Weak error analysis for a nonlinear SPDE approximation of the Dean-Kawasaki equation

We consider a nonlinear SPDE approximation of the Dean-Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order $N^{-1-1/(d/2+1)}\log (N)$. Along the way we show well-posedness, a comparison principle and an entropy estimate for a class of nonlinear regularized Dean-Kawasaki equations with It\^o noise. Keywords: Dean-Kawasaki equation, weak error analysis, Laplace duality


Introduction
The Dean-Kawasaki (DK) equation, named after [Dea96] and [Kaw94], is a stochastic partial differential equation (SPDE) for the evolution of the empirical measure of particles following Langevin dynamics with pairwise interaction.It is formally given by where ξ is a vector-valued space-time white noise and α > 0. In [Dea96], (1.1) is formally derived as a closed equation for the dynamics of the empirical measure of mean-field interacting diffusions: Let (X i ) N i=1 solve the Langevin dynamics with interaction potential W , that is for N independent Brownian motions (B i ) N i=1 .Dean [Dea96] argues that the empirical measure µ N t = 1 N N i=1 δ X i t solves (1.1) with α = N .This is based on an ad-hoc replacement of the (stochastic integral against the) Brownian noise by a (stochastic integral against a) space-time white noise which formally has the same law.Due to the singular gradient noise and because of the square root nonlinearity, the mathematical meaning of (1.1) is dubious, although there has been a lot of progress in recent years, see the literature review below.Nonetheless, the Dean-Kawasaki equation is very useful for applications and SPDE models of DK-type are very popular in physics (e.g.[MT00, VCCK08, GNS + 12, DOL + 16, DCG + 18]), where they are also known as (stochastic) dynamical density functional theory, see the recent survey [tVLW20].The motivation for this work is the successful use of the Dean-Kawasaki equation as a computational tool for simulations of fluid dynamics [DFVE14] or social dynamics [DCKD22, HCD + 21].If N is large and if the particles live in a low-dimensional space, a discretization of (1.1) can be computationally much cheaper than simulating N particles with their interactions.This is particularly interesting for moderate sizes of N , where the hydrodynamic limit (N → ∞) would miss important stochastic fluctuations; for example, in social dynamics it is interesting to consider N = O(10 4 − 10 5 ).However, because of the singular nature of the DK equation and because of its fragile solution theory (see the literature section below), it is far from obvious why a discretization of (1.1) should give meaningful results for (1.2).To understand this on a conceptual level, we focus on the case without interaction (W = 0) and we introduce a regularized version of the DK-equation, where we truncate the noise, we mollify the square root nonlinearity and we consider on R + × T d the equation for the Fourier-basis (e k ) k∈Z d , for a Lipschitz function f δ N which approximates the square root, and for d-dimensional complex Brownian motions (B k ) k∈Z d .Under appropriate conditions we prove the well-posedness of this approximate DK-equation.The equation is not locally monotone in the sense of [LR15] and the nonlinear Itô gradient noise is quite tricky (much more so than Stratonovich noise), therefore we prove well-posedness through a suitable transformation of the equation in combination with a priori energy bounds.Furthermore, we prove a comparison principle which yields non-negativity of the solution for non-negative initial data and the conservation of the L 1 -norm.This is important because μN should represent the empirical distribution µ N of particles, which is a probability measure by the mass conservation of the particle system.Our main result concerns the quality of approximation of µ N by μN .We do not expect that μN and µ N are pathwise close and we do not even couple the noises in their dynamics to each other.Instead, we focus on the weak approximation quality, and we show that for suitable test functions Since the equation for μN should be amenable to discretization, this gives a partial justification for the use of DK-type equations in numerical simulations.Although our methods crucially rely on the absence of interactions because then we can compute E[F (µ N t )] explicitly by duality arguments (cf.