Global-in-time mean-field convergence for singular Riesz-type diffusive flows

We consider the first-order mean-field dynamics of stochastic interacting particle systems of the form

Above, $x_{i}^{0}\in{\mathbb{R}}^{d}$ are the pairwise distinct initial positions, ${\mathbb{M}}$ is a $d\times d$ matrix such that

and $W_{1},\ldots,W_{N}$ are independent standard Brownian motions in ${\mathbb{R}}^{d}$, so that the noise in (1.1) is of so-called additive type. There are several choices for ${\mathbb{M}}$. For instance, choosing ${\mathbb{M}}=-\mathbb{I}$ yields gradient-flow/dissipative dynamics, while choosing ${\mathbb{M}}$ to be antisymmetric yields Hamiltonian/conservative dynamics. Mixed flows are also permitted. The potential ${\mathsf{g}}$ is assumed to be repulsive, which, as we shall later show in Section 4, ensures that there is a unique, global strong solution to the system (1.1). In particular, with probability one, the particles never collide. The model case for ${\mathsf{g}}$ is either a logarithmic or Riesz potential indexed by a parameter $0\leq s<d-2$, according to

The above restriction on $s$ means that we are considering potentials that are sub-Coulombic: their singularity is below that of the Coulomb potential, which corresponds to $s=d-2$. As explained precisely in the next subsection, we can consider a general class of potentials ${\mathsf{g}}$ which have sub-Coulombic-type behavior.

Systems of the form (1.1) have numerous applications in the physical and life sciences as well as economics. Examples include vortices in viscous fluids [Ons49, Cho73, Osa87c, MP12], models of the collective motion of microscopic organisms [OS97, TBL06, Per07, GQn15, LY16, FJ17], aggregation phenomena [BGM10, CGM08, Mal03], and opinion dynamics [HK02, Kra00, MT11, XWX11]. For more discussion on applications, we refer the reader to the survey of Jabin and Wang [JW17] and references therein.

Establishing the mean-field limit consists of showing the convergence in a suitable topology as $N\to\infty$ of the empirical measure

associated to a solution $\underline{x}_{N}^{t}\coloneqq(x_{1}^{t},\dots,x_{N}^{t})$ of the system (1.1). We remark that for fixed $t$, the empirical measure is a random Borel probability measure on ${\mathbb{R}}^{d}$. If the points $x_{i}^{0}$, which themselves depend on $N$, are such that $\mu_{N}^{0}$ converges to some regular measure $\mu^{0}$, then a formal calculation using Itô’s lemma leads to the expectation that for $t>0$, $\mu_{N}^{t}$ converges to the solution of the Cauchy problem with initial datum $\mu^{0}$ of the limiting evolution equation

as the number of particles $N\rightarrow\infty$. While the underlying $N$-body dynamics are stochastic, we emphasize that the equation (1.5) is completely deterministic, and the noise has been averaged out to become diffusion. Proving the convergence of the empirical measure is closely related to proving propagation of molecular chaos (see [Gol16, HM14, Jab14] and references therein): if $f_{N}^{0}(x_{1},\dots,x_{N})$ is the initial law of the distribution of the $N$ particles in ${\mathbb{R}}^{d}$ and if $f_{N}^{0}$ converges to some factorized law $(\mu^{0})^{\otimes N}$, then the $k$-point marginals $f_{N,k}^{t}$ converge for all time to $(\mu^{t})^{\otimes k}$.

The mean-field problem for the system (1.1) with $\sigma>0$ and interactions which are regular (e.g. globally Lipschitz) has been understood for many years now [McK67, Szn91, M9́6, Mal03] (see also [BGM10, BCnC11, MMW15, Lac21, DT21] for more recent developments still in the regular case). The classical approach consists in comparing the trajectories of the original system (1.1) to those of a cooked-up symmetric particle system coupled to (1.1). Subsequent work has focused on treating the more challenging singular interactions — initially by compactness-type arguments that yield qualitative convergence [Osa86, Osa87b, Osa87c, FHM14, GQn15, LY16, FJ17, LLY19] and later by more quantitative methods that yield an explicit rate for propagation of chaos [Hol16, JW18, BJW19a, BJW20]. To our knowledge, the best results in the literature can quantitatively prove propagation of chaos for singular interactions up to and including the Coulomb case $s=d-2$ for conservative dynamics [JW18] and arbitrary $0\leq s<d$ in for dissipative dynamics [BJW19a, BJW20]. Unlike the previous aforementioned works which utilize the noise in an essential way, the methods of [JW18, BJW19a] allow for taking vanishing diffusion: $\sigma=\sigma_{N}\geq 0$, where $\sigma_{N}\rightarrow 0$ as $N\rightarrow\infty$. We also mention that the recent preprint [WZZ21] has gone beyond the mean-field limit and shown the convergence of the fluctuations of the empirical measure to (1.1) to a generalized Ornstein-Uhlenbeck process for singular potentials including the two-dimensional Coulomb case. These state-of-the-art works are limited to the periodic setting.

When there is no noise in the system (1.1) (i.e. $\sigma=0$), much more is known mathematically about the mean-field limit thanks to recent advances that are capable of treating the full potential case $s<d$. Approaches vary, but they all typically involve finding a good metric to measure the distance between the empirical measure and its expected limit and then proving a Gronwall relation for the evolution of this metric. The $\infty$-Wasserstein metric allowed to treat the sub-Coulombic case $s<d-2$ [Hau09, CCH14]. A Wassertein-gradient-flow approach [CFP12, BO19] can also treat the one-dimensional case using the convexity of the Riesz interaction in (and only in) dimension one. The modulated-energy approach of [Due16, Ser20], inspired by the prior work [Ser17], managed to treat the more difficult Coulomb and super-Coulombic case $d-2\leq s<d$ for the model potential (1.3). In very recent work by the authors together with Nguyen [NRS21], this modulated-energy approach has been redeveloped to allow to treat the full range $s<d$ and under fairly general assumptions for the potential ${\mathsf{g}}$. In these works, the modulated energy is a Coulomb/Riesz-based metric that can be understood as a renormalization of the negative-order homogeneous Sobolev norm corresponding to the energy space of the equation (1.5). More precisely, it is defined to be

where we remove the infinite self-interaction of each particle by excising the diagonal $\triangle$.

Contemporaneous to the development of the modulated-energy approach, Jabin and Wang [JW16, JW18] introduced a relative-entropy method capable of treating the mean-field limit of (1.1) when the interaction is moderately singular and which works well with or without noise. The relative-entropy and modulated-energy approaches were recently combined into a modulated free energy method [BJW19b, BJW19a, BJW20] that allows for treating the mean-field limit of (1.1) in the dissipative case, but not the conservative case, set on the torus and under fairly general assumptions on the interaction, impressively allowing even for attractive potentials (e.g. Patlak-Keller-Segel type).

In this article, we show for the first time that the modulated-energy approach of [Due16, Ser20, NRS21] can be extended to treat the mean-field limit of (1.1) in the sub-Coulombic case $0\leq s<d-2$.¹¹1Our ability to use the modulated-energy approach in the sub-Coulombic case crucially relies on our recent work [NRS21] with Nguyen, since the previous works [Due16, Ser20] could only treat this way the Coulomb/super-Coulombic case. No incorporation of the entropy, as in the modulated-free-energy approach of [BJW19b, BJW19a, BJW20] is needed. Moreover, the modulated-energy approach is well-suited for exploiting the dissipation of the limiting equation (1.5) to obtain rates of convergence in $N$ which are uniform over the entire interval $[0,\infty)$. In other words, mean-field convergence holds globally in time. At the time of completion of this manuscript, this is, to the best of our knowledge, the first instance of such a result for singular potentials. Obtaining a uniform-in-time convergence is important in both theory and practice — for instance, when using a particle system to approximate the limiting equation or its equilibrium states and for quantifying stochastic gradient methods, such as those used in machine learning for other interaction kernels (for instance, see [CB18, MMN18, RVE18]).

