Concentration bounds for stochastic systems with singular kernels

Let $N,d\in\mathbb{N}$ and $K:\mathbb{T}^{d}\rightarrow\mathbb{R}^{d}$ be a vector field defined on the $d-$dimensional flat torus $\mathbb{T}^{d}$. We consider the system of $N$ particles described by the dynamics

where the $W^{i}$ are $N$ independent standard Wiener processes defined on a standard filtered probability space $\big{(}\Omega,\mathbb{F}=(\mathcal{F}_{t})_{0\leq t\leq T},\mathbb{P})$. We work on a finite time horizon $[0,T]$, and we assume that the initial positions of the particles are i.i.d., i.e.

Thus the data used to define our particle system consists of the time horizon $T>0$, the maps $F,K:\mathbb{T}^{d}\to\mathbb{R}^{d}$, and the initial distribution $\rho_{0}\in\mathcal{P}(\mathbb{T}^{d})$. Under these conditions, the law $\rho^{N}_{t}=\mathcal{L}(\bm{X}_{t})$¹¹1Here and throughout the note we use the notation $\rho^{N}_{t}$ to indicate a curve $[0,T]\;\reflectbox{$\in$}\;t\mapsto\rho^{N}_{t}\in\mathcal{P}(\mathbb{T}^{dN})$, and we abuse notation by writing $\rho^{N}(t,\cdot)$ for the density of $\rho^{N}_{t}$ when it exists. of the particles is formally described by the Liouiville equation

where $\bm{x}=(x^{1},...,x^{N})\in\mathbb{T}^{dN}$. When $F$ and $K$ are smooth (or at least Lipschitz), it is well known that the asymptotic behaviour of the particle system (1.1) is described by the non-local Fokker-Planck equation

More precisely, it is known that in an appropriate sense we have

The first statement above confirms that $\overline{\rho}_{t}$ is the mean field limit of the empirical measures $m_{\bm{X}_{t}}^{N}$, while the second statement is referred to as propagation of chaos. Quantitative versions of these statements, as well as finer results like central limit theorems, large deviations principles, and concentration bounds for the empirical measures $m_{\bm{X}_{t}}^{N}$ have also been obtained for regular $K$. We do not make any attempt to summarize this literature, but refer to survey articles like [JW17, CD22] for an introduction. Extending such results to more singular kernels $K$, which are often met in applications, is a very active area of research. We mention in particular the recent flurry of activity around singular kernels which derive from Riesz potentials (see e.g. [Ser20, BJW20, RS23, CdCRS23] and the references therein) and some recent efforts aimed at singular attractive kernels (see e.g. [BJW23, CdCRS23]). More relevant to the present note are the slightly less recent contributions of [JW16] and [JW18], where quantitative propagation of chaos was established by estimating the relative entropy between a solution of the Liouiville equation (1.2) and the tensor product $\overline{\rho}^{N}=\overline{\rho}^{\otimes N}$ of the solution of (1.3). More precisely, [JW18] established the following quantitative version of “entropic propagation of chaos”:

for some constant $C$ independent of $N$, where $H$ denotes the rescaled relative entropy

We note that (1.4) is established under a general initial condition for (1.2), but if $\rho^{N}_{0}=\overline{\rho}_{0}^{\otimes N}$ then the first term on the right-hand side of (1.4) vanishes, leading to $H_{N}(\rho^{N}_{T}|\overline{\rho}^{N}_{T})=O(1/N)$, which by a classical subadditivity property of relative entropy gives $H(\rho^{N,k}_{T}|\overline{\rho}^{\otimes k}_{T})=O(k/N)$, where $\rho^{N,k}$ denotes the $k$-particle marginal of $\rho^{N}$, and in particular $H(\rho^{N,k}_{T}|\overline{\rho}^{\otimes k}_{T})\to 0$ as $N\to\infty$ for each fixed $k$, which is what we mean by entropic propagation of chaos. We note also that establishing (1.4) is not the only way to obtain quantitative entropic propagation of chaos - we refer to [Lac23] for a more detailed discussion and for an approach which leads to optimal bounds on $H(\rho_{T}^{N,k}|\overline{\rho}_{T}^{\otimes k})$.

Our goal in this note is to point out a connection between entropic propagation of chaos and concentration inequalities for the empirical measure of the particle system (1.1). In particular, for $1\leq p<\infty$, we are interested in bounds of the form

where ${\bf d}_{p}$ denotes the $p$-Wasserstein distance. That is, we want to show that the empirical measure for the particle system admits concentration bounds of the same type available for i.i.d. samples, as established in [FG15]. When written in terms of the Liouville equations (which is often necessary for technical reasons when $K$ is singular), (1.5) becomes

Such concentration inequalities were obtained for regular interactions by several authors. We highlight in particular [BGV05], where concentration inequalities for the system (1.1) were inferred from concentration inequalities for i.i.d. random variables via the so-called “synchronous coupling” method originally due to McKean [McK69] and popularized by Sznitman [Szn91], and [DLR18], where a more general concentration of measure result was obtained by exploiting certain (uniform in $N$) functional inequalities for the law of $\bm{X}=(X^{1},...,X^{N})$. When $K$ is singular, the approaches in [BGV05] and [DLR18] break down, and the only estimate similar to (1.5) we are aware of is Proposition 5.3 of [HC23], where the structure of the repulsive Riesz interactions is leveraged to get bounds somewhat similar to (1.5).

To explain our main idea, we must introduce the “controlled Liouville equation”:

where $\bm{\alpha}=(\alpha^{1},...,\alpha^{N}):[0,T]\times(\mathbb{T}^{d})^{N}\to(\mathbb{R}^{d})^{N}$ is a measurable map which we view as a control. For the so-called $W^{-1,\infty}$ kernels of [JW18], one can (as is explained in more detail in the proof of Proposition 1.6) easily generalize the original entropy estimate (1.4) of Jabin and Wang to get an estimate of the form²²2The factor of $1/4$ in (1.8) is purely aesthetic, and is included to make (1.8) more consistent with Proposition 3.1 below.

for any solution $f^{N}$ to (1.7). We emphasize that here $f^{N}_{0}$ may be different from $\rho_{0}^{\otimes N}$. We think of (1.8) as a “controlled” or “perturbed” version of the entropic propagation of chaos estimate (1.4) of Jabin and Wang. Our main observation is that an estimate of the form (1.8) in fact implies a concentration bound of the form (1.5). For technical reasons, we first make this observation precise through the following a-priori estimate, which requires $K$ and $\overline{\rho}^{0}$ to be bounded.

As mentioned above, we can verify that (1.8) holds under roughly the same conditions as in [JW18]. Thus we make use of the following assumption (and refer to the notation section below for the definition of $W^{-1,\infty}$):

Throughout the paper $T$ is a positive real number and $\mathbb{T}^{d}$ is the $d-$dimensional flat torus. $f*g$ is the convolution between the functions $f,g$. