Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $\epsilon$ is proportional to $N^{-1/\theta}$ for $\theta>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Fa\`a di Bruno's formula.


Introduction
Fluctuating hydrodynamics is a class of models describing fluctuations around the hydrodynamic limit of a many-particle system; a particular example is the Dean-Kawasaki model [10,17], which describes the evolution of finitely many particles governed by over-damped Langevin dynamics. At its core, this model is a stochastic PDE for the empirical density, comprising a diffusion equation that is stochastically perturbed by a mass-preserving multiplicative spacetime white noise; see (6) below. Equations of fluctuating hydrodynamics are widely used in physics and other sciences (e.g., in the description of active matter [24,5], thermal advection [19], neural networks [22], and agent based models [11]), and are currently being investigated numerically [16]. Still, the mathematical analysis of these equations is in its infancy. A truly remarkable recent result [18] shows that a solution for the original Dean-Kawasaki model (as derived in [10] and given in (6) below) only exists when the initial datum is a superposition of a finite-number N of Dirac delta functions and the diffusion coefficient is 1 2 N ; if such an initial datum is ever so slightly mollified, then no solution exists. Given the numerous applications of equations of fluctuating hydrodynamics, this apparent mathematical instability is particularly puzzling.
In light of this, several regularised Dean-Kawasaki models (featuring smooth noise coefficient and coloured driving noise) have been proposed and studied [13,11,14,7,8]. In recent work [7,8], the authors have derived and analysed stochastic PDE models for the empirical density of N -particles following second-order Langevin dynamics and interacting weakly. The models are derived from particles as entities of finite size rather than Dirac delta functions and this regularisation is crucial for the mathematical theory. We refer to this PDE as the Regularised Inertial Dean-Kawasaki (RIDK) model. In particular, we have established that RIDK has a well-defined mild solution in one-dimension with probability converging to one in the limit as N → ∞ and the particle width ǫ → 0, subject to particles being wide enough (as given by the scaling condition N ǫ θ = 1 for a given θ). In this paper, we establish well-posedness for RIDK in any finite spatial dimension and significantly improve the scaling condition (relax conditions on θ) in the one-dimensional case. To the best of our knowledge, this is the first proof of well-posedness for RIDK or any Dean-Kawasaki model in several space dimensions.