Lastly, we mention that previous uses of the modulated energy in the stochastic setting [Ros20b, NRS21] were limited to the case of multiplicative noise, which behaves very differently in the limit as $N\rightarrow\infty$. Most notably, the limiting evolution equation is stochastic.

Let us sketch our main proof in the model case (1.3). For simplicity of exposition, let us also restrict ourselves to the simpler range $d-4<s<d-2$. We note that the modulated energy (1.6) is a real-valued continuous stochastic process. Formally by Itô’s lemma (see Section 6 for the rigorous computation), it satisfies the stochastic differential inequality (cf. [Ser20, Lemma 2.1])

where we have set $u^{t}\coloneqq{\mathbb{M}}\nabla{\mathsf{g}}\ast\mu^{t}$. The third term in the right-hand side (formally) has zero expectation and may be ignored for the purposes of this discussion. The first term is the contribution of the drift and also appears in the deterministic case, but the second term is new and due to the nonzero quadratic variation of Brownian motion. Observe that

Since $0\leq s<d-2$ by assumption, we see that $\Delta{\mathsf{g}}$ is superharmonic and equals a constant multiple of $-\tilde{{\mathsf{g}}}$, where $\tilde{{\mathsf{g}}}$ is the kernel of the Riesz potential operator $(-\Delta)^{\frac{s+2-d}{2}}$. We would like to conclude that the second term in the right-hand side of (1.7) is nonpositive by Plancherel’s theorem and therefore may be discarded, but the excision of the diagonal $\triangle$ obstructs this reasoning. Fortunately, prior work of the second author [Ser20, Proposition 3.3] gives the lower bound

for all choices of parameters $\eta_{i}>0$. Here, $C$ is a constant that just depends on $s,d$. We emphasize that (1.9) is a functional inequality which holds independently of any underlying dynamics. The choice of $\eta_{i}$ that balances the decay in $N$ between the two terms in the right-hand side of the inequality (1.9) is the typical interparticle distance $N^{-1/d}$. Since the $L^{\infty}$ norm of $\mu^{t}$ is a source of decay and we wish to distribute it between terms, we instead choose

so that the right-hand side of (1.9) is bounded from below by

providing a bound from above for the corresponding term in (1.7). Note that since $\mu^{t}$ is time-dependent, our choice for $\eta_{i}$ above depends on time, a trick previously used by the first author [Ros20a, Ros20b] to study the mean-field limit for point vortices with possible multiplicative noise when $\mu^{t}$ belongs to a function space which is invariant or critical under the scaling of the equation.

It remains to consider the first term in the right-hand side of (1.7), which, as previously mentioned, also appears in the deterministic case. This expression has the structure of a commutator which has been renormalized through the exclusion of diagonal in order to accommodate the singularity of the Dirac masses. As shown in [Ros20a, NRS21], one can make this commutator intuition rigorous (see Propositions 5.7 and 5.15 below) and, revisiting the estimates there together with some elementary potential analysis, we are able to optimize the dependence in $\|\mu\|_{L^{\infty}}$ of the estimate and show the pathwise and pointwise-in-time bound

where again $C,\beta>0$ are constants depending only $s,d$.

Now taking expectations of both sides of (1.7), integrating with respect to time, and using Remark 5.2 below to control $|F_{N}(\underline{x}_{N}^{t},\mu^{t})|$ by $F_{N}(\underline{x}_{N}^{t},\mu^{t})$, we find that

The structure of the right-hand side of (1.13) allows us to leverage the decay rate of the solution to (1.5). In Proposition 3.8, we show the decay rate

This is done by revisiting work of Carlen and Loss [CL95] on the optimal decay of nonlinear viscously damped conservation laws, which was essentially restricted to divergence-free vector fields, and adapting it to treat the case of (1.5).

Once this is done, an application of the Gronwall-Bellman lemma to (1.13) yields a uniform-in-time bound if $s>0$, while if $s=0$, long-range effects only allow us to obtain an $O(t^{\frac{C}{\sigma}})$ growth estimate.

