Rescaled Whittaker driven stochastic differential equations converge to the additive stochastic heat equation

We study SDEs arising from limiting fluctuations in a $(2+1)$-dimensional surface growth model called the Whittaker driven particle system, which is believed to be in the anisotropic Kardar--Parisi--Zhang class. The main result of this paper proves an irrelevance of nonlinearity in the surface growth model in the continuum by weak convergence in a path space; the first instance of this irrelevance is obtained recently for this model in terms of the covariance functions along certain diverging characteristics. With the same limiting scheme, we prove that the derived SDEs converge in distribution to the additive stochastic heat equation in $C(\Bbb R_+,\mathcal S'(\Bbb R^2))$. The proof addresses the solutions as stochastic convolutions where the convolution structures are broken by discretization of the diverging characteristics.


Introduction and main results
In this paper, we consider rescaled limits of the Whittaker driven stochastic differential equations (SDEs) obtained in the recent work [7]. These SDEs arise as limiting fluctuations of an interacting particle system modeling (2 + 1)-dimensional surface growth. The model is named the Whittaker driven particle system for its connections with methods in integrable probability (see [4] and the references therein) and is originally introduced in [9] to study the anisotropic Kardar-Parisi-Zhang class in (2 + 1) dimensions. Mathematical results for this class are possible, but very little is known compared to the (1 + 1)-dimensional class.
In [7], the SDEs derived from the Whittaker driven particle system obey the following linear system indexed by sites in a two-dimensional discrete torus R m with size m 2 : (1.1) Here, W = {W (x); x ∈ R m } is an m 2 -dimensional standard Brownian motion and v is a constant defined by the parameters of the particle system. In addition to the particular geometry of R m as a certain parallelogram in Z 2 subject to periodic boundary conditions, the main characteristics of the derived SDEs come from the constant drift coefficient matrix A. See Proposition 2.1 and Example 2.3 for the precise forms. In more detail, these SDEs arise from the site-wise fluctuation fields of the Whittaker driven particle system mentioned above by taking central limit theorem type limits. Since the jump rates of this particle system are defined by total asymmetry and algebraic complexity in using the planarity of the space, the SDEs inherit these properties by the matrix A as well as the coefficient v. In particular, in terms of formal connections between the SDEs and stochastic heat equations, we note that the drift terms of the SDEs do not take the form of a discretization of the Laplacian (the matrix has zero row sums but is not even a generator matrix). See [9] and [7, Sections 1-3] for more details of this planar particle system and further connections with the SDEs.
The anisotropic Kardar-Parisi-Zhang class. The Whittaker driven particle system is introduced in [9] to study the anisotropic Kardar-Parisi-Zhang class in (2 + 1) dimensions. This class goes back to Villain [31]. It consists of height functions in the continuum of generic surface growth models where growths along the two directions of a spatial coordinate frame are not related by symmetry. In this case, the time evolution of a height function H(x, t) obeys the following singular stochastic partial differential equation (SPDE): for (x, t) ∈ R 2 × R + , whereẆ is a space-time white noise and the three terms on the right-hand side physically capture surface tension, lateral surface growth, and random fluctuation, respectively, in the surface growth. Anisotropy refers to the property that the eigenvalues of the 2 × 2 symmetric matrix Λ in (1.2) have different signs. This complements the case in (2 + 1)-dimensions studied earlier by Kardar, Parisi and Zhang [20], which defines the isotropic case where the eigenvalues of Λ have the same signs. Note that the case of two spatial dimensions is singled out in [20] for its criticality leading to a notion of marginal relevance of nonlinearity. See, for example, the lectures of Kardar [19] for more on the physical developments of the Kardar-Parisi-Zhang equations in one and two spatial dimensions and the monograph of Barabási and Stanley [3] for an introduction to these equations in all dimensions.
The most studied case of the Kardar-Parisi-Zhang class in one spatial dimension now leads to many-faceted mathematical investigations. See [1,2,14,16,17,21], to name but a few. This class and the isotropic class both feature predicted nonlinearity in the roughness of height functions. By contrast, the anisotropic class is noted for the prediction by Wolf [33] on the irrelevance of nonlinearity. The prediction states that in the limit of large time, the coefficient σ of the space-time white noise in (1.2) is not pulled along significantly by the nonlinear term ∇H, Λ∇H that is responsible for the singularity of the SPDE in (1.2). The overall effect is that the expected noise should behave like the expected noise in the corresponding Edwards-Wilkinson equation [12], that is, a (2 + 1)-dimensional additive stochastic heat equation (e.g. Walsh's lectures [32,Chapter 5]): Here, the SPDE in (1.3) was originally introduced in [12] for (2 + 1)-dimensional surface growth without the asymmetry from lateral growth leading to the nonlinear term in (1.2). (To obtain (1.3), [12] imposed Langevin equations for the Fourier modes of the height function, which is reminiscent of the approach for the Whittaker drive SDEs discussed below.) In stark contrast to the additive stochastic heat equations, the anisotropic SPDE in (1.2) remains mathematically out of reach for several basic aspects including the existence of solutions. Accordingly mathematical results are very few. See [30] for a broad discussion of Wolf's prediction and the mathematical progress.
Expected noise in the Whittaker driven SDEs. Our main object of this paper is a connection, among several other things, proven in [7]. By the Whittaker driven particle system, it gives the first instance to prove rigorously Wolf's prediction on the irrelevance of nonlinearity in the form of expectations. The connection is established for the SDEs in (1.1) subject to general noise coefficients v ∈ (0, ∞) and matrices A satisfying only key features of the drift coefficient matrices in the Whittaker driven SDEs (Assumption 2.2). The main quantitative assumption states that the Taylor expansion of the Fourier transform takes the following form: for a real vector U and a strictly negative definite matrix Q. The matrix A thus deviates from a "Laplacian" additively in its Fourier transform by the pure imaginary translation −i k, U as well as the error term O(|k| 3 ). In the rest of this section, the SDEs in (1.1) are assumed to be under this general setup unless otherwise mentioned. The connection from [7] states that, with V = √ −Q, the limiting covariance function lim δ→0+ lim m→∞ Cov[X m s (x); X m t (y)], 0 < s < t, x, y ∈ R 2 , (1. 6) of the two-parameter processes exists. Moreover, the limit coincides with the covariance function κ s,t (x, y) of the S (R 2 )-valued solution X to an additive stochastic heat equation as in (1.3): (1. 8) In addition to the usual diffusive rescaling (δ −1/2 z, δ −1 t) of space and time in (1.7), as pointed out in [7], the main feature of the limit scheme in (1.6) is a discretization of space by the following sets of time-adaptive meshes: (1.9) These meshes naturally induce distinguished characteristics in space and time that diverge as δ → 0+. See also [5,6,8] for rescaled limits of closely related growth models in (2 + 1) dimensions and [23,15] for convergences to the Edwards-Wilkinson equations in three and higher spatial dimensions. The convergence in (1.6) brought to the process level is not a consequence given the convergence of the covariance functions already obtained, although the limiting SPDE is very simple. This is attributable to several features in the SDEs (1.1) and the time-dependent nature of the spatial discretization in (1.9). They begin with the fact that the useful positivity in matrix exponentials solving the mean functions of the rescaled densities does not hold for the SDEs derived from the Whittaker driven particle system (see (2.5) and Example 2.3). Further issues arise since the rescaled densities in (1.7) appear to have irregular discontinuity due to the diverging spatial mesh points and it is wellknown that the limiting covariance kernel defined in (1.8) explodes at equal times and equal spatial points leading to non-solvability of the stochastic heat equation by mild solutions. It is neither clear to us whether X δ obeys useful exact dynamics. We will give more detailed discussions below when explaining the proof of the main theorem.
Main theorem. We follow the same double limit scheme in (1.6) and prove that solutions to the generalized Whittaker driven SDEs (1.1) converge weakly to the solution of an additive stochastic heat equation as distribution-valued processes. This proves in particular the pathwise Edwards-Wilkinson fluctuation in the Whittaker driven particle system via the SDEs, and hence, may suggest the possibility of further pathwise investigations of the anisotropic SPDE (1.2). Note that [7, Theorem 1] proves weak convergence of the fluctuations of the particle system to these SDEs.
To carry out the double limit scheme in (1.6), we first embed R m increasingly into Z 2 so that they fill the whole space as m → ∞. Then the weak convergence proven in this paper is established by the following two separate results: for the distribution-valued processes X δ defined by (1.12) Here in (1.10), x → ξ m (x) is understood to be zero outside R m and ξ ∞ is a Gaussian process with explicitly defined mean and covariance functions in terms of Fourier transforms (Proposition 2.6). Note that the density of X δ in (1.12) is subject to the same rescaling of both space and time as in (1.6). Also, (1.10) and (1.11) can be integrated in the obvious way for the weak convergence of the distribution-valued processes with densities X m defined by (1.7) if one passes the double limits in (1.6).
The main theorem of this paper is given by Theorem 3.1 for the proof of (1.11). We use Mitoma's conditions [24] on the tightness of probability measures on S (R 2 )-valued path spaces. The major argument here is devoted to proving tightness of the laws of the family {X δ (φ)} δ∈(0,1) defined in (1.12) for a Schwartz function φ. In particular, the proof of Theorem 3.1 does not use the asymptotics in (1.6) as δ → 0+ obtained in [7].
To prove tightness of the family {X δ (φ)} δ∈(0,1) , we first notice that the expected moduli of continuity in the densities of X δ (φ)'s are complicated by the Fourier character of their covariance functions (defined by the Gaussian process ξ ∞ in (1.10)). We have to carefully address by precise calculations the feature of the density of X δ that the time-adaptive spatial mesh points in (1.9) are in use and they are defined by mixtures of space and time subject to different scalings.
