Exponent Inequalities in Dynamical Systems

In this letter we derive exponent inequalities relating the dynamic exponent $z$ to the steady state exponent $\mathrm{\Gamma }$ for a general class of stochastically driven dynamical systems. We begin by deriving a general exact inequality, relating the response function and the correlation function, from which the various exponent inequalities emanate. We then distinguish between two classes of dynamical systems and obtain different and complementary inequalities relating $z$ and $\mathrm{\Gamma }$. The consequences of those inequalities for a wide set of dynamical problems, including critical dynamics and Kardar-Parisi-Zhang-like problems are discussed.