[KLvR19]), we expect similar results to hold in the interacting case, and we expect that our general approach is quite powerful and can possibly be generalized to the interaction case.The paper is structured as follows.Below we review some mathematical literature on the Dean-Kawasaki equation and we discuss why its solution theory is subtle and some recent breakthroughs.Section 2 introduces the notation and assumptions for our approximate Dean-Kawasaki equation, and we state the main results.In Sections 3.1 and 3.2 we prove the wellposedness and a comparison principle for regularized Dean-Kawasaki-type equations, which in particular applies to our approximation (1.3).Section 4 provides the weak error estimates.Some mathematical literature on Dean-Kawasaki type equations Giving a pathwise meaning to the DK equation (1.1) is an open problem even in the case without interaction potential, W = 0.The equation is highly singular because the noise is effectively a derivative of the space-time white noise and thus extremely irregular, and therefore the solution can only be a distribution and not a function and thus the square root µ N t and the product µ N t ξ have no meaning.Even in dimension d = 1, the equation is scaling supercritical in the language of regularity structures ( [Hai14]) and paracontrolled distributions ( [GIP15]), which are techniques to tackle subcritical singular SPDEs.Entropy solutions of approximate DK-type equations with Stratonovich noise k ∇ • σ k (x, u) • dB k t were studied in [DG20].Using a pathwise approach via rough paths techniques, [FG19] established well-posedness for porous media equations (replacing ∆u by ∆(|u| m−1 u) with exponent m ∈ (0, ∞)), again for Stratonovich noise.Both of these results require very regular diffusion coefficients σ.The well-posedness of the Dean-Kawasaki equation with truncated Stratonovich space-time white noise and with square-root diffusion coefficient was recently shown in [FG21], where the authors establish existence and uniqueness of stochastic kinetic solutions.The Stratonovich noise enables them to obtain a priori entropy-type estimates on the kinetic solutions (cf.[FG21, Section 5.1]), which yield the relative compactness of solutions with regularized square root nonlinearity.We refer to [FG21] and the references therein for more details on the well-posedness of related equations.Furthermore, the Dean-Kawasaki equation is related to scaling limits of interacting particle systems, see e.g.[GLP98].[FG22] show that the Stratonovich DK equation correctly predicts the large deviation rate function for non-equilibrium fluctuations of the zero range process.Dean-Kasawaki type equations with regularized square root and truncated Itô noise have been considered also in [Bec21], where the existence of weak solutions is shown.Regularized Dean-Kawasaki equations for underdamped kinetic particles were investigated in [CSZ19,CSZ20].The authors impose a cutoff on the noise, which is formally justified by an approximation of the Dirac deltas in µ N t = N i=1 δ X i t with mollifying kernels.They establish the existence of mild solutions with high probability.The mathematically correct interpretation of the "full" (not truncated) Dean-Kawasaki equation was found only recently [KLvR19,KLvR20].The authors show that (1.1) should be interpreted as a martingale problem, and that by similar duality arguments as for superprocesses [Eth00] uniqueness holds for this martingale problem.In fact, the authors prove much more.Namely, at least for bounded interaction potentials, the Dean-Kawasaki equation has a unique martingale solution if and only if the parameter N is a natural number and the initial condition is an atomic probability measure (given by the sum of N Dirac measures).If we replace µ N 0 by an initial condition µ 0 ∈ M that is not of the form 1 N N i=1 δ y i , or if we replace N by a constant α ∈ R >0 \ N, then there exists no martingale solution.In particular, the well-posedness is very fragile with respect to changes of the parameters, and as such, the equation is not suitable for a stable numerical approximation.From a numerical perspective, this instability in the parameter N is worrisome.That is, slight changes in N yield an ill-posed problem and possibly to large numerical errors.Nonetheless, numerical schemes for Dean-Kawasaki-type equations were considered in [CS22,CF21].[CS22] introduce a discontinuous Galerkin scheme for the regularized DK equation from [CSZ19,CSZ20].The work most related to ours is Cornalba-Fischer [CF21], who consider finite element and finite difference approximations of (1.1) without interaction (W = 0) and who also prove weak error estimates.Their weak distance is parametrized by the Sobolev regularity of the test functions and the rate measured in their distance can be arbitrarily high, only limited by the numerical error and the error coming from the negative part of the approximation.However, the authors do not prove positivity for the approximations (hence the consideration of the negative part).Additionally they impose a strong assumption ([CF21, Assumption FD4]) on the existence of lower and upper bounds for the solution of a discrete heat equation, which means that initially the particles must be fairly spread out.Our results are orthogonal in the sense that we consider an SPDE approximation and not a discrete model, and we do not have any restrictions on the initial distribution of particles.Our convergence rate is upper bounded by N −5/3 in d = 1 and it gets worse in higher dimensions, which is due to constraints on the choice of parameters coming from the solution theory of the SPDE.Our methods are very different (Laplace duality/Kolmogorov backward equation compared to a clever recursive scheme and choice of distance in [CF21]).

