Graphical construction Preliminary notation

are given for the contact process. For the voter process, once the approximate martingale problem is derived, almost all of the remaining steps are left to the reader. Many arguments of our proof are similar to [13] but as they did not provide details for the long-range voter model and as additions and adaptations are needed due to our broader setup, we shall not omit the details. In Subsection 3.2 we shall introduce a graphical construction for each approximating model ξ N . This allows us to write out the time-evolution of our models. By integrating it against a test function and summing over x ∈ ZN we finally obtain an approximate martingale problem for the Nth-process in Subsection 3.4. We defined the approximate density A ξ N t x as the average density of particles of type 1 on Z N in an interval centered at x of length 2 p N recall 4. By choosing a specific test function, the properties of which are under investigation at the beginning of Subsection 3.5, an approximate Green’s function representation for the approximate densities A ξ N t · is derived towards the end of Subsection 3.5 and bounds on error-terms appearing in it are given. Making use of the Green’s function representation, tightness of A ξ N t · is proven in Subsection 3.6. Here the main part of the proof consists in finding estimates on pth-moment differences. In Subsection 3.7 the tightness of the approximate densities is used to show tightness of the measure corresponding to the sequence of configurations ξ N t . Finally, in the special case with no short-range competition, every limit is shown to solve a certain SPDE. In Subsection 3.8 we additionally prove that this SPDE has a unique weak solution if R u xd x ∞. In this case, weak uniqueness of the limits of the sequence of approximate densities follows.

3.2 Graphical construction

Recall that the rates of change of the approximating processes ξ N t that we consider in Theorem 2.10 are given in 18. Note in particular that by Hypothesis 2.5 the coefficients q k,m,N i j are non-negative. We shall first derive a graphical construction and evolution in time of our approximating processes ξ N t . The graphical construction uses independent families of i.i.d. Poisson processes: € P t x; y : x, y ∈ N −1 Z Š i.i.d. Poisson processes of rate N − θ N 2cN N 1 2 , 22 and for m ≥ 2, i, j, k = 0, 1, Q m,i, j,k t x; y 1 , . . . , y m ; z : x, y 1 , . . . , y m , z ∈ N −1 Z i.i.d. Poisson processes of rate q k,m,N i j 2cN m N m 2 pN x − z. Note that we suppress the dependence on N in the family of Poisson processes P t x; y and Q m,i, j,k t x; y 1 , . . . , y m ; z. At a jump of P t x; y the voter at x adopts the opinion of the voter at y provided that y is a neighbour of x with opposite opinion. At a jump of Q m,i, j,k t x; y 1 , . . . , y m ; z the voter at x adopts the opinion 1 − k provided that y 1 , . . . , y m are neighbours of x, y 1 has opinion j, all of y 2 . . . , y m have opinion 1 − k and z has opinion i. 631 This yields the following stochastic integral equation to describe the evolution in time of our ap- proximating processes ξ N t : ξ N t x = ξ N x + X y ∼x Z t ¦ 1 € ξ N s − x = 0 Š 1 € ξ N s − y = 1 Š − 1 € ξ N s − x = 1 Š 1 € ξ N s − y = 0 Š© d P s x; y 23 + X k=0,1 1 − 2k X m ≥2,i, j=0,1 X y 1 ,..., y m ∼x X z Z t 1 € ξ N s − x = k Š 1 € ξ N s − y 1 = j Š × m Y l=2 1 € ξ N s − y l = 1 − k Š 1 € ξ N s − z = i Š dQ m,i, j,k s x; y 1 , . . . , y m ; z for all x ∈ N −1 Z . An explanation of why 23 has a unique solution can be found at the beginning of Section 4.3 of the author’s thesis, Kliem [10]. There it is further shown that the solution is the spin-flip system with rates cx, ξ N given by 18. In what follows we shall often drop the superscripts w.r.t. N to simplify notation.

3.3 Preliminary notation

In what follows we shall consider e λ x = exp λ|x| for λ ∈ R and we let C = { f : R → [0, ∞ continuous with | f xe λ x| → 0 as |x| → ∞ for all λ 0} be the set of non-negative continuous functions with slower than exponential growth. Define k f k λ = sup x | f xe λ x| and give C the topology generated by the norms k·k λ : λ 0. Remark 3.1. We work on the space C instead of C 1 because in Subsection 3.5 we shall introduce functions 0 ≤ ψ z t x ≤ C N 1 2 and shall show in Lemma 3.9b that they converge in C to the Brownian transition density p € t 3 , z − x Š . Finally, in Subsection 3.6 we shall derive estimates on pth-moment differences of ˆ A ξ t z ≡ Aξ t z − 〈ξ , ψ z t 〉, where Aξ → u in C to finally establish the tightness claim for the sequence of approximate densities A ξ N x. Notation 3.2. For x ∈ N −1 Z , f : N −1 Z → R and δ 0 we shall write D f , δx = sup{| f y − f x| : | y − x| ≤ δ, y ∈ N −1 Z }, 24 ∆ f x = N − θ N 2cN N 1 2 X y ∼x f y − f x, where we suppress the dependence on N . 632

3.4 An approximate martingale problem