We now state the precise assumptions for the class of interaction potentials we consider. This class corresponds to the sub-Coulombic sub-class of the larger class of potentials considered by the authors in collaboration with Nguyen in [NRS21]. In the statement below and throughout this article, the notation $\mathbf{1}_{(\cdot)}$ denotes the indicator function for the condition $(\cdot)$.

For $d\geq 3$ and $0\leq s<d-2$, we assume the following:

1. (i)

   $${\mathsf{g}}(x)={\mathsf{g}}(-x)$$

2. (ii)

   $$\lim_{x\rightarrow 0}{\mathsf{g}}(x)=\infty$$

3. (iii)

   $$\text{$\exists r_{0}>0$ such that}\quad\Delta{\mathsf{g}}\leq 0\quad{\text{in $B(0,r_{0})$}}$$

4. (iv)

   $$\forall k\geq 0,\qquad|\nabla^{\otimes k}{\mathsf{g}}(x)|\leq C\left\lparen\frac{1}{|x|^{s+k}}+|\log|x||\mathbf{1}_{s=k=0}\right\rparen\quad\forall{x\in{\mathbb{R}}^{d}\setminus\{0\}}$$

5. (v)

   $$|x||\nabla{\mathsf{g}}(x)|+|x|^{2}|\nabla^{\otimes 2}{\mathsf{g}}(x)|\leq C{\mathsf{g}}(x)\quad\forall x\in B(0,r_{0}){\setminus\{0\}}$$

6. (vi)

   $$\frac{C_{1}}{|\xi|^{d-s}}\leq\hat{\mathsf{g}}(\xi)\leq\frac{C_{2}}{|\xi|^{d-s}}\quad{\forall\xi\in{\mathbb{R}}^{d}\setminus\{0\}}$$

   where $\hat{\cdot}$ denotes the Fourier transform.

7. (vii)

   $$\begin{cases}\text{$\exists c_{s}<1$ such that}\quad{\mathsf{g}}(x)<c_{s}{\mathsf{g}}(y)\quad\forall x,y\in B(0,r_{0})\ \text{with $|y|\geq 2|x|$},&{s>0}\\ \text{$\exists c_{0}>0$ such that}\quad{\mathsf{g}}(x)-{\mathsf{g}}(y)\geq c_{0}\quad\forall x,y\in B(0,r_{0})\ \text{with $|y|\geq 2|x|$},&{s=0}\end{cases}$$

8. (viii)

   If $d-4<s<d-2$, then we also assume that there is an $m\in{\mathbb{N}}$ and ${\mathsf{G}}:{\mathbb{R}}^{d+m}\rightarrow{\mathbb{R}}$ such that

   $$-\Delta{\mathsf{g}}(x)={\mathsf{G}}(x,0)\quad\forall(x,0)\in{\mathbb{R}}^{d+m}$$

   (1.14)

   $${\mathsf{G}}(X)={\mathsf{G}}(-X)$$

   (1.15)

   $$\text{$\exists r_{0}>0$ such that}\quad\Delta{\mathsf{G}}(X)\leq 0\quad\text{in $B(0,r_{0})\subset{\mathbb{R}}^{d+m}$}$$

   (1.16)

   $$\forall k\geq 0,\quad|\nabla^{\otimes k}{\mathsf{G}}(X)|\leq\frac{C}{|X|^{s+2+k}}\quad\forall X\in B(0,r_{0})$$

   (1.17)

   $$\hat{{\mathsf{G}}}(\Xi)\geq 0\quad\forall\Xi\in{\mathbb{R}}^{d+m}\setminus\{0\}.$$

9. (ix)

   In the cases $s=d-2k\geq 0$, for some positive integer $k$, we also assume that the $({\mathbb{R}}^{d})^{\otimes(2k+2)}$-valued kernel

   $$\mathsf{k}(x-y)\coloneqq(x-y)\otimes\nabla^{\otimes(2k+1)}{\mathsf{g}}(x-y)$$

   is associated to a Calderón-Zygmund operator.³³3Sufficient and necessary conditions for this Calderón-Zygmund property are explained in [Gra14a, Section 5.4]. The reader may check that this condition is satisfied if ${\mathsf{g}}$ is the Riesz potential $|x|^{2k-d}$.