Setting and main result
We consider N -weakly interacting particles on the d-dimensional torus T d := [0, 2π) d . The particles are identified by position and momentum (q i , p i ) N i=1 ∈ T d × R d , and satisfy the stochastic differential equatioṅ where γ, σ are positive constants, U : T d → R is a smooth pairwise interaction potential, and {b i } N i=1 is a family of independent standard d-dimensional Brownian motions. We work under the key modelling assumption that the particles have a finite size. Specifically, we describe their spatial occupancy by means of a kernel w ǫ : T d → [0, ∞) indexed by ǫ > 0, which may be thought of as the particle width. We propose RIDK as a model for the particle density and momentum density (ρ ǫ (x, t), j ǫ (x, t)) : In particular, RIDK defines an approximate particle and momentum density (ρ ǫ , ǫ ) : subject to (ρ ǫ (·, 0), ǫ (·, 0)) = (ρ 0 , 0 ) for initial densitiesρ 0 andj 0 , where {ξ ℓ } d ℓ=1 are independent space-time white noises, and P ǫ is the convolution operator P ǫ : L 2 (T d ) → L 2 (T d ) : f → P ǫ f (·) = T d w ǫ (· − y) f (y) dy. The operator P ǫ describes the spatial correlation of the stochastic noise and is intrinsically linked to the spatial occupancy of the particles through the regularising kernel w ǫ . This model is of inertial type (meaning that it keeps track of both density and momentum density), and is a generalisation of the models studied in [7,8] to higher dimensions. For w ǫ , we choose the von Mises kernel which is the toroidal equivalent of a multivariate Gaussian with mean zero and diagonal covariance matrix ǫ 2 I d .
For regularity purposes which will become clear later, it is convenient to replace the squareroot in (2) with a smooth function h δ : R → R such that h δ (z) = |z| for |z| ≥ δ/2, for some small and fixed δ > 0. Following this change, the RIDK equation (2) is rewritten in the abstract stochastic PDE notation where X ǫ,δ = (ρ ǫ,δ , ǫ,δ ), X 0 = (ρ 0 , 0 ), A is a linear operator describing the deterministic drift excluding the interaction-potential term α U (X ǫ,δ ), B N,δ is the stochastic integrand associated with the introduction of h δ , and W per,ǫ is a Q-Wiener representation of the noise P More details concerning (4), as well as a sketch of its derivation from the Langevin particle dynamics (1), are given in Section 2.
Throughout the paper, we work under the general scaling From a modelling point of view, (5) imposes the particle size (comparatively to N ), where increasing θ implies increasing particle size. From an analytical perspective, (5) affects the spectral properties of the noise W per,ǫ in (4) through the operator P ǫ . We state the main result of this paper.
, and a set F ν of probability at least 1 − ν such that min x∈T d ,s∈[0,T ]ρǫ,δ (x, s) ≥ δ on F ν , and X ǫ,δ solves (4) pathwise on F ν in the sense of mild solutions [9,Chapter 7]. As a consequence, X ǫ,δ also solves the RIDK equations (2) pathwise on F ν in the sense of mild solutions.
The proof exploits a small-noise analysis, by obtaining the solution to (4) as a small perturbation of the strictly positive solution of the noise-free dynamics (i.e., the damped wave equation). When the perturbations are small and the initial data is everywhere larger than δ, the solution to (4) remains outside the regularisation regime (−∞, δ/2) for h δ and the regularisation is bypassed, resulting in a well-defined solution of (2). The C 0 -norm is used to measure the perturbations and keep track of whether the solution falls into the regularisation region. To do this, the parameter s is chosen so the mild solutions take values in the Sobolev space W s , which is embedded continuously in With this in mind, the proof of Theorem 1.1 (see Section 4) is built upon three conceptual blocks, developed in Section 3. Firstly, A is proven to generate a C 0 -semigroup with respect to the W s -norm (see Subsection 3.1, Lemma 3.1). Secondly, the stochastic integrand B N,δ is shown to be locally Lipschitz and sublinear (d = 1) or locally Lipschitz and locally bounded (d > 1) with respect to the Hilbert-Schmidt L 0 2 (W s )-norm (Subsection 3.3, Lemma 3.4). These two blocks give rigour to the application of the mild solution theory. Thirdly, sharp bounds for the trace of W per,ǫ with respect to the W s -norm are provided via spectral analysis of P √ 2ǫ (see Subsection 3.2, Lemma 3.2). In combination with Lemma 3.4, this guarantees the vanishing-noise regime for (4) in the W s -norm as N → ∞. Theorem 1.1 carries two significant contributions. Firstly, it provides a well-posedness theory for the multi-dimensional RIDK model; to the best of our knowledge, this is the first paper to give an existence and uniqueness theory for such a model. Secondly, it improves an analogous one-dimensional result [7,8] by significantly relaxing the scaling threshold in (5) from θ 0 = 7 to θ 0 = 2. The more restrictive threshold for θ resulted from a suboptimal analysis with respect to the W 1 norm. The θ 0 = 7 scaling is inconveniently restrictive, as it only allows for rather large particles (comparatively to N ). Specifically, θ is significantly away from the null value (formally corresponding to representing particles by Dirac delta functions) and from the unitary value (associated with volume preservation of the particle system).
The main technical novelties that we introduce in the proof of Theorem 1.1 are the following. First, we deploy improved estimates for the spectral properties of P √ 2ǫ , which rely on refined bounds for modified Bessel functions of the first kind. Secondly, we set the analysis in the 'least restrictive' Sobolev space W s that embeds continuously in the space of continuous functions, and this corresponds to considering s = d/2 + η for arbitrarily small positive η. Thirdly, we extend the analysis to higher dimensions by relying on the separability of the kernel w ǫ in its d variables and the considerations from the one-dimensional case. Technical tools related to fractional Sobolev spaces and Faà di Bruno's formula are deferred to Appendix B, while relevant elementary algebraic tools are summarised in Appendix A. Additionally, the proof of Theorem 1.1 in Section 4 is finalised with a localisation procedure argument. Crucially, the same techniques adopted to deal with the superlinear interaction α U (analogous to those developed in [8,Section 4]) also allow to deal with the locally bounded noise in d > 1.

