  Consider the stochastic heat equation $\partial_t u =
(\frac{\varkappa}{2})\Delta u+\sigma(u)\dot{F}$, where the solution $u:=u_t(x)$
is indexed by $(t,x)\in (0, \infty)\times\R^d$, and $\dot{F}$ is a centered
Gaussian noise that is white in time and has spatially-correlated coordinates.
We analyze the large-$|x|$ fixed-$t$ behavior of the solution $u$ in different
regimes, thereby study the effect of noise on the solution in various cases.
Among other things, we show that if the spatial correlation function $f$ of the
noise is of Riesz type, that is $f(x)\propto \|x\|^{-\alpha}$, then the
"fluctuation exponents" of the solution are $\psi$ for the spatial variable and
$2\psi-1$ for the time variable, where $\psi:=2/(4-\alpha)$. Moreover, these
exponent relations hold as long as $\alpha\in(0, d\wedge 2)$; that is precisely
when Dalang's theory implies the existence of a solution to our stochastic PDE.
These findings bolster earlier physical predictions.
