  We consider critical one dimensional quantum systems initially prepared in
their groundstate and perturbed by a smooth noise coupled to the energy
density. By using conformal field theory, we deduce a universal description of
the out-of-equilibrium dynamics. In particular, the full time-dependent
distribution of any $2$--pt chiral correlation function can be obtained from
solving two coupled ordinary stochastic differential equations. In contrast
with the general expectation of heating, we demonstrate that the system reaches
a non-trivial and universal stationary state characterized by broad
distributions. As an example, we analyse the local energy density: while its
first moment diverges exponentially fast in time, the stationary distribution,
which we derive analytically, is symmetric around a negative median and
exhibits a fat tail with $3/2$ decay exponent. We obtain a similar result for
the entanglement entropy production associated to a given interval of size
$\ell$. The corresponding stationary distribution has a $3/2$ right tail for
all $\ell$, and converges to a one-sided Levy stable for large $\ell$. Our
results are benchmarked via analytical and numerical calculations for a chain
of non-interacting spinless fermions with excellent agreement.