Remark 1.2.
The justification of the scaling assumptions of Theorem 1.1 is found in Lemma 3.2. There, each index s is associated to a relevant value θ c (s) := 2s + d, and the trace of W per,ǫ with respect to the W s -norm is bounded by ǫ −θc(s) . In combination with Lemma 3.4, this implies that the W s -norm of the stochastic noise of (4) vanishes as N → ∞ for any θ > θ c (s). As our well-posedness theory relies on the embedding W s ⊂ C 0 ×[C 0 ] d , we require the equality s = d/2 + η = (θ c (s) − d)/2 to hold, giving θ > 2d + 2η. As η may be chosen arbitrarily small, we obtain the threshold θ 0 = 2d.
Furthermore, for each s, the value θ c (s) is optimal, in the sense that θ c (s) is also the minimum value for which E[ ρ ǫ (·, t) 2 H s ] (where, we recall, ρ ǫ denotes the true particle density) is uniformly bounded in N and ǫ, at least in the case of independent particles given by U ≡ 0. Namely, it is easy to proceed as in [7] and argue that, under reasonable assumptions on the law of the particle dynamics, for some constants C 1 , C 2 . Crucially, we obtain scaling agreement for fluctuations on microscopic and mesoscopic scale; here microscopic means particle-level dynamics, see (1) above, while mesoscopic means the Dean-Kawasaki dynamics (2). As a result, the value θ 0 is also optimal, as lim s→d/2 θ c (s) = θ 0 .
For θ ≤ 2d, the embedding W s ⊂ C 0 × [C 0 ] d is lost, and one needs to consider a less restrictive notion of solution to (4), possibly related to a measure-valued martingale formulation. This is a matter for future works. Remark 1.3. The RIDK model (2) may be regarded as the regularised inertial analogue of the original over-damped Dean-Kawasaki model [10,17] where ρ is the particle density and ξ is a space-time white noise.

Basic notation
We work with periodic functions on the d-dimensional torus T d = [0, 2π) d for d ∈ N. We never specify the dependence of any function space on d, as this is always clear from the context. Bold face characters always denote vectors. For m ∈ N 0 and p ∈ [1, ∞], we denote by W m,p the standard Sobolev space of periodic functions on T d with derivatives up to order m belonging to L p . For 0 < s / ∈ N and p ∈ [1, ∞), we define the fractional spaces W s,p via the norm where ⌊s⌋ := max{n ∈ N 0 : n ≤ s}. We also set ⌈y⌉ := min{n ∈ N : y ≤ n}. We consider the fractional Hilbert spaces H s and H s := [H s ] d identified by the Fourier-type inner products and we define the norm on The norms · H s and · W s,2 are equivalent; see [3,Proposition 1.3]. We define the space V s+1 := {v ∈ H s : ∇ · v ∈ H s } ⊃ H s+1 , and recall the integration-by-parts formula In dimension d = 1, we trivially have V s+1 ≡ H s+1 . We denote by L(W s ) (respectively, L 0 2 (W s )) the set of continuous linear functionals mapping W s into itself (respectively, the set of Hilbert-Schmidt operators from P where # denotes the number of elements in a set. Furthermore, for every partition π ∈ Π α , we set As an immediate consequence of the definitions, we have j∈J(π) j β j (π) = α.
We use C as a generic constant whose value may change from line to line (with dependence on relevant parameters highlighted whenever necessary, for example C(s)). In addition, we denote the embedding constant of H s ⊂ C 0 by K H s →C 0 . Finally, we use the subscript notation to link specific constants with the lemmas where they are defined; for example, K B.1 is the constant introduced in Lemma B.1.

