  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
$\partial_t u=\frac{1}{2}\Delta u+u\dot{W}_\alpha$, where $\Delta$ is the
hypoelliptic Laplacian on $\mathbf{H}^{n}$ and $\{\dot{W}_\alpha; \alpha>0\}$
is a family of Gaussian space-time noises which are white in time and have a
covariance structure generated by $(-\Delta)^{-\alpha}$ in space. Our aim is
threefold: (i) Give a proper description of the noise $W_\alpha$; (ii) Prove
that one can solve the stochastic heat equation in the It\^{o} sense as soon as
$\alpha>\frac{n}{2}$; (iii) Give some basic moment estimates for the solution
$u(t,x)$.
