Spin-glass stochastic stability:: a rigorous proof

We prove the property of stochastic stability previously introduced as a consequence of the (unproved) continuity hypothesis in the temperature of the spinglass quenched state. We show that stochastic stability holds in beta-average for both the Sherrington-Kirkpatrick model in terms of the square of the overlap function and for the Edwards-Anderson model in terms of the bond overlap. We show that the volume rate at which the property is reached in the thermodynamic limit is V-1. As a byproduct we show that the stochastic stability identities coincide with those obtained with a different method by Ghirlanda and Guerra when applied to the thermal fluctuations only.