Derivation of RIDK
We now derive the RIDK model (4) by following the methodology outlined in [8]. Consider the second-order Langevin system (1), as well as the quantities (3). Simple Itô computations imply that ρ ǫ and j ǫ satisfy the system where the ℓth component of terms on the right are defined by The terms j 2,ǫ , I U , andŻ N are not closed in the leading quantities (ρ ǫ , j ǫ ), and approximations are used to close the system of equations. We now sketch how the approximations in [7,8] extend to the multi-dimensional case.
The term j 2,ǫ is dealt with under a local-equilibrium assumption [12,Corollary 3.2]. In this situation, the probability density function of (q i (t), p i (t)) is approximately separable in the position variable q i (t) and momentum variable p i (t) due to the structure of the Gibbs invariant measure. In addition, the momentum variable is distributed according to a Gaussian of mean zero and diagonal covariance matrix (σ 2 /2γ) I d . Furthermore, under the additional assumption σ 2 ≪ 2γ, the approximation is legitimate. All these considerations imply that E j 2,ǫ ≈ σ 2 /(2γ) E[∇ρ ǫ ] and this motivates the replacement j 2,ǫ ≈ (σ 2 /2γ) ∇ρ ǫ .
Finally, one may substitute the noiseŻ N ( where P ǫ is the convolution operator P ǫ : , which relies on the two noises being approximately equivalent in distribution, is justified as in the one-dimensional case [8], due to the separability of variables in the kernel w ǫ . In addition, the stochastic independence of the d components of each member of the family Taking all into account, we obtain our multi-dimensional RIDK system (2).
The noiseẎ N (x, t) can be explicitly expanded using the spectral properties of the operator P ǫ , which, due to the separability of the kernel w ǫ , are readily available from the one-dimensional case [7,Section 4.2]. More specifically, with {e j } j∈Z being the trigonometric system is, for some choice of normalisation constant C(d), an H s -orthonormal basis of eigenfunctions for P √ 2ǫ for any ǫ > 0. Furthermore, the eigenvalue of P √ 2ǫ corresponding to the eigenfunction f j,s is where the eigenvalues from the one-dimensional case are given by with I j denoting the modified Bessel function of first kind and order j [1, Eq. (9.6.26)]. As a result, the stochastic process with iid families {β ℓ,j } d ℓ=1 of independent Brownian motions, is a W s -valued Q-Wiener process representation of the R × R d -valued stochastic noise (0,Ẏ N (x, t)). It follows that, upon swappingŻ N (x, t) withẎ N (x, t), we can write (10) in the abstract stochastic PDE form where X ǫ = (ρ ǫ , ǫ ), A is the wave-type differential operator given by the interaction potential is α U (X ǫ,δ ) := −ρ ǫ (∇U * ρ ǫ ), and the stochastic integrand B N is given by .
in (16), and we finally obtain the following equation in which is exactly (4).

Main technical results for the proof of Theorem 1.1
We develop the three main technical tools upon which we base the main proof in Section 4. We investigate the cases d = 1 and d > 1 separately.

Semigroup analysis of operator
The proof is identical to the one provided in [7,Lemma 4.2], simply with all relevant spaces H α being replaced by H α−1+s . We assume σ 2 /(2γ) := 1 for simplicity, even though the proof is analogous for the general case σ 2 /(2γ) > 0.
We verify the assumptions of the Hille-Yosida Theorem, as stated in [21, Theorem 3.1].
Step 1: A is a closed operator, and D(A) is dense in W s . This is easily checked.
Step 2: The resolvent set of A contains the positive half line. Let λ > 0. We show that the operator A λ := A − λI is injective. Assume that A λ (ρ, j) = (0, 0). We take the H s -inner product of the first component of A λ (ρ, j) with ρ and of the second component of A λ (ρ, j) with j, and we obtain where we have used (9). Since λ, γ > 0, we deduce that (ρ, j) = (0, 0). We now show that Taking the H s -inner product of (18) (respectively, of (19)) with ρ (respectively, with j), we get We use the Cauchy-Schwartz and Young inequalities to deduce which promptly gives A Fourier series expansion argument provides existence of a unique solution ρ ∈ H s+2 for (21).
Proof of Lemma 3.1 in dimension d > 1. Steps 2 and 3 are readily adapted, as the Fourier analysis is unchanged. We only need need to justify the validity of Step 1. As for the density of D(A) in W s , this is implied by the density of H s+1 into H s and H s+1 into H s , as well as by the inclusion H s+1 ⊂ V s+1 . As for the closedness of the operator A, this follows from the consistency of the first component of A and of the definition of V s+1 . More specifically, consider a sequence D(A) ∋ (ρ n , j n ) → (ρ, j) in W s , such that A (ρ n , j n ) → (x, y) in W s . This immediately implies that ∇ · j n converges in H s−1 to both −x and ∇ · j, forcing them to agree. In particular, j ∈ V s+1 . Similarly, ∇ρ n converges in H s−1 to both ∇ρ and −γ j − y, forcing them to agree. In particular, ρ ∈ H s+1 . Therefore, (ρ, j) ∈ D(A) and A (ρ, j) = (x, y).