Preliminaries and statement of the main results
Consider the particle system of N ∈ N independent standard d−dimensional Brownian motions (X i ) N i=1 projected onto the torus T d = R d /Z d (with generator being the Laplacian with periodic boundary conditions) started at X i 0 = x i ∈ T d .We are interested in the empirical measure (2.1) ), we see that µ N formally solves the Dean-Kawasaki equation without interaction and with atomic initial condition, where ξ := (ξ j ) j=1,...,d with independent space-time white noise processes ξ j , see [Dea96].As in [KLvR19, Definition 2.1] we will interpret (2.2) as a martingale problem.The state space of the solution process is the space M of probability measures on T d , equipped with the topology of weak convergence.For µ ∈ M and ϕ ∈ C(T d ), we write µ(ϕ) = µ, ϕ = T d ϕdµ.
Definition 2.1.We call a stochastic process (µ N t ) t 0 on a complete filtered probability space (Ω, F , (F t ) t 0 , P) with values in C(R + , M ) a solution to (2.2) if for any test function ϕ ∈ C ∞ (T d ), the process By [KLvR19, Theorem 2.2] this martingale problem has a unique (in law) solution given by (2.1).And if in (2.2) we replace µ N 0 by an initial condition µ 0 ∈ M that is not of the form then there exists no martingale solution to the equation.In particular, the well-posedness is very fragile with respect to changes of the parameters.Our goal is to approximate the equation (2.2) in a controlled manner with an equation that has good stability properties and that preserves the physical constraints of µ N , i.e. positivity and mass conservation.We aim for a bound of the form for t > 0 and for suitable nonlinear test functions F : M → R. As usual, a b means that there exists a constant C > 0 (not depending on the relevant parameters; above the parameters are δ, x), such that a Cb.If we want to indicate the dependence of the constant C(κ) on a parameter κ, we write a κ b.We will see that for ] can be computed in closed form and therefore we consider such F .Due to the factor 1/N in front on the quadratic variation, a direct computation shows that the hydrodynamic/mean-field limit, that is, the solution ρ of the heat equation achieves a rate of convergence α d = 1.However, ρ is deterministic and it does not capture random fluctuations in the particle system µ N .For the Gaussian approximation ρ of the fluctuations of the particle system around the hydrodynamic limit, that is, with the same vector-valued space-time white noise ξ as above, we can prove that ρ is not a measure, let alone positive, and it does not conserve the mass of the initial condition (in fact Therefore, we consider a nonlinear approximation of (2.2):We replace the non-Lipschitz square root function with a Lipschitz approximation that will depend on a parameter δ and we replace the noise by its ultra-violet cutoff at frequencies of order M .The parameters δ, M will influence the order of the approximation and they will be chosen subsequently in the error estimate, depending on N .Let for now δ = δ N > 0 and M = M N ∈ N. We then define the Lipschitz function f = f δ as follows (2.4) The smooth interpolation should be such that f ∈ C 1 (R) satisfies Any C 1 approximation of the square root satisfying those bounds works for our analysis and a particular example of such a function is given in the following example.
Example 2.2.Consider, for example, (2.7) It is not hard to see that f ∈ C 1 and that f satisfies the bounds (2.5) and (2.6).
We denote by μN the solution of the approximated equation where (ρ N ) N is an appropriate approximation of the identity, that we will choose later.Moreover, the truncated noise (W N t ) t is given by , and a truncation parameter M N ∈ N, that will be chosen depending on N for the error estimate.
In the next section we prove the strong well-posedness of slightly more general equations than (2.8), as well as the non-negativity of the strong solution and the mass conservation property.But first let us formulate a summary of our main results: Theorem 2.3 (Summary of the main results).Let μN 0 ∈ L 2 be positive and let M N and δ N be such that Then there exists a unique solution μN to (2.8), which is positive and which satisfies μN t L 1 (T d ) = μN 0 L 1 (T d for all t 0, as well as the entropy bound for λ := 1 4 (1 < 1, and if µ N is the martingale solution of (2.2) and if μN 0 is an approximation of µ N 0 as in Lemma 4.3, then for any t > 0, ϕ ∈ C ∞ (T d ) and F (µ) := exp( µ, ϕ ) for µ ∈ M , the following weak error bound holds: (2.11) Proof.All this is shown in Section 4. See Proposition 4.1 for the well-posedness, see Proposition 4.2 for the entropy estimate, and see Theorem 4.4 for the weak error estimate.
