  We investigate the dynamical properties of low dimensional systems, driven by
external noise sources. Specifically we consider a resistively shunted
Josephson junction and a one dimensional quantum liquid in a commensurate
lattice potential, subject to $1/f$ noise. In absence of nonlinear coupling, we
have shown previously that these systems establish a non-equilibrium critical
steady state [Nature Phys. 6, 806 (2010)]. Here we use this state as the basis
for a controlled renormalization group analysis using the Keldysh path integral
formulation to treat the non linearities: the Josephson coupling and the
commensurate lattice.
  The analysis to first order in the coupling constant indicates transitions
between superconducting and localized regimes that are smoothly connected to
the respective equilibrium transitions. However at second order, the back
action of the mode coupling on the critical state leads to renormalization of
dissipation and emergence of an effective temperature. In the Josephson
junction the temperature is parametrically small allowing to observe a
universal crossover between the superconducting and insulating regimes. The IV
characteristics of the junction displays algebraic behavior controlled by the
underlying critical state over a wide range. In the noisy one dimensional
liquid the generated dissipation and effective temperature are not small as in
the junction. We find a crossover between a quasi-localized regime dominated by
dissipation and another dominated by temperature. However since in the thermal
regime the thermalization rate is parametrically small, signatures of the
non-equilibrium critical state can be seen in transient dynamics.
