  Consider the stochastic heat equation $\dot{u}=\frac12 u"+\sigma(u)\xi$ on
$(0\,,\infty)\times\mathbb{R}$ subject to $u(0)\equiv1$, where
$\sigma:\mathbb{R}\to\mathbb{R}$ is a Lipschitz (local) function that does not
vanish at $1$, and $\xi$ denotes space-time white noise. It is well known that
$u$ has continuous sample functions; as a result, $\lim_{t\downarrow0}u(t\,,x)=
1$ almost surely for every $x\in\mathbb{R}$.
  The corresponding fluctuations are also known: For every fixed
$x\in\mathbb{R}$, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of
fractional Brownian motion (fBm) with index $1/4$. In particular, an
application of Fubini's theorem implies that, on an $x$-set of full Lebesgue
measure, the short-time behavior of the peaks of the random function $t\mapsto
u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to
possibly a suitable normalization constant. By contrast, the main result of
this paper claims that, on an $x$-set of full Hausdorff dimension, the
short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law,
and that those peaks contain a rich multifractal structure a.s.
  Large-time variations of these results were predicted in the physics
literature a number of years ago and proved very recently in Khoshnevisan, Kim
and Xiao (2016). To the best of our knowledge, the short-time results of the
present paper are observed here for the first time.
