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We introduce a two-dimensional, distribution-valued field, which we call the quadratic field, associated with the one-dimensional Ornstein-Uhlenbeck process and we prove that the stationary quadratic fluctuations of the simple exclusion process, in the diffusive scaling, converge to this quadratic field. Moreover, we prove that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.

Quadratic fluctuations of the symmetric simple exclusion