In summary, the nonlinear approximation μN achieves a weak rate α d > 1, which is always better than the error of the deterministic approximation ρ.In d = 1 the nonlinear SPDE has a smaller weak error than the Gaussian approximation ρ + 1 √ N ρ (rate N −5/3 log N compared to N −3/2 ), and in d = 2 the rates are nearly the same (N −3/2 log N respectively N −3/2 ).In higher dimensions we get a worse error bound for μN than for the Gaussian approximation, which achieves N −3/2 independently of the dimension.But on the positive side μN is a probability density, while ρ + 1 √ N ρ is not positive and only a Schwartz distribution and not even a signed measure.The main obstruction towards reaching a better rate for μN is that the solvability conditions of Section 3.1 impose constraints on δ N and M N .

Well-posedness
In this section, we prove strong well-posedness for SPDEs of the type Similar stochastic conservation laws with Stratonovich noise have been intensely studied in the past years, see for example [DG20, FG19, FG21], among many others.However, due to the gradient noise the solution theory for Itô noise is more subtle and we need to make an additional smallness assumption to even get the existence of solutions; see the discussion before Assumption 3.4 below.Also, the coefficients of (3.1) are not locally monotone in the sense of [LR15, Theorem 5.1.3],and therefore the solution theory does not follow from standard theory.There is the related work [Bec21] by Bechtold on equations resembling (3.1), and we are in the case of "critical unboundedness" from [Bec21, Section 4].However, [Bec21] only proves the existence of a probabilistically weak solution with paths in C([0, T ], H −ε ) ∩ L 2 ([0, T ], H 1 ) for any ε > 0. By a pathwise uniqueness argument using the a priori energy bound that we derive below, it should be possible to show the strong existence and uniqueness of a solution with paths in C([0, T ], H −ε )∩ L 2 ([0, T ], H 1 ) for any ε > 0. Instead, we directly show the stronger statement of strong existence and uniqueness of a solution with paths in C([0, T ], L 2 ) ∩ L 2 ([0, T ], H 1 ).We make the following assumptions.
Assumption 3.1 (Assumption on the noise).
We assume that the noise (W t ) t is given by For some results we only require C W 1 < ∞, and we call this the relaxed Assumption 3.1.
Remark 3.2.An example for a noise expansion satisfying the assumption is the Fourier expansion with cut-off M N ∈ N from (2.9).The summability assumptions are trivially satisfied due to the finite cut-off.Similar to [FG21, Remark 2.3], we can also consider a noise for a real sequence (a k ) with k∈Z d |k| 2 a 2 k < ∞, which also satisfies our assumption.
Assumption 3.3 (Assumptions on the diffusion coefficient).We assume that the diffusion In particular, b is of linear growth: There exist Assumption 3.4 (Stochastic parabolicity).We assume that the parameters from the previous assumptions satisfy The well-posedness theory of this section will apply to the approximating Dean-Kawasaki equation (2.8), where b = 1 √ N f with f from (2.4) and L = C b 1 = 1/ √ N δ.Furthermore, the noise expansion is given with respect to the Fourier basis (φ k ) k with cut-off M N ∈ N. In that case, we have that Note that due to the gradient noise term the local monotonicity condition is violated for SPDEs of the form (3.1) and the variational approach of [LR15] cannot be applied directly.Instead, we transform the equation (by applying (∆ − 1) 1/2 ) into an equation for which the variational theory can be applied and we deduce well-posedness of the original equation by using a priori energy bounds.The setting for the variational theory is defined as follows.Let V = H 1 (T d ) with V * = H −1 (T d ) and H = L 2 (T d ), such that we have the Gelfand triple V ⊂ H ⊂ V * .We consider the Laplacian with periodic boundary conditions, that is, ∆ : For the duality pairing, we have that ∆u, v V * ,V = − ∇u, ∇v H = − ∇u • ∇v when u, v ∈ V .Then, we define a solution to the equation (3.1) as follows.
