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For the one dimensional Burgers equation with a random and periodic forcing,it is well-known that there exits a family of invariant measures, eachcorresponding to a different average velocity. In this paper, we consider thecoupled invariant measures and study how they change as the velocity parametervaries. We show that the derivative of the invariant measure with respect tothe velocity parameter exists, and it can be interpreted as the steady state ofa diffusion advected by the Burgers flow.

Jointly stationary solutions of periodic Burgers flow