协方差结构。我们的目标是三倍：（i）对噪声$W_α$的正确描述； （ii）证明可以在$α> \ frac {n} {2} $的情况下立即在ITôSense中求解随机热方程; （iii）给出解决方案$ u（t，x）$的一些基本力矩估计。
In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}Δu+u\dot{W}_α$, where $Δ$ is the hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_α; α>0\}$ is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by $(-Δ)^{-α}$ in space. Our aim is threefold: (i) Give a proper description of the noise $W_α$; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as $α>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution $u(t,x)$.