Definition 3.5.A stochastic process (u t ) t 0 with paths in is a (probabilistically strong and analytically weak) solution to the equation (3.1) for the initial condition First, we study well-posedness of a transformed equation with the variational approach: Lemma 3.6.Let Assumptions 3.3 and 3.4 hold, as well as the relaxed Assumption 3.1, and consider an initial condition v 0 ∈ L 2 (T d ).Then, there exists a unique probabilistically strong Proof.We will check the conditions of [LR15, Section 4] to apply the variational theory.Let us define G(v)w ) and v ∈ H 1 .Using (3.4), we obtain the following coercivity bound (that is, (H3) from [LR15, Section 4]): , where we used the Assumption 3.2.Here, the first inequality follows from the Plancherel theorem for the Fourier transform F T d on the torus (F and similarly we get The coercivity then follows by the Assumption 3.4 on the parameters, since C W 1 C b 1 < 1.The weak monotonicity condition (that is, condition (H2) from [LR15, Section 4]) follows from the analogue estimate, using the global Lipschitz bound L due to (3.3): L 2 , using C W 1 L 2 < 1 in the last step.Thus, we obtain the weak monotonicity.The hemicontinuity from [LR15, Section 4, H1] follows by linearity of the Laplacian.The boundedness condition from [LR15, Section 4, H4] is trivially satisfied due to continuity, that is, ∆u V * v V .
Remark 3.7.If u is a solution to (3.5), then v := (1 − ∆) −1/2 u is a solution to (3.6).As (3.6) has a unique strong solution, it follows that the solution to (3.5) is pathwise unique.Moreover, if in Definition 3.5 we formally integrate ∇(b(u t )φ k ), ϕ = b(u t )φ k , ∇ϕ by parts, the definition makes sense even for u with paths in . With this definition the equation is equivalent to Equation (3.6) for v := (1 − ∆) −1/2 u.Therefore, under Assumptions 3.3 and 3.4 and the relaxed Assumption 3.1, for any u 0 ∈ H −1 (T d ) there exists a pathwise unique strong solution u with paths in to (3.5).We only need the full Assumption 3.1 to obtain better regularity for u.
Note that the relaxed Assumption 3.1 corresponds to a subcriticality condition in the sense of regularity structures [Hai14]: Indeed, this is precisely what we need from the noise so that the solution to the linearized equation dZ t = 1 2 ∆Z t dt + ∇ •dW t is a function in the space variable and not a distribution.However, under such weak assumptions Z t will not be in H 1 and therefore also u t should not be in H 1 .So, to obtain better regularity for u t , we make stronger assumptions on the noise.Besides the subcriticality condition we also need the smallness condition Assumption 3.4 which is due to the Itô noise.We expect that for Stratonovich noise the relaxed Assumption 3.1 together with Assumption 3.3 is sufficient and that with our transform we can solve the Stratonovich version of (3.5) in the entire subcritical regime.
To prove existence of a solution to (3.5) in the sense of Definition 3.5, we proceed as follows: Given the solution v of (3.6), we can define u almost surely and, by the equation for v, u is a solution to the ("very weak") equation The lemma below proves an energy estimate which yields tightness of the sequence of Galerkin projected solutions (u R ) R .That is, for the orthogonal projection Π R : for the Fourier basis (e r ) r∈Z d , we let By the uniqueness of the solution v and the construction of the solution in the variational theory (cf.[LR15, Theorem 5.1 Lemma 3.8.Let Assumptions 3.1, 3.3 and 3.4 hold.Let u 0 ∈ L 2 (T d ) and let v 0 := (1 − ∆) −1/2 u 0 .Let v R be the solution of the equation (3.6) with G k replaced by G R k , let v R 0 = Π R v 0 , and let u R be defined as in (3.8).Then, the following energy bound holds true: By applying Itô's formula to (u R t (x)) 2 , we then obtain Taking the expectation, the martingale vanishes and using Π R v L 2 v L 2 and (3.3), we obtain Letting λ := 1 − C W 1 L 2 > 0 and using the linear growth assumption on b given by (3.4), we obtain (3.12) Using Gronwall's inequality, we thus obtain 1 and hence, plugging (3.13) in (3.12), yields which implies (3.9), as u R 0 2 Remark 3.9 (Energy estimate).If we take b = 1 √ N f , where f is given by (2.4), we can improve the energy estimate by using that |f (x)| |x| in order to estimate the L 2 -norm of b(u R s ) in (3.11).Utilizing also the mass conservation of the solution u, that we later prove in Proposition 3.15, we then obtain the following a priori energy bound Theorem 3.10.Let Assumptions 3.1, 3.3 and 3.4 hold and let u 0 ∈ L 2 (T d ).Then there exists a unique solution u with paths in of the equation (3.1) in the sense of Definition 3.5.
