  We propose new Degroot-type social learning models with feedback in a
continuous time, to investigate the effect of a noisy information source on
consensus formation in a social network. Unlike the standard Degroot framework,
noisy information models destroy consensus formation. On the other hand, the
noisy opinion dynamics converge to the equilibrium distribution that
encapsulates correlations among agents' opinions. Interestingly, such an
equilibrium distribution is also a non-equilibrium steady state (NESS) with a
non-zero probabilistic current loop. Thus, noisy information source leads to a
NESS at long times that encodes persistent correlated opinion dynamics of
learning agents. Our model provides a simple realization of NESS in the context
of social learning. Other phenomena such as synchronization of opinions when
agents are subject to a common noise are also studied.