The key issue here arises from the presence of the floor function z → z in (1.9). This function already defines discontinuity in the density of X δ , and so it becomes natural to expect that the test function φ in X δ (φ) would help smooth things out. We use the following stochastic integral representation of X δ (φ) after re-centering to make explicit the smoothing effect as well as the whole process under consideration: (1.14) and W 1 and W 2 are independent space-time white noises on R + × R 2 (Section 4.2). Then (1.14) shows that the floor function interferes cancellation of the two growing, time-dependent factors δ −1 U t in the Fourier transform of φ, since there is a discretization of the first of them by the floor function. Nevertheless, if this cancellation were viable, then the space-time stochastic integrals in (1.13) would reduce to convergent stochastic convolutions. We develop several methods to address this property which may be extended for proving convergence of more general stochastic integrals where convolution structures are broken by discretization. By the stochastic integrals in (1.13), the proof of Theorem 3.1 leads to martingale problem characterizations for limits of the re-centered processes. As the reader may have already noticed, it gives an alternative explanation why the choice of the time-adaptive meshes (1.9) under the diffusive scaling (δ −1/2 z, δ −1 t) is necessary. Moreover, the natural limit of (1.13) as δ → 0+ arises under the assumption (1.5) and satisfies (1.13) with Φ δ t replaced by The characteristic of the corresponding stochastic integral as a solution to an additive stochastic heat equation then follows upon Fourier inversions.
Organization of this paper. In Section 2, we discuss the explicit solutions of the system (1.1) and the proof of (1.10) in Proposition 2.6. In Section 3, we state Theorem 3.1. The steps of its proof are explained in more detail at the end of Section 3. In Section 4, details for the above discussions consist in the proof of the convergence of X δ after re-centering. The convergence of the mean functional of X δ is a real-analysis result and is proven in Section 5. As we need more complicated notation after Section 2, the reader can find a list of frequent notations for Sections 3-5 at the end of Section 6.

Fourier representations of the solutions
In this section, we describe the SDEs studied in [7] in more detail and discuss the Fourier transforms of the solutions. This section ends with a Fourier characterization of the solutions in the limit of infinite volume. First, let us describe in more detail the discrete torus R m that parameterizes the SDEs (see [7,Section 2]). Given two positive integers m 2 and m such that m 2 /m ∈ (0, 1), the torus R m is defined to be the quotient group Z 2 /∼, where the equivalence relation ∼ is given by: x ∼ y ⇐⇒ x + (j 1 m, j 2 m) = y + (j 2 m 2 , 0) for some j 1 , j 2 ∈ Z. (2.1) The quotient group Z 2 / ∼ can be identified with a discrete parallelogram subject to the periodic boundary conditions to be defined in (2.2), which is suitable for the purpose of this paper. Whenever R m is used as a set, we always refer to this discrete parallelogram unless otherwise mentioned. See Figure 1 for an example.
Proposition 2.1. The quotient group Z 2 /∼ is isomorphic to the quotient group with points in the discrete parallelogram subject to the pasting rule ≡ defined as follows: (1) Points on the lower and upper edges are pasted together by the following rule: that is, along the direction defining the lateral edges.
(2) Points on the left and right edges are pasted together horizontally.
Proof. Write P m for the discrete set defined in (2.2). For x, y ∈ P m , x ∼ y implies that j 2 = 0 since −m/2 ≤ x 2 , y 2 < m/2, and hence, x 2 = y 2 . Similarly, j 1 = 0 and x 1 = y 1 . Also, any point in Z 2 is ∼-equivalent to a point in P m . We conclude that there is a natural isomorphism between equivalence classes in Z 2 /∼ and those in P m /≡.

Assumption 2.2 (Coefficients of the SDEs)
. From now on, we assume unless otherwise mentioned that, the coefficients of the SDEs in (1.1) are given by a constant v ∈ (0, ∞) and a constant matrix A indexed by Z 2 × Z 2 such that, for some integer m 0 ≥ 2, the following five conditions are satisfied for every m ≥ m 0 : (1) The matrix A is translation-invariant on the quotient group R m : (2) The Fourier transform A(k) (1.4) of A is 2π-periodic and in C ∞ (R 2 ).
(4) The function expands as where Q(k) = k, Qk for a strictly negative definite matrix Q.
(5) The function R(k) defined by (2.3) is nonpositive and its only zero in T 2 is k = 0. Here and throughout this paper, T d = [−π, π] d for d ≥ 1 is a set and no periodic boundary conditions are imposed.
