I'm interested in a physical problem that I have been modelling with a Markov process. $\psi(z)$ is randomly varying and can be modelled by a stationary and ergodic process whose stationary distribution is uniform over $[0,2\pi].$ One can then assume that $\psi(z)$ is a Markov process on the torus with generator satisfying the Fredholm alternative.

I have an integral of the form

$$\int_{0}^{\infty} \mathbb{E}[\sin 2 \psi(0) \sin 2 \psi(z)] d z.$$

Is the quantity $\psi(z=0)$ random?



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