  We evaluate the RC-time of edge states capacitively coupled to a gate located
away from a QPC which allows for partial transmission of an edge channel. At
long times or low frequencies the RC-time governs the relaxation of charge and
current and governs the fluctuations of the equilibrium electrostatic
potential. The RC-time in mesoscopic structures is determined by an
electrochemical capacitance which depends on the density of states of the edge
states and a charge relaxation resistance. In the non-equilibrium case, in the
presence of transport, the shot noise leads to charge fluctuations in proximity
of the gate which are again determined by the equilibrium electrochemical
capacitance but with a novel resistance. The case of multiple edge states is
discussed and the effect of a dephasing voltage probe on these resistances is
investigated. The potential fluctuations characterized by these capacitances
and resistances are of interest since they determine the dephasing rate in
Coulomb coupled mesoscopic conductors.