Proof.From Lemma 3.6 it follows that for By the regularity of v, we obtain that almost surely Furthermore, from the equation of v, testing against ϕ ∈ C ∞ (T d ), we obtain that u solves the "very weak" equation (3.7).By Lemma 3.8, the Galerkin projected solutions (u R ) R satisfy the energy bound (3.9).Since L 2 (Ω × [0, T ], H 1 ) is reflexive, we thus obtain that, along a subsequence, (u L 2 ).Thus, the limit of each such subsequence is given by u and we can conclude that the whole sequence (u R ) R converges to u, weakly in L 2 (Ω × [0, T ], H 1 ).In particular, the limit u satisfies u ∈ L 2 ([0, T ], H 1 ) almost surely.Due to u ∈ L 2 ([0, T ], H 1 ) ∩ C([0, T ], H −1 ) a.s., the mapping t → u t ∈ L 2 is almost surely weakly continuous.Since u ∈ L 2 ([0, T ], H 1 ) a.s. and u 0 ∈ L 2 , and because (3.7) is equivalent to u solving we can apply [LR15, Theorem 4.2.5] to obtain an Itô formula for d u t 2 L 2 .Almost sure continuity of the integrals in time then implies almost sure continuity of the mapping (3.15) From continuity of (3.15) and continuity of t → u t ∈ L 2 in the weak topology we get that u ∈ C([0, T ], L 2 ) almost surely.Hence, overall, we indeed have that u ∈ L 2 ([0, T ], H 1 ) ∩ C([0, T ], L 2 ) almost surely.By the regularity of u and as u solves (3.7), it follows that u solves (3.5) (for all ϕ ∈ C ∞ (T d ) and thus, by density for all ϕ ∈ H 1 (T d )).Uniqueness of the solution follows from Remark 3.7.
Remark 3.11.Using the Itô formula for u R t − u t 2 L 2 (cf.[LR15, Theorem 4.2.5]) and as H 1 ) and that the energy estimate holds true for the limit u.

A comparison principle for regularized DK-type SPDEs
In this section we prove a comparison principle for the class of SPDEs (3.1), which will in particular imply positivity and mass conservation of the solution.
Theorem 3.13.Let Assumptions 3.1, 3.3 and 3.4 hold.Furthermore, let u + and u − be two solutions of (3.5) with initial conditions u + 0 , u − 0 ∈ L 2 , respectively, such that u + 0 (x) u − 0 (x) for Lebesgue-almost all x ∈ T d .Then where Leb is the Lebesgue measure on T d .
Proof.We follow the proof of [DMP93, Theorem 2.1].The main idea is an application of Itô's formula to a suitable C 2 approximation of the map x → max(x, 0) 2 , applied to the difference of the solutions.More precisely, let for p > 0, ϕ p ∈ C 2 (R, R) be defined by ]dzdy.Note that ϕ p satisfies 0 ϕ ′ p (x) 2 max(x, 0) and 0 ϕ ′′ p (x) 2 1 x 0 .
Proof.Since b(0) = 0, it follows that the zero function is a solution of (3.1).Then the claim directly follows from Theorem 3.13.Proof.Using non-negativity of the solution u obtained by Corollary 3.14 and testing the equation against ϕ = 1 ∈ C ∞ (T d ), we have that, for almost all ω ∈ Ω, The claim follows from the continuity of t → u t (x)dx.