(i) The following bound holds
Step 1. There exists K > 0 such that, for any j and any ǫ, it holds λ ǫ,j < K. This follows from (14) together with the monotonicity of {λ ǫ,j } j (see [20,Introduction]).
Step 2. Let x ≥ 1. Picking k = 2 and m = 0 in [20, Theorem 2, bound (a)], we have We show that the inequality holds when x for suitable x(α, β) > 0 and C > 0 to be discussed below. Simple algebraic rearrangements imply that (24) is equivalent to which is in turn satisfied (taking (25) into account), at least under the sufficient condition Take C > 0 in (25) if α + β > 1, otherwise take C > 1 if α + β = 1. Then, for x large enough (i.e., for x large enough in (26)), inequality (28) is satisfied, and therefore so is inequality (24).

Proof of Lemma 3.2 in dimension d > 1. The result promptly follows from the bound
and the validity of (22) for d = 1. The optimality of the scaling under (α, β) = (1/2, 1/2) has already been dealt with in the one-dimensional case. Firstly, the bound on {λ j,ǫ } j is now uniform in ǫ and j (i.e., we no longer bound λ j,ǫ using ǫ −1 ). Secondly, the exponential decay of the eigenvalues 'kicks in' earlier, namely around Cǫ −2β rather than around ǫ −2 . This leads to a sharper estimate concerning the sum on the region A 1 .
These improvements bring the threshold θ 0 down from 7 to 3 for the suboptimal choice s = 1 (see [7,Lemma 4.3]). In addition, the switch to fractional Sobolev spaces, i.e., the choice s = 1/2 + η instead of s = 1 as in [7], where η can be chosen arbitrarily small, grants a further decrease of θ 0 from 3 to 2.

Proof of Lemma 3.4 for d = 1. We limit ourselves to proving Statements (ii) and (iii).
Statement (15) and (17) we have that We use the fact that {f j,s } j are orthonormal in H s , the equivalence of the norms · H s and · W s,2 (see Subsection 1.2), the boundedness of h ′ δ , and Lemma B.1 to write We bound the numerator of (33). If either u 1 (x) = u 2 (x) or u 1 (y) = u 2 (y), then simply Otherwise, we use the embedding H s ⊂ C 0 and write We now focus on T 2 . We define the auxiliary function We write In the above, we perform a first-order Taylor expansion (with respect to the first variable of r only for T 3 , and with respect to the second variable of r only for T 4 ). This is possible because r has partial derivatives defined everywhere (as a consequence of h δ being C 2 (R)). In addition, the partial derivatives of r are uniformly bounded by sup z∈R |h ′′ δ (z)| ≤ C(δ). This implies We plug (34), (35) and (36) into (33) and take into account the assumption ( We can go back to (32) and deduce the local Lipschitz property where we have used Lemma B.1, the sublinearity of h δ at infinity, the boundedness of h ′ δ , and Lemma 3.2. This completes the proof.
Proof of Lemma 3.4 for d > 1. In this proof, we need to analyse quantities associated with derivatives of the distinctive nonlinearity h δ (u), u ∈ H d/2+η . For this purpose, we make heavy use of the contents of Appendix B (integrability properties of the Faà di Bruno representation of derivatives of h δ (u)) and Appendix A (factorisation of differences of two distinct instances of the same derivative).
We again focus on points (ii) and (iii) only.
Upon adding and subtracting terms of the type h (|πα|) δ (u 2 )P πα,α u 1 , for α ∈ {0, . . . , ⌊d/2⌋}, the term A 1 is bounded (up to a constant) by where we have used a Taylor expansion for (and boundedness of) derivatives of h δ and the Sobolev embedding H s ⊂ C 0 in (39), and Lemma B.3 in (40). We may now apply Lemma A.1-(i) to factorise P πα,α u 1 − P πα,α u 2 into a sum of terms, each of which can then be dealt with using Lemma B.3. We obtain More generally, each application of Lemma A.1 below is, at least conceptually, identical to the one illustrated above. Namely, it is used to factorise a difference of objects into a sum of terms which in turn can be estimated using either Lemma B.3 or Lemma B.4.
Following simple algebraic rewritings, the term A 2 can be bounded (up to a constant) by Term T 1 is dealt with using (35) The proof is similar to that of Statement (ii). Take (u, v) ∈ W s , such that (u, v) W s ≤ k. We only need to bound for any choice π α ∈ Π α , α ∈ {0, . . . , ⌊d/2⌋}, and π ∈ Π ⌊d/2⌋ . The term A 3 is easily settled using the boundedness of derivatives of h δ and Lemma B.3. Furthermore, A 4 is bounded (up to a constant) by Term T 5 is bounded using a Taylor expansion of h Putting the bounds obtained for (45) into (44) and using Lemmas B.1 and 3.2-(ii), we deduce The assumption (u, v) W s ≤ k gives the desired local boundedness property. The proof is complete.