Conditions (2)-(5) in Assumption 2.2 are imposed for the Fourier transform of the sub-matrix of A restricted to R m × R m for every m ≥ m 0 . The Fourier transform does not depend on the set representation of the quotient group R m , and so the choice in Proposition 2.1 applies. Moreover, according to the applications of these assumptions in [7], it is understood that the Fourier transform of A restricted to R m × R m is identical to the Fourier transform of the full matrix A on Z 2 × Z 2 defined by (1.4) with R m replaced by Z 2 . It follows that x → A x,0 has a finite support. Example 2.3. In [7], the SDEs derived from the Whittaker driven particle system on R m are defined by (1.1) with the following coefficients: and Obviously, this matrix A satisfies Assumption 2.2 (1). By this translation invariance of A and the property R m = −R m , the functions A(k) and R(k) defined by (1.4) and (2.3) take the following simple forms: for all k ∈ R 2 , (1,−1) cos(k 1 − k 2 ) + A 0,(0,−1) cos(k 2 ) + A 0,(−1,0) cos(k 1 ), which clearly give (2)-(3) in Assumption 2.2. The strict negative definiteness of Q in (4) and the conditions in (5) need some algebra to verify [7,Appendix B]. See [7,Proposition 2] for these five properties.
Recall that the explicit solution to the system (1.1) is given by [18,Eq.(6.6) in Section 5.6]). Here in (2.5), e tA is understood to be the usual matrix exponential of the sub-matrix of A restricted to R m × R m . Henceforth, we decompose the Gaussian process ξ m into where η m t (x) and ζ m t (x) are defined by the first and second sums in (2.5) and called the deterministic part and stochastic part of ξ m , respectively.
To apply Assumption 2.2, we turn to the Fourier transform of ξ m . Define and Then Assumption 2.2 (1) and the definition (1.4) of A(k) imply that for any analytic function F , the usual multiplier formula holds: To represent the processes η m and ζ m by their Fourier transforms η m (k) and ζ m (k), it is enough to require k be points in the following set: (2.10) The additional properties that we need are summarized in Lemma 2.4 below (see [29, Chapter 1] or [7, Section 3.1]). For any subset E of Z 2 , write and K m be defined by (2.7) and (2.10), respectively. Then the following properties hold: (1) For any k ∈ K m , f k is well-defined on the quotient group (R m , ∼), where the equivalence relation ∼ is defined by (2.1).
(2) The set {f k } k∈Km forms an orthonormal basis of C Rm with respect to the inner product · , · Rm defined by (2.11).
(3) The inversion formula holds: Corollary 2.5. With respect to the decomposition in (2.6), it holds that Proof. By (2.9) and the inversion formula in (2.12), (2.13) follows and, for (2.14), we have The following theorem uses (2.13) and (2.14) to characterize the limit of ξ m as m → ∞. In particular, the limiting mean function immediately implied by [7, (6.4)] and the limiting covariance function in [7, (6.8) Similar extension applies to η m and ζ m . Also, we write Cov[ for complexvalued random variables X and Y . and, for some continuous function µ on K ∞ , In particular, ξ ∞ admits a natural extension, still denoted by ξ ∞ , which is a jointly continuous realvalued Gaussian process indexed by R + × R 2 .
Proof. We compute the mean function and covariance function of ξ m in the limit m → ∞ first. By (2.13), (2.16) and dominated convergence, where the last equality follows from the 2π-periodicity of the integrand. As for the covariance function of ξ m in the limit m → ∞, notice that by Lemma 2.4 (2), the complex-valued Brownian motion in (2.15) satisfies Hence, for any x, y ∈ Z 2 and m large such that x, y ∈ R m , (2.14) gives by dominated convergence and the 2π-periodicity of the integrand as above in (2.20).
We are ready to prove the weak convergence of ξ m ; then (2.18) and (2.19) will follow from (2.20) and (2.23), respectively, by the closure of centered Gaussians under weak convergence. By [13, Proposition 3.2.4], it suffices to show that for any fixed x ∈ Z 2 , the sequence of laws of the real-valued processes ξ m (x), m ≥ m 0 , is weakly relatively compact in C(R + , R). For this purpose, by Kolmogorov's criterion [28, Theorem XIII.1.8] and the convergence of the mean functions of ξ m 's in (2.20), the following uniform modulus of continuity is enough: For any fixed T ∈ (0, ∞), we can find some constants C 2.24 and ε > 0 such that Recall the function R(k) defined by (2.3). To obtain (2.24), first we use (2.22) with x = y to compute the second moments of the (real) Gaussian variables in (2.24): by the following inequality: The required inequality in (2.24) thus follows upon applying to (2.25) Assumption 2.2 (2) and the fact that the fourth moment of a centered, real-valued Gaussian with variance σ 2 is given by 3σ 4 . Next, we show that ξ ∞ admits an extension to a jointly continuous Gaussian process as defined in the statement of the present proposition. The extension to a two-parameter real-valued Gaussian process, say ζ ∞ , follows readily from the standard reproducing kernel argument for Gaussian processes. In more detail, we use the Hilbert space L 2 (R + × T 2 , drdk) and the real and imaginary parts of the following functions to construct ζ ∞ : (See also Section 4.2.) To obtain a jointly continuous modification of ζ ∞ , notice that, for 0 ≤ s ≤ t < ∞ and x, y ∈ R 2 , (2. 19) gives where the next to the last equality uses Assumption 2.2 (5) and the last equality follows from the same assumption and (2.26). From (2.28) and the Gaussian property of ζ ∞ , we deduce from Kolmogorov's continuity theorem [28, Theorem I. 2.1] that ζ ∞ admits a jointly continuous modification. The proof is complete.