Weak error estimate
In this section we estimate the weak error between the martingale solution µ N of the Dean-Kawasaki equation and the strong solution μN of the approximate Dean-Kawasaki equation.For that purpose we first have to discuss the solution theory of the approximate Dean-Kawasaki equation: Proposition 4.1.Consider the equation and where Proof.This follows from the results in Section 3.1 once we verify that our equation satisfies the assumptions of that section.For that purpose, note that and, since f (0) = 0, we also have N δ N by assumption, and C b 2 = 0. Therefore, Assumptions 3.1 and 3.3 hold.Moreover, the condition ) Proof.Let γ > 0 and g γ (y) := (γ + y) log(γ + y), y ∈ [0, ∞).Then g γ ∈ C ∞ ([0, ∞), R), with (g γ ) ′ (y) = log(γ + y) + 1 and (g γ ) ′′ (y) = 1 γ+y , and we extend g γ to R so that the extension is in C 2 .But recall that μN is positive by the comparison principle, and therefore we only evaluate g γ on [0, ∞) below.By the Itô formula from [Par80, Theorem 1.2] we have where M denotes the local martingale term.We estimate the Itô correction term for κ > 0 (applying Young's inequality in the same way as in the proof of the comparison principle) by Summing over k and estimating f ′ 2 ∞ C/δ N , we obtain for some K > 0 and by choosing κ > 0 appropriately we can achieve that 1 Next, we show that the local martingale is a true martingale: Since |f (µ)| 2 |µ|, its quadratic variation satisfies  Proof.Positivity is obvious because μN 0 is a convolution of positive measures.The statement about the L 1 norm follows by Fubini's theorem.For the entropy, we first trivially bound Since x log x is negative on [0, 1), we get together with the trivial L ∞ bound: To compare µ N 0 , ϕ and μN 0 , ϕ , note that .5] to construct a metric for the topology of weak convergence of probability measures on M and then replace the left-hand side of (4.7) by the distance of μN t and µ N t in this metric.Proof of Theorem 4.4.To prove the weak error bound (4.7), we apply the duality argument of [KLvR19].For that purpose, let v solve the Hamilton-Jacobi equation with initial condition ϕ ∈ C ∞ (T d ),

. 4 )
From the assumption 3.3, it follows that b is Lipschitz continuous with bound L, that is |b(x) − b(y)| ≤ L|x − y| for all x, y ∈ R. If we would consider Stratonovich noise, these two assumptions would be sufficient.For Itô noise we need an additional smallness condition for the gradient noise, because otherwise it could break the parabolic nature of the equation; see [Par21, Section 2.4.2].
d T d N d ρ(N (x i + k − y))ϕ(y)dy x i ) − ψ n * ϕ(x i )) ϕ − ψ N * ϕ ∞ ,where ψ N (x) = N d ρ(N x) for x ∈ R d , and where the convolutions in the second line are on R d and not on T d .Using the symmetry of ρ, we get for allx ∈ R d |(ϕ − ψ N * ϕ)(x)| = R d ρ(N y)N d ϕ(x) − ∇ϕ(x) • y − ϕ(y) dy ϕ C 2 b R d ρ(N y)N d |y| 2 dy = N −2 ϕ C 2 b R d ρ(y)|y| 2 dy,which concludes the proof.Note that the entropy of μN 0 is much smaller than its L 2 norm, which in general we can only bound by N d .Theorem 4.4.Let µ N be the martingale solution of the Dean-Kawasaki equation in the sense of Definition 2.1 with initial condition µ N 0 := 1 N N i=1 δ x i .Let μN 0 be as in Lemma 4.3 and let f and W N be as in Proposition 4.1.Assume that with the notation of Proposition 4.1 sup N C N (2M N +1) d N δ N < 1.Let μN be the solution of the approximate Dean-Kawasaki equation (4.1) with initial condition μN 0 .Then for any t > 0, ϕ ∈ C ∞ (T d ) and F (µ) := exp( µ, ϕ ) for µ ∈ M , the following weak error bound holds:|E[F (μ N t )] − E[F (µ N t )]| ϕ N −2 + (which is the optimal choice under the coercivity condition) we have|E[F (μ N t )] − E[F (µ N t )]| ϕ N −1− 1 d/2+1 (t + log(N )).(4.7) Remark 4.5.With the functions F k (µ) = exp( µ, ϕ k ) for a suitable dense set (ϕ k ) k∈N ⊂ C ∞ (T d ), we could use [EK86, Theorem 3.4 also Assumption 3.4 holds.This well-posedness result together with the energy estimate of Section 3.1 would be sufficient to derive a weak error estimate for the approximation of µ N with μN .However, through the energy estimate the weak error would depend on μN Proposition 4.2 (Entropy estimate).Let μN be a solution of the approximate Dean-Kawasaki equation (3.5) with the initial condition μN 0