Proof of Theorem 1.1
This is an adaptation of [8,Theorem 4.4], and we heavily rely on the tools developed in Section 3. The functional α U is locally Lipschitz and locally bounded in the W s -norm. This is a consequence of the following simple bound for u ∈ H s and ℓ ∈ {1, . . . , d} These properties of α U , together with Lemmas 3.1 and 3.4, allow us to use [23,Theorem 4.5] and deduce the existence and uniqueness of a local W s -valued mild solution to (4) in the sense of [9,Chapter 7]. Specifically, there is a stopping time τ > 0 and a unique W s -valued predictable process X ǫ,δ = (ρ ǫ,δ , ǫ,δ ) defined on [0, τ ] such that P( τ 0 X ǫ,δ (z) 2 W s dz < ∞) = 1, and satisfying, for each t > 0 where {S(t)} t≥0 is the C 0 -semigroup generated by A. Using [23, Theorem 4.5 and Remark 4.6], the continuous embedding H s ⊂ C 0 , and the assumption min x∈T dρ 0 (x) > δ, we deduce that there exists T = T (ρ 0 ) and a unique deterministic W s -valued mild solution Z δ = (ρ Z , j Z ) to the noise-free equivalent of (4) up to T , such that min We compare X ǫ,δ and Z δ . As X ǫ,δ is a local mild solution, it is well-defined up to the first exit time from the W s -ball of radius k. In particular, X ǫ,δ and Z δ are well-defined up to the stopping time We consider the difference Let q > 2. For some for some C 2 = C 2 (σ, δ, T, q, k, η, d, K B.1 , K B.3 , K B.4 ). Crucially, the last inequality is not affected by the superlinear nature of the noise for d > 1, as X ǫ,δ lives on a bounded set of W s up to τ δ,k . Applying the Gronwall Lemma to (50)-(51) gives The choice of θ for (5) given in the assumption and a Chebyshev-type argument imply that for any β ∈ (0, 1). It is now a standard routine (see [8,Theorem 4.4]) to pick β small enough, N big enough, and deduce the existence of a set F ν such that P(F ν ) > 1−ν, on which τ δ,k ≡ T , on whichρ ǫ ≥ δ, and on which (4) is satisfied by X ǫ,δ the sense of mild solutions. Going back to (47), this implies and this concludes the proof.

A Factorisation of products
We recall the following simple factorisation for differences of products.
where we have used the shorthand notations b <k := k−1 j=1 b j and a >k := N j=k+1 a j (with the usual convention of the product over an empty set being unitary). We have Proof. Point (i) is easily proven by induction. As for Point (ii), we use Point (i) twice and obtain and the proof is complete.

