Dean-Kawasaki Dynamics: Ill-posedness vs. Triviality

We prove that the Dean-Kawasaki SPDE admits a solution only in integer parameter regimes, in which case the solution is given in terms of non-interacting particles.


Introduction
In this paper we show a dichotomy of non-existence vs. triviality of solutions to a certain class of nonlinear SPDE which arise e.g. in macroscopic fluctuation theory in physics. As a prototype one might consider the model ∂ t ρ = T ∆ρ + ∇ · T ρẆ , (1.1) which is a particular instance of the more general class of formal Ginzburg-Landau stochastic phase field models (c.f. [17,22,32]) of the form ∂ t φ + ∇ · −L(φ)∇ δH δφ (φ) + T L(φ)Ẇ = 0, (1.2) where H is a Hamiltonian, L is an Onsager coefficient andẆ is vector valued space-time white noise. The particular equation (1.1) was proposed independently in [8] and [20] as mesoscopic description for interacting particles and is referred to as Dean-Kawasaki equation today. Since then, together with several variants, it has been an active research topic in various branches of non-equilibrium statistical mechanics over the last years (c.f. [9,12,16,18,21,26,34]). Mathematically, interest in equations like (1.1) comes from the fact that it appears to describe an 'intrinsic' random perturbation of the gradient flow for the entropy functional on Wasserstein space by a noise which is locally uniformly distributed in terms of dissipated energy, e.g. by a noise that is aligned with Otto's formal Riemannian structure [30] for optimal transportation. To see this, consider the rescaled Hamiltonian H(µ, φ) := lim →0 e − 1 µ,φ G e 1 µ,φ , with G being the formal generator associated to the evolution (1.1) with = T . Following [10], one expects the short time asymptotics of ρ to be governed by the large deviation rate function for regular curves ν in the Wasserstein-space. But since H(µ, φ) = 1 2 µ, |∇φ| 2 , we see that A is precisely the energy functional which determines the Wasserstein-metric by means of the Benamou-Brenier-formula (c.f. [3] and Appendix D in [14]).
In spite of significant continuing interest in Dean-Kawasaki type models both in physics and mathematics, rigorous results on existence and uniqueness exist so far only for equations, which, apart from variants in the drift term, all share some regularisation of the white noise, see e.g. [7,13,27]. On the other hand, in [35] Sturm and the third author succeeded in constructing a process having the Wasserstein distance as its intrinsic metric and which can formally be seen as a solution to the SPDE where Ξ is some nonlinear operator. Particle approximations of these dynamics as well as analytic and geometric properties of the corresponding entropic measure were investigated afterwards in [1,33] and [5,11], respectively. Based on Arratia flows, a new candidate for a process with Wasserstein-short-time asymptotics, but with different drift component as in [35] was recently studied by the latter two authors in [25,24] and subsequently, in [28].
The main contribution of this note asserts that some correction term Ξ is in fact necessary for the existence of (nontrivial) solutions to these DK-type models. More precisely, in Theorem 2.2 below we find that the (uncorrected) generalized Dean-Kawasaki equation corresponding to a Ginzburg-Landau model with H = α 2 µ log µ, T = 1 and L = identity operator, admits solutions only for a discrete spectrum of parameters α and atomic initial measures. Moreover, for these particular choices solutions are trivial, in being just 'measure-valued lifts' of the martingale-problem for α 2 ∆. Finally, given the apparent similarity of (DK) α µ0 to the SPDE description of the Dawson-Watanabe ('Super Brownian Motion') process admitting unique in law solutions for every β > 0, our result is interesting also from an independent SPDE point of view.
As for notation, given a Polish space E, we will write M 1 (E) for the set of all probability measures on E and for any function f on E, we write as usual µ, f = E f (x)µ(dx), whenever the integral is well-defined. Also, on M 1 (E) we shall always consider the weak topology. By C b (E) we denote the space of real-valued, bounded continuous functions on E.
Let us briefly motivate, what we will refer to as a solution to the Dean-Kawasaki equation. Typically, one would call a time-continuous process t → µ t , which takes values in absolutely continuous measures on R d , a solution to (DK) α µ0 , provided that for all with W i being mutually independent space-time white noises (for instance in the sense of Walsh [36]).
Of course, for such a process µ we knew that Rather then the weak formulation in (2.1), it is this martingale characterisation that we will study in the following, however in slightly more general setup.
Let E be a Polish state-space and (E, π, Γ) be the Markov-triple associated to some symmetric Markov Diffusion operator L in the classical sense, as e.g. in [2]. In this setting, we know L has the diffusion property, i.e.
for every ψ ∈ C 2 and f ∈ D(L). Here, Γ is the carré du champs operator, defined on some algebra Additionally, we impose henceforth the following two hypotheses on boundedness and regularity for the Markov semigroup P t belonging to L: 1. There exists an exhaustion {A n } n∈N ∈ B(E), A n ↑ E and real numbers (c n ) n∈N such that for every n ∈ N 2. P t satisfies a gradient bound, i.e. for some ρ ∈ R, we have