10. (x)

    In all cases,

    $${\mathbb{M}}:\nabla^{\otimes 2}{\mathsf{g}}(x)\geq 0\quad\forall x\in{\mathbb{R}}^{d}\setminus\{0\},$$

    where $:$ denotes the Frobenius inner product.

We shall say that any potential ${\mathsf{g}}$ satisfying assumptions i – x is an admissible potential. We refer to [NRS21, Subsection 1.3] for a discussion of the types of potentials permitted under these assumptions. Compared to that work, only assumptions viii and x are new. The former is to ensure that our modulated-energy method can be applied to $\tilde{\mathsf{g}}$ and thus to the diffusion term in (1.7), while the latter is to ensure that the solutions of (1.5) satisfy the temporal decay bounds of the heat equation. Note that x is automatically satisfied if ${\mathbb{M}}$ is antisymmetric. Additionally, if ${\mathbb{M}}=-\mathbb{I}$ so that we consider gradient-flow dynamics, then x amounts to requiring that ${\mathsf{g}}$ is globally superharmonic (i.e. $r_{0}=\infty$ in iii). In general, though, we do not require global superharmonicity except where explicitly stated.

We assume that we are given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ on which a countable collection of independent standard $d$-dimensional Brownian motions $(W_{n})_{n=1}^{\infty}$ are defined. Moreover, $(\mathcal{F}_{t})_{t\geq 0}$ is the complete filtration generated by the Brownian motions. All stochastic processes considered in this article are defined on this probability space.

Let $\underline{x}_{N}^{0}\in({\mathbb{R}}^{d})^{N}$ be an $N$-tuple of distinct points in ${\mathbb{R}}^{d}$. As shown in Proposition 4.5 (more generally Section 4), there exists a unique, global strong solution $\underline{x}_{N}$ to the Cauchy problem for (1.1). Moreover, with probability one, the particles $x_{i}^{t}$ and $x_{j}^{t}$ never collide on the interval $[0,\infty)$. Let $\mu^{0}\in\mathcal{P}({\mathbb{R}}^{d})\cap L^{\infty}$ be a probability measure with $L^{\infty}$ density with respect to Lebesgue measure. We abuse notation here and throughout the article by using the same symbol to denote both the measure and its density.

As shown in Proposition 3.8 (more generally Section 3), there is a unique, global solution to the Cauchy problem for (1.5) in the class $C([0,\infty);\mathcal{P}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d}))$. In the case of logarithmic interactions (i.e. $s=0$), we also assume that $\mu^{0}$ satisfies the logarithmic growth condition $\int_{{\mathbb{R}}^{d}}\log(1+|x|)d\mu^{0}(x)$, which is propagated locally uniformly by the evolution (see Remark 3.6).

Since $\underline{x}_{N}$ is stochastic, $\{F_{N}(\underline{x}_{N}^{t},\mu^{t})\}_{t\geq 0}$ is a real-valued stochastic process. It is straightforward to check from the non-collision of the particles and Hölder’s inequality that $F_{N}(\underline{x}_{N}^{t},\mu^{t})$ is almost surely finite on the interval $[0,\infty)$ and a continuous process. Our main theorem is a quantitative estimate for the expected magnitude of $F_{N}(\underline{x}_{N}^{t},\mu^{t})$.

The first result of this article is the following functional inequality for the expected magnitude of the modulated energy. In the case $0<s<d-2$, we get a linear growth estimate, while in the case $s=0$, we have superlinear growth of size $O(t^{\frac{\sigma+C}{\sigma}})$ as $t\rightarrow\infty$.

Assume now that $r_{0}=\infty$ in assumption iii. In other words, the potential ${\mathsf{g}}$ is globally superharmonic, as opposed to just in a neighborhood of the origin. The second result of this article is a functional inequality for the expected magnitude of the modulated energy which yields a global bound in the case $0<s<d-2$. In the case $s=0$, we have an almost global bound, in the sense that the growth is $O(t^{{\frac{1}{\sigma}}^{+}})$ as $t\rightarrow\infty$, which can be arbitrarily small by choosing the diffusion strength $\sigma$ arbitrarily large.