B Technical lemmas on fractional Sobolev spaces
We recall a useful lemma about the multiplication of functions in fractional Sobolev spaces, which is a direct consequence of the Sobolev embedding [ The following lemma is an adaptation of the classical multivariate Faà Di Bruno's formula [6] in the context of weak rather than classical derivatives. We derive it under some restrictive assumptions, which are however satisfied by the nonlinearity h δ in our regularised Dean-Kawasaki noise (4).
to be chosen later. We use a multi-factor Hölder inequality on the |π| + 1 terms making up (60). The exponents we use are q j := d/(2(j − η)) for the each of the β j terms associated with the product over the set {b ∈ B(π) : |b| = j}, and q := d/(d − 2⌊d/2⌋ + 2η j∈J β j ) for the remaining term. We obtain We now impose conditions on η and γ so that C j , D j , and E are suitably bounded. The integrals C j may be dealt with using a standard change of variables in spherical coordinates, and they are bounded if and only if −γα j β −1 The terms D 1/q j j,b are bounded, as in the case of Lemma B.3, by using the Sobolev embedding H d/2+η−j ⊂ L 2q j [3, Corollary 1.2]. We now turn to E. We rewrite the exponent of |x − y| according to the notation of the space W r,2q , for some r to be determined. More precisely, the rewriting is solved in r, giving r = (d − γ)/2 + η(1 − j∈J β j ). The restriction r ∈ (0, 1) gives the condition The term E may be bounded using the Sobolev embedding W d/2+η,2 ⊂ W r,2q , and this embedding is true under the condition [2, Theorem 5.1] which is equivalent to If we pick γ := 2⌊d/2⌋ − 3η j∈J β j and η small enough, then (64) and (63) are satisfied. Furthermore, summing the right-hand side of (62) over j, we obtain j∈J 2β j (j − η) 2⌊d/2⌋ − 3η j∈J β j = 2⌊d/2⌋ − 2η j∈J β j 2⌊d/2⌋ − 3η j∈J β j > 1.
The above inequality implies that the α j 's can be chosen so that (61) and (62) are satisfied. As a result of the bounds for C j , D j,b , E, the inequality (58) follows and Point (i) is settled.
Point (ii). The case j∈J β j = 1 uniquely corresponds to having = ⌊d/2⌋ and β = 1. Therefore, the only term surviving in the product of integrands in the left-hand side of (59) is the last term, and the result is trivial.
We consider all the other cases, where necessarily j∈J β j > 1. We rewrite (59) as where the second curly brackets is understood to be equal to 1 should β = 1, for some appropriate γ > 0 and {α j } j∈J ⋆ such that to be chosen later, where J ⋆ = J if β > 1, and J ⋆ = J \ otherwise. We use a multi-factor Hölder inequality on the |π| terms making up (66). The exponents we use are q j := d/(2(j −η)) for the each of the β j terms associated with the product over the set {b ∈ B(π) : |b| = j} for j ∈ J \, then q := d/(2( − η)) for the each of the β − 1 terms associated with the product over the set {b ∈ B(π) : |b| =, b =b}, and finally q := d/(d − 2⌊d/2⌋ + 2η j∈J β j + 2( − η)) for the remaining term. We obtain Bounding the above involves similar discussions as per Point (i). More specifically, the boundedness of the spherical integrals (the C j 's above) is granted under the conditions with the last condition only imposed if ∈ J ⋆ . The bound for the terms D 1/q j j,b is settled exactly as in Point (i). As for E, we solve the equation in the variable r, thus getting r := (d − γ)/2 + η(1 − j∈J β j ) − ( − η). The constraint r ∈ (0, 1) results in the requirement We control E using the embedding H d/2+η ⊂ W +r,2q , which is valid under the constraint which is equivalent to If we take γ := 2⌊d/2⌋ + 3η(1 − j∈J β j ) − 2 and η small enough, then (69) and (70) are satisfied. Furthermore, summing all the right-hand sides in (68) gives where the last inequality is valid because j∈J β j > 1. Therefore the α j 's can be chosen so that (67) and (68) are satisfied. As a result of the bounds for C j , D j,b , E, the inequality (59) follows and Point (ii) is settled.