Information-response inequality in the small noise limit

The invariant response was defined from a formulation of the fluctuation-response theorem in the space of probability distributions. An inequality which sets the mutual information as a limiting factor to the invariant response is here derived in the small noise limit based on Stam's isoperimetric inequality. Beyond the small noise limit, numerical violations exclude its general validity, however, a strong distribution bias is observed. Applications to the thermodynamics of feedback control and to estimation theory are discussed.