We close this subsection with some remarks on the statements of Theorems 1.1 and 1.2, further extensions, and interesting questions for future work.

At the time of completion of this manuscript, our Theorem 1.2 is, to the best of our knowledge, the first time that a quantitative rate of convergence for the mean-field limit of (1.1) has been shown to hold uniformly on the interval $[0,\infty)$. Additionally, in the case $s=0$, the rate of convergence as $N\rightarrow\infty$ is (up to a $\log N$ factor) optimal. Existing results in the literature for singular potentials (e.g. [JW18, BJW19a]) have at least an exponential growth in time due to a reliance on a Gronwall-type argument without exploiting the dissipation of the limiting equation. Those works (e.g. [Mal03, CGM08, DEGZ20, DT21]) that do have a uniform convergence result are restricted to regular potentials and require confinement.

Additionally, our theorem is at the level of the empirical measure for the original SDE dynamics for (1.1), as opposed to their associated Liouville/forward Kolmogorov equations for the joint law of the process $\underline{x}_{N}^{t}=(x_{1}^{t},\ldots,x_{N}^{t})$,

Namely, no randomization of the initial data is needed, although as discussed in Remark 1.5, our result implies convergence of this form as well. This stands in contrast to previous work [JW18, BJW19a] whose starting point is the Liouville equation (1.22).

The costs of the strong bounds we obtain with Theorems 1.1 and 1.2 are two-fold. First, we need somewhat stronger assumptions on the potential ${\mathsf{g}}$ than in [JW18, BJW19a]–especially the latter work. The reader may find a detailed comparison in [NRS21, Subsection 1.3]. Second, and more importantly, our results are limited to the sub-Coulombic range $0\leq s<d-2$ and dimensions $d\geq 3$. The Coulomb case is barely just out of the reach. Indeed, the diligent reader will note that if ${\mathsf{g}}$ is the Coulomb potential, then $\Delta{\mathsf{g}}=-\delta$, which is obviously no longer a function. Thus, the argument we described in the previous subsection to bound the second term in the right-hand side of (1.7) no longer applies. The situation is even worse when $s>d-2$ as $\Delta{\mathsf{g}}>0$, meaning what was previously a dissipation term should now cause the modulated energy to grow in time. We mention again that it is an open problem to prove the mean-field limit of (1.1) in the conservative case and when $d-2<s<d$.

During the final proofreading of the manuscript of this article, Guillin, Le Bris, and Monmarché posted to the arXiv their preprint [GBM21] showing uniform-in-time propagation of chaos in $L^{1}$ norm in the periodic setting ${\mathbb{T}}^{d}$ for singular interactions in the range $0\leq s\leq d-2$ and $d\geq 2$ with $N^{-\frac{1}{2}}$ rate. In particular, they can treat the viscous vortex model corresponding to the $d=2$ Coulomb case. Their method is very different from ours, as it is based on the relative-entropy method of Jabin-Wang [JW18]. Moreover, they assume random initial data and work at the level of the Liouville equation, and their interaction kernel is assumed to be integrable, have zero distributional divergence, and equal to the divergence of an $L^{\infty}$ matrix field. Their result is limited to the periodic setting, due to a need for compactness of domain, and to conservative flows. We also note that they impose much stronger regularity assumptions on their solutions to (1.5) than we do.

In Section 2, we review some estimates for Riesz potential operators and the interaction potential ${\mathsf{g}}$ that are used frequently in the paper.

Section 3 is devoted to the study of the limiting equation (1.5), showing that it is globally well-posed and moreover the $L^{p}$ norms of solutions satisfy the optimal decay bounds (see Propositions 3.1 and 3.8).

In Section 4, we show that the $N$-particle system (1.1) has well-defined dynamics (see Proposition 4.5) in the sense that there exists a unique strong solution and with probability one, the particles never collide. We also introduce in this section a truncation and stopping time procedure that will be used again later in Section 6.