4)
is an F t -adapted martingale, 2. whose quadratic variation is given by With this notation, our main results reads as follows.
where the X i are n independent diffusion processes, each with generator 1 2 L and starting point x i .
Certainly, the Dean-Kawasaki dynamic fits in as the special case of taking L = ∆ on E = R d . Theorem 2.2 will be proven in several steps. We start by the almost trivial observation that the empirical measures (2.6) provide solutions to (MP) n µ0 . Consider first the case n = 1. Plugging µ t = δ Xt into the martingale problem, immediately yields that Lφ(X s )ds is a martingale, plainly since X is the solution of the martingale problem for 1 2 L. Also necessarily, the quadratic variation of M satisfies For the general case, denote by M i the martingale associated to X i . Then Γφ(X ns )ds = Our next aim is to show that these solutions are unique in law. In fact, the statement that we prove is much stronger, namely that any solution, provided its existence, must be unique in law. The proof adopts the argument which is used in order proof weak uniqueness for super-Brownian motion, by Laplace-duality to some reaction-diffusion equation (c.f. [15,29,31]).
Before we present the duality statement in our context, we provide some preliminary considerations on viscous Hamilton-Jacobi equations. That is, we regard for some initial where L and Γ are as before generator and carré du champs operator of some symmetric Markov diffusion. Of course, the unique solution is just the classical Cole-Hopf-solution given by where P t is now the heat semi-group associated to α 2 L. Indeed, plugging in the definition of V t f into the PDE, we see Moreover, one can easily check that this solution satisfies maximum/minimum-principles, We can now formulate our duality result, which will be crucial not only for the uniqueness, but also for the discussion of non-existence later on. (2.8)

Proof.
Assume for now f ∈ A and let v(t, x) = V t f (x) be the Cole-Hopf solution to (vHJ) f . For 0 ≤ s ≤ t ≤ T , by Itō's formula Hence for t ∈ [0, T ] and s ≤ t the map s → e − µs,v(t−s) is a local martingale. Since by it is in fact a martingale. Therefore, upon choosing s = t we find E µ0 e − µt,f = e − µ0,Vtf .
By density of A in C b (E) the previous equation also holds for f ∈ C b (E), which yields the claim.
Observe in particular, that the duality in Theorem 2.3 determines the Laplacetransform of µ t and by the same argument as in Theorem 11 of [19], also the finite dimensional distributions of µ uniquely. Hence, we obtain uniqueness (in law) for solutions to the Dean-Kawasaki equation.
In order to develop our non-existence statement, we insert a short intermezzo on generating functions.
Whereas of course, for the probability-generating function of a some discrete random variable X with values in N 0 = {0, 1, 2, . . . }, one knows that p k = P (X = k), we are interested in the opposite direction and establish Lemma 2.4. Let X be a non negative random variable, such that for each n ∈ N 0 g(s) = Es X = n k=0 s k p k + o(s n ), as s → 0+, (2.10) for some sequence {p k , k ∈ N 0 }. Then X ∈ N 0 a.s., p k ≥ 0 and P{X = k} = p k for each k ∈ N.
Proof. We will prove the statement by induction. Due to (2.10), On the other hand, since s X → 1 {X=0} a.s. as s → 0+, we infer by dominated convergence Es X → P{X = 0} as s → 0 + .
This finishes the proof of the lemma.
We can now return to our main question and prove, that the trivial solutions we found for α ∈ N and atomic µ 0 must in fact be the only possible ones.
Proof of Theorem 2.2. Take α > 0, µ 0 ∈ M 1 (E) and a solution µ to (MP) α µ0 . To ease notation, let us also introduce C := {A ∈ B(E), A ⊂ A n for some n ∈ N} (2.11) and for A ∈ C and fixed t ∈ [0, T ], we abbreviate h(x) = P t 1 A (x). Note that by assumption (2.2), we can find δ > 0 with 0 ≤ h(x) ≤ 1 − δ for all x ∈ E. Now, for A ∈ C and fixed t > 0, consider the generating function g of the real-valued random variable X = αµ t (A). The Laplace-duality of Theorem 2.3 yields (2.13) By the boundedness of h, the function g is well-defined on (−δ, ∞). Moreover, it is infinitely differentiable. Thus, for each n ∈ N 0 g(s) = n k=0 p k s k + o(s n ) on (−δ, δ), by Taylor's theorem. Consequently, by Lemma 2.4 we know that αµ t (A) ∈ N 0 a.s. So, we have proved that for each A ∈ C and t > 0 µ t (A) ∈ 0, 1 α , . . . , α α a.s.
Here, we also used the fact that µ t is a probability measure a.s. Next, making A ↑ E, we obtain that 1 = µ t (E) ≤ α α ≤ 1 a.s.
This implies that α ∈ N. In order to make a conclusion about µ 0 , we take again any A ∈ C with µ 0 -zero boundary and use the continuity of the process µ. So, we obtain µ 0 (A) ∈ 0, 1 α , . . . , α α .
Hence, there exist x k , k ∈ {1, . . . , α} such that We close the considerations with the following final remark concerning generalisations of our result. In fact, the dichotomic nature which we proved in this paper remains unaffected for Dean-Kawasaki dynamics driven by a free energy functional H = α 2 µ log µ + F , with some smooth potential F . In that case however, particles which govern the evolution exhibit mean-field interaction. This result is the content of our forthcoming paper [23]. ECP 24 (2019), paper 8.