  Inspired by the works in [1] and [8] we introduce what we call $k$-th-order
fluctuation fields and study their scaling limits. This construction is done in
the context of particle systems with the property of orthogonal self-duality.
This type of duality provides us with a setting in which we are able to
interpret these fields as some type of discrete analogue of powers of the
well-known density fluctuation field. We show that the weak limit of the $k$-th
order field satisfies a recursive martingale problem that formally corresponds
to the SPDE associated with the $k$th-power of a generalized Ornstein-Uhlenbeck
process.

Probability