I have a linear system with input $ x(t) $ and output $ y(t)$ given by $$ y(t) = \int_0^\infty K(t')x(t-t')dt', $$ where $K(t)$ is a known kernel, with some parameters.

The functional form of $K(t)$ is simple enough such that when $x(t)$ is an Ornstein-Uhlenbeck process, I can analytically compute the input-output correlation $$ \langle x(t)y(t) \rangle. $$

My question is: how is this related to the mean or maximum output amplitude $\langle|y(t)|\rangle$ or $\max|y(t)|$? For example, can we say something like, the amplitude is maximal, for a given set of parameters, when the correlation is maximal?


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