Abstract
Inspired by the works in [2] and [11] 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 corresponds to the SPDE associated with the kth-power of a generalized Ornstein-Uhlenbeck process.
Highlights
In the context of interacting particle systems with a conserved quantity in [6, 12] one studies the time-dependent density fluctuation field X (n)(φ, η(n2t)) =1 nd/2 φ(x/n)(ηx(n2t) − ρ). x∈ZdHere φ denotes a test-function, and ηx the number of particles at site x ∈ Zd
In this paper we show exactly the emergence of a scenario of this type: within a general class of models with orthogonal polynomial self-duality we consider the fluctuation fields associated to orthogonal polynomials and prove that they converge, in the scaling limit, to the solution of a recursive system of martingale problems
Ωf = Ωk of configurations with a finite number of particles, the self-duality functions that we consider in this paper are functions Dρ : Ωf × Ω → R parametrized by the density ρ > 0 satisfying the following properties
Summary
In the context of interacting particle systems with a conserved quantity (such as the number of particles) in [6, 12] one studies the time-dependent density fluctuation field. In this paper we show exactly the emergence of a scenario of this type: within a general class of models with orthogonal polynomial self-duality we consider the fluctuation fields associated to orthogonal polynomials and prove that they converge, in the scaling limit, to the solution of a recursive system of martingale problems. We believe that this can be a first step in the direction of defining non-linear fields, such as the square of the density field, via approximation of the identity, i.e., via a singular linear observable (cf [11]) of the field constructed in our paper. The rest of the sections are devoted to the proof of Theorem 5.2
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