  We consider the multidimensional generalised stochastic Burgers equation in
the space-periodic setting:
  $ \partial \mathbf{u}/\partial t+$ $(\nabla f(\mathbf{u}) \cdot \nabla)$
$\mathbf{u} -\nu \Delta \mathbf{u}=$ $\nabla \eta,\quad t \geq 0,\ \mathbf{x}
\in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d,$ under the assumption that
$\mathbf{u}$ is a gradient. Here $f$ is strongly convex and satisfies a growth
condition, $\nu$ is small and positive, while $\eta$ is a random forcing term,
smooth in space and white in time. For solutions $\mathbf{u}$ of this equation,
we study Sobolev norms of $\mathbf{u}$ averaged in time and in ensemble: each
of these norms behaves as a given negative power of $\nu$. These results yield
sharp upper and lower bounds for natural analogues of quantities characterising
the hydrodynamical turbulence, namely the averages of the increments and of the
energy spectrum. These quantities behave as a power of the norm of the relevant
parameter, which is respectively the separation $\mathbf{l}$ in the physical
space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not
depend on the initial condition and hold uniformly in $\nu$. We generalise the
results obtained for the one-dimensional case in \cite{BorW}, confirming the
physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates
does not depend on the dimension: the powers of $\nu, |\mathbf{k}|, \mathbf{l}$
are the same in the one- and the multi-dimensional setting.