In Section 5, we review properties of the modulated energy and renormalized commutator estimates from the perspective of our recent joint work [NRS21]. We also prove refinements (see Section 5.2) of the results from that work in the case where ${\mathsf{g}}$ is globally superharmonic.

Finally, in Sections 6 and 7, we prove our main results, Theorems 1.1 and 1.2. Section 6 gives the rigorous computation of the Itô equation (cf. (1.7)) satisfied by the process $F_{N}(\underline{x}_{N}^{t},\mu^{t})$. The main result is Proposition 6.3, which establishes an integral inequality for ${\mathbb{E}}(|F_{N}(\underline{x}_{N}^{t},\mu^{t})|)$. Using this inequality together with the decay bound of Proposition 3.8 and the results of Section 5, we close our Gronwall argument in Section 7, completing the proofs of Theorems 1.1 and 1.2.

We close the introduction with the basic notation used throughout the article without further comment.

Given nonnegative quantities $A$ and $B$, we write $A\lesssim B$ if there exists a constant $C>0$, independent of $A$ and $B$, such that $A\leq CB$. If $A\lesssim B$ and $B\lesssim A$, we write $A\sim B$. To emphasize the dependence of the constant $C$ on some parameter $p$, we sometimes write $A\lesssim_{p}B$ or $A\sim_{p}B$. Also $(\cdot)_{+}$ denotes the positive part of a number.

We denote the natural numbers excluding zero by ${\mathbb{N}}$ and including zero by ${\mathbb{N}}_{0}$. Similarly, we denote the positive real numbers by ${\mathbb{R}}_{+}$. Given $N\in{\mathbb{N}}$ and points $x_{1,N},\ldots,x_{N,N}$ in some set $X$, we will write $\underline{x}_{N}$ to denote the $N$-tuple $(x_{1,N},\ldots,x_{N,N})$. Given $x\in{\mathbb{R}}^{n}$ and $r>0$, we denote the ball and sphere centered at $x$ of radius $r$ by $B(x,r)$ and ${\partial}B(x,r)$, respectively. Given a function $f$, we denote the support of $f$ by $\operatorname{supp}f$. We use the notation $\nabla^{\otimes k}f$ to denote the tensor with components ${\partial}_{i_{1}\cdots i_{k}}^{k}f$.

We denote the space of Borel probability measures on ${\mathbb{R}}^{n}$ by $\mathcal{P}({\mathbb{R}}^{n})$. When $\mu$ is in fact absolutely continuous with respect to Lebesgue measure, we shall abuse notation by writing $\mu$ for both the measure and its density function. We denote the Banach space of complex-valued continuous, bounded functions on ${\mathbb{R}}^{n}$ by $C({\mathbb{R}}^{n})$ equipped with the uniform norm $\|\cdot\|_{\infty}$. More generally, we denote the Banach space of $k$-times continuously differentiable functions with bounded derivatives up to order $k$ by $C^{k}({\mathbb{R}}^{n})$ equipped with the natural norm, and we define $C^{\infty}\coloneqq\bigcap_{k=1}^{\infty}C^{k}$. We denote the subspace of smooth functions with compact support by $C_{c}^{\infty}({\mathbb{R}}^{n})$. We denote the Schwartz space of functions by ${\mathcal{S}}({\mathbb{R}}^{n})$ and the space of tempered distributions by ${\mathcal{S}}^{\prime}({\mathbb{R}}^{n})$.

The second author thanks Eric Vanden-Eijnden for helpful comments.

We start by proving local well-posedness for the equation (1.5) for initial data $\mu^{0}$ in the Banach space $X\coloneqq L^{1}({\mathbb{R}}^{d})\cap L^{\infty}({\mathbb{R}}^{d})$. That is, we show existence, uniqueness, and continuous dependence on the initial data. The proof proceeds by a contraction mapping argument for the mild formulation of (1.5). In the next subsection, we will upgrade this local well-posedness to global well-posedness through estimates for the temporal decay of the $L^{p}$ norms of the solution.

Let us introduce the mild formulation of equations (1.5), on which we base our notion of solution. With $e^{t\Delta}$ denoting the heat flow, we write