Setup for the main theorem
In this section, we recall the rescaling from [7, Corollary 3.1] for the limiting Gaussian process defined in Proposition 2.6 and then state the main theorem of this paper.
Let U be a real vector defined by and V be the square root of −Q −1 so that (Recall that Q is the strictly negative definite matrix in Assumption 2.2 (4).) Then for any δ ∈ (0, 1), we define an S(R 2 )-valued process X δ by where ξ ∞,δ is the limiting Gaussian process in Proposition 2.6 and has a constant initial condition µ δ . Our goal in the rest of this paper is to prove the full convergence of X δ to the solution of a stochastic heat equation. The main result is stated in the following theorem. Theorem 3.1 (Main theorem). Let Assumption 2.2 be in force and write V = −Q −1 for Q chosen in Assumption 2.2 (4). In addition, let a family of functions {µ δ } δ∈(0,1) in 1 (Z 2 ) be given such that for some µ 0 ∈ S (R 2 ). Then the rescaled processes X δ defined by (3.3) satisfy The limiting process X 0 is the pathwise unique solution to the following additive stochastic heat equation: In the case that µ δ (x) ≡ ψ(δ 1/2 x) for some ψ ∈ S(R 2 ), the assumed convergence in (3.4) holds and we have For the proof of Theorem 3.1, we decompose the Gaussian process ξ ∞,δ according to its deterministic part and stochastic part as before in Section 2: (2.18) and ζ ∞,δ is a centered Gaussian process with a covariance function given by (2.19). The analogous decomposition of X δ (φ) is defined by: We also define a counterpart of Z δ where the floor function is removed: Organization of the proof of Theorem 3.1. We study the convergence of Z δ in Section 4 and the convergence of Y δ in Section 5. The main result of Section 4 (Proposition 4. 19) shows that the family of laws {Z δ } δ∈(0,1) is tight as probability measures on C(R + , S (R 2 )). Moreover, its distributional limit as δ → 0+ is unique and is given by the law of a C(R + , S (R 2 ))-valued random element Z 0 which satisfies the following equation. For some space-time white noise W (dr, dk) with covariance measure drdk on R + × R 2 , Then the main result of Section 5 (Proposition 5.1) shows that Y δ converges to the solution Y 0 of a heat equation in C(R + , S (R 2 )) as δ → 0+: In summary, writing for convergence in distribution as δ → 0+, we obtain from (3.10) and (3.11) that and X 0 solves the additive stochastic heat equation defined in (3.5).

Convergence of the stochastic parts
This section is devoted to the proof of weak convergence of the stochastic parts Z δ defined in (3.8) as δ → 0+. We will verify Mitoma's conditions for weak convergence in the space of probability measures on C(R + , S (R 2 )) (cf. [24, Theorem 3.1]) and characterize all the subsequential limits. For the present setup, the first of Mitoma's conditions requires that Z δ is C(R + , S (R 2 ))-valued for every δ ∈ (0, 1). This is satisfied by the following proposition.
Proposition 4.1. The stochastic part ζ ∞ of the Gaussian process ξ ∞ in Proposition 2.6 continuously extended to R + × R 2 satisfies the following growth bounds: Hence, for every δ ∈ (0, 1), Z δ and Z δ,c take values in C(R + , S (R 2 )) almost surely.
Proof. We partition Here, in the second inequality, we use the spatial translation invariance of ζ ∞ by the analogous property of the covariance function in (2.19). By (2.28), the Gaussian property of ζ ∞ and Kolmogorov's criterion for continuity [28,Theorem I.2.1], we deduce that the expectation of ζ ∞ in (4.2) is finite. Then (4.1) follows.
The required properties of Z δ and Z δ,c follow from the almost surely polynomial growth of ζ ∞ implied by (4.1) (see [

27, Example 4 on page 136]).
The other condition of Mitoma requires that the laws of {Z δ (φ)} δ∈(0,1) is tight in the space of probability measures on C(R + , R) for any φ ∈ S(R 2 ). The proof is carried out in Sections 4.1-4.5. Before proving the stochastic integral representations of Z δ (φ)'s in (1.13) for this purpose, we derive in Section 4.1 a semi-discrete integration by parts for functions taking the following form: (recall the integrands in (1.14)). The semi-discrete integration by parts has an obvious analogue for the integration by parts of the usual Fourier transform dzφ(z)e i k,V −1 z . It will handle the discontinuity of the floor function · in cancelling the two large factors δ −1 U t. Then in Section 4.2, we prove a slightly more detailed form of (1.13) by representing as a vector of stochastic integrals with respect to space-time white noises. The convergence of D δ (φ) to zero in probability uniformly on compacts and the convergence of Z δ,c to the space-time stochastic integral in (1.13) with Φ δ t replaced by Φ 0 (1.15) occupy Sections 4.3 and 4.4. The characterization of the limit of Z δ is given in Section 4.5.
Proposition 4.2. For any f ∈ 1 (Z), n ∈ Z + , δ ∈ (0, 1) and k 1 ∈ δ −1/2 T \ {0}, we have where S δ is defined in (4.4) and ∆ δ is the ordinary difference operator defined by Proof. It suffices to prove (4.6) for n = 1, and then the case of general n follows from iteration. By summation by parts, we can write Since Then by telescoping and the assumption that f ∈ 1 (Z 2 ), we get from the last two equalities that Applying the notations S δ and ∆ δ to the last equality proves (4.6) for n = 1. This completes the proof.
Proof. The integral on the left-hand side of (4.11) can be written as (4.12) Below we prove the required formula (4.11) for j = 1 by (4.12). Now, we partition R 2 by the semi-closed squares I δ δ 1/2 x−δ −1/2 U t for x ranging over Z 2 , where These squares I δ δ 1/2 x−δ −1/2 U t are chosen such that Then by the foregoing display, the right-hand side of (4.12) can be written as where (4.15) By Proposition 4.2, (4.12) and (4.14), we get Our next step is to rewrite the last sum as an integral. We claim that, for all n ∈ Z + , where · δ,t and ∆ δ,1 are defined in (4.8) and (4.10), respectively. We first show by an induction on n that ∆ n δ Φ δ (x 1 ) = δ −|α|/2 where the following change of variables for x ∈ Z 2 is in use: First, (4.17) for n = 0 follows immediately from the definition (4.15) of Φ δ : where the last equality uses the definition in (4.8). In general, if (4.18) holds for some n ∈ Z + , we write which gives (4.18) for n replaced by n + 1. Hence, by mathematical induction, (4.18) holds for all n ∈ Z + . In summary, from (4.18) and the definition in (4.8), we get which gives the required identity in (4.17). The proof of (4.11) with j = 1 is complete upon combining (4.16) and (4.17).

Stochastic integral representations
Our goal in this subsection is to obtain joint stochastic integral representations of the two-dimensional Gaussian process D δ (φ), Z δ,c (φ) , which is defined by (3.8), (3.9) and (4.3). By definition, the process D δ (φ) can be written as To lighten the stochastic integral representations to be introduced below, we use the following ad hoc notation: Here, W 1 and W 2 are independent space-time white noises. The covariance measure of W j is given by drdk: E W j s (φ 1 )W j t (φ 2 ) = min{s, t} φ 1 , φ 2 L 2 (R 2 ,dk) . When using the notation in (4.22), we always let V act on the whole function before W(dr, dk). Also, we define a change-of-variable operator T V on S(R 2 ) by (4.23) Proposition 4.4. For fixed φ ∈ S(R 2 ), the two-dimensional process D δ (φ), Z δ,c (φ) defined by (4.3) and the following two-dimensional process D δ (φ), Z δ,c (φ) have the same law: where ϕ δ t (k) and Fφ V (k) are defined by (4.27) The notation defined in (4.22) and (4.23) is used here.
Proof. First, we show that for all δ ∈ (0, 1), 0 ≤ s ≤ t < ∞ and φ ∈ S(R 2 ), (4.28) By the change of variables z → V z, it follows from (4.21) that Recall the definition (2.3) of R(k). By (2.19), κ 1 s,t (z, z ) defined by the last equality admits the following integral representation: where the third equality follows by changing variables to δ 1/2 k = k and δ −1 r = r. That is, we apply the usual diffusive scaling to exchange the scales of time and space in the last equality.
Next, integrating both sides of the last equality against dzφ V (z)dz φ V (z ) gives With respect to the other kernels κ j s,t (z, z ) defined by (4.29), similar integral representations hold for the minor differences are about whether one should remove the floor functions in (4.30) or not. The formula (4.28) follows from (4.30) and the analogous identities for the integrals in (4.31).
To see that D δ (φ), Z δ,c (φ) has the same law as D δ (φ), Z δ,c (φ) , we first note that for all 0 ≤ s ≤ t < ∞, the definition of D(φ) in (4.24) implies by (4.28) since D δ (φ) is a real-valued process so that the imaginary part of the integral in (4.28) vanishes. Similarly, along with the definition in (4.25), we get for all 0 ≤ s, t < ∞. Since D δ (φ), Z δ,c (φ) and D δ (φ), Z δ,c (φ) are both two-dimensional Gaussian processes with càdlàg paths, (4.32) and the last display show that they have the same law. The proof is complete.
Our next step is to introduce decompositions of D δ (φ) and Z δ,c (φ) which will be used for the rest of Section 4. For the decomposition of D δ (φ), we use the following representations of the function ϕ δ t defined by (4.26). They show the precise decay rate of the function.

Removal of remainders: dampening oscillations
Our goal in this subsection is to show that the processes D δ,3 (φ) and Z δ,c,2 (φ) in (4.44) and (4.47) converge weakly to zero as δ → 0+. The proofs mainly handle the differences of exponentials in (4.44) and (4.47), and for (4.44), dampen oscillations in the functions ϕ δ t (k) arising from the floor function (recall (4.26)); the effect we also need is that the convergences to zero stay regularly in C(R + , R). Handling the differences of the exponentials amounts to removing the remainders in the following equations: Note that (4.48) follows from the Taylor expansion of A obtained by combining Assumption 2.2 (4) and the definition (3.1) of U . We set some notation for the moduli of continuity of D δ,3 (φ) and Z δ,2 (φ). By polarization, the metrics ρ D δ and ρ Z δ induced by their covariance functions are given as follows: for 0 ≤ s ≤ t < ∞, where Note that sup δ∈(0,1) ρ D δ (s, t) is a-priori finite for the following two reasons. First, k → sup 0≤s≤T |ϕ δ s (k)| decays polynomially of any order by (4.5) and Assumption 4.6. Second, we have the following bounds for the real and imaginary parts of the left-hand side of (4.48). To bound the real part, we use for some C 4.51 ∈ (0, 1), which follows from Assumption 2.2 (4) and (5). For the imaginary part, we set so that and then use the following bound from the definition (3.1) of U : Since Fφ V ∈ S(R 2 ), (4.51)-(4.53) applied to sup δ∈(0,1) ρ Z δ (s, t) shows that this supremum is also a-priori finite.
We are ready to prove the bound in (4.54) for the metric ρ D defined by (4.49). We apply (4.57), (4.58) and (4.59) to (4.55) and then use the mean-value theorem. By (4.49), this leads to sup δ∈(0,1) for some constant C 4.60 depending only on (φ, A, T ). The required inequality in (4.54) for ρ D follows. The bound for sup δ∈(0,1) ρ Z δ (s, t) 2 in (4.54) can be obtained by a simpler argument if we use (4.50), since Fφ V is in place of the functions ϕ δ s and ϕ δ t in (4.49). The proof is complete.
Proof. By dominated convergence, it follows from (2.26), (4.51) and (4.53) that D δ,3 t (φ) and Z δ,c,2 t (φ) converge to zero in L 2 (P) for all t ∈ R + . We also have the weak compactness of the laws of
We start with a slightly more general framework and bound expectations of the following form in the next few lemmas: for δ ∈ (0, 1), where W (dr, dk) is a space-time white noise on R + × R 2 . The proofs of these preliminary results use the standard factorization method (cf. [11,Section 5.3.1]) and a factorization of Brownian transition densities. We write (q t (w 1 , w 2 )) t>0 for the transition densities of a centered two-dimensional Brownian motion with covariance matrix −Q (chosen in Assumption 2.2 (4)) and q t (w) = q t (0, w). Then for Borel measurable functions (s, w 1 ) → v(s, w 1 ) : R + × R → R and v : δ −1/2 T 2 → R, we define two integral operators J a−1 and J −a : for s, t ∈ R + and w 1 ∈ R 2 , (4.66) See also [11] and [25, Appendix A] for these integral operators.
Lemma 4.11. Let a be chosen as in Assumption 4.10. For v ∈ L 2 (δ −1/2 T 2 , dk), J −a v(s, w 1 ) and J a−1 J −a v(t) are well-defined integrals and we have Proof. By the Chapman-Kolmogorov equation, we can write dw 2 q s−r (w 1 , w 2 )e i k,w 2 , ∀ 0 < r < s < t. Note that (4.68) gives On the other hand, it follows from (4.64) and (4.66) that where the second equality follows from the stochastic Fubini theorem (see [32, Theorem 2.6 on page 296]) and the third equality follows from the identity: , ∀ 0 ≤ r ≤ t, α ∈ (0, 1) and (4.69). The last term in (4.71) is the same as the right-hand side of (4.70), and so the required identity (4.67) is proved.
The next two lemmas give bounds for J a−1 .
Proof. In this proof, we write C for a constant depending only on (p 1 , p 2 , a), T and λ, which may change from line to line. By the definition of J a−1 v(t) in (4.64), it holds that where the second and last inequalities follow from Hölder's inequality and the last inequality also uses (4.62) so that the first integral on its left-hand side is finite. The last inequality proves (4.72).
Lemma 4.12 will be used in the following form.
Proof. We use (4.72) with λ replaced by 2λ. Writing C = C 4.72 , we get where the second inequality follows from Hölder's inequality. We bound the right-hand side of (4.74) in two different ways according to q 2 /q 1 < 1 or not. If q 2 /q 1 < 1, then applying Hölder's inequality twice gives where C 4.75 depends only on T and (p 1 , p 2 ). If q 2 /q 1 ≥ 1, then we apply Hölder's inequality to the integral in (4.74) with respect to w 1 and get where C 4.76 depends only on (p 1 , p 2 ) and λ. Then we obtain (4.73) by applying the last two inequalities to (4.74) and using the notation q = max{q 1 , q 2 }.
The convergence of the processes D δ,2 (φ) defined in (4.43) follows from a more refined argument. We also need the following two lemmas.
Let q ∈ [1, ∞) and T ∈ (0, ∞). To obtain the analogue of (4.80) with ϕ δ,1 replaced by ϕ δ,2 t , we use by dominated convergence. Note that the sum in n in (4.82) starts with 1 so that (4.87) is applicable. Then we can apply dominated convergence and Lemma 4.13 to the above two displays as before in the proof of Proposition 4.14 and get (4.89). The proof is complete.

Characterization of limits
Let us summarize the results proven so far in Section 4. By Propositions 4.9, 4.14, and 4.17, D δ (φ) converges in distribution to zero in the space of probability measures on C(R + , R). (Recall the decomposition of D δ (φ) in (4.41).) By Propositions 4.7 and 4.9, the family of laws Z δ,c (φ) is tight in the space of probability measures on C(R + , R). (Recall the decomposition of Z δ,c (φ) in (4.45).) By (4.3), these two combined show that the family of laws Z δ (φ) is tight in the space of probability measures on C(R + , R). Since Z δ is C(R + , S (R 2 ))-valued by Proposition 4.1, it follows from Mitoma's theorem [24,Theorem 3.1] that the family of laws of Z δ for δ ranging over (0, 1) is tight in the space of probability measures on C(R + , S (R 2 )). Moreover, it is plain from (4.46) that the distributional limit of Z δ in C(R + , S (R 2 )) can be written as and so is unique. Our goal in this subsection is to show that Z 0 defined above in (4.90) solves an additive stochastic heat equation (driven by a single space-time white noise). We start with an application of Duhamel's principle.
Lemma 4.18. Write 0 −1/2 T 2 for R 2 . Then for δ ∈ [0, 1) and any bounded continuous complex-valued function ϕ defined on δ −1/2 T 2 , the continuous process solves the following SPDE: Proof. We write out the right-hand side of (4.91) and then use the stochastic Fubini theorem [32, Theorem 2.6 on page 296] in the second equality below to get: which is (4.91).
Proposition 4.19. The unique distributional limit Z 0 defined in (4.90) of Z δ as δ → 0+ solves the following SPDE: for some space-time white noise W (dr, dk) with covariance measure drdk on R + ×R 2 , Proof. Recall φ V and T V defined in (4.23). Using the bijectivity of F and T V on S(R 2 ), we define . Then Lemma 4.18 implies that is a continuous centered Gaussian process, and its covariance across times 0 ≤ s ≤ t < ∞ is given by where the first two equalities follow from the change of variables V z = z (for the Fourier transforms) and k = V −1 k, respectively, and the last equality follows from Plancherel's identity (we use the normalization of Fourier transforms as in [27, Section IX.1]). To rewrite the Riemann-integral term in (4.93) in terms of φ, we recall V = −Q −1 and then change variables to get From the last three displays, we deduce that, for a space-time white noise W with covariance measure drdk, it holds that v| det(V )|  ]. See also [22] for uniqueness theorems for stochastic equations.

Convergence of the deterministic parts
In this section, we prove convergence of the S (R 2 )-valued processes Y δ defined by (3.7) as δ → 0+.
Step 1. We begin with the observation that all the functionals in (3.4) are in S (R 2 ) and the convergence holds uniformly on compact subsets of S(R 2 ). To see the former, simply note that, by the assumption that µ δ ∈ 1 (Z 2 ), each functional in (3.4) for δ ∈ (0, 1) is in S (R 2 ). Hence, the convergence in (3.4) is with respect to the weak topology of S (R 2 ). Since S(R 2 ) is a Frechét space [27,Theorem V.9], it follows from [27,Theorem V.8] that the tempered distributions in (3.4) converge uniformly on compact subsets of S(R 2 ) as δ → 0+.
Step 2. Let us start the proof of (5.1) in this step and derive an explicit formula of Y δ (φ) for a fixed φ ∈ S(R 2 ).
Step 4. We prove (5.4) in this step and make two observations before that.
The second observation for the proof of (5.4) is that we can use (4.51), (4.52) and (4.53) to get the following bound: Note that Assumption 2.2 (4) and (5) are used to obtain (5.7). The two observations (5.5) and (5.7) can be applied to the integrals in (5.4) indexed by δ by integration by parts with respect to y j , |β| times for each j ∈ {1, 2}. Indeed, integration by parts with respect to k j once brings out a multiplicative factor 1/[−i(V −1 y) j ] from e −i k,V −1 y (whenever (V −1 y) j = 0) and the boundary terms vanish as δ → 0+ by (5.5) and (5.7). This proves (5.4), and hence, the convergence in (5.3).
Step 5. In this step, we evaluate the limit of Y δ t (φ) as δ → 0+ for fixed t. The limiting integral in (5.3) with respect to k over R 2 can be simplified as follows: with the change of variables k = V j/ √ t, and so R 2 dzφ(z) 1 (2π) 2 R 2 dke tQ(k)/2−i V −1 y,k +i k,V −1 z = | det(V )|P t φ(y), (5.8) where (P t ) is the semigroup of the two-dimensional standard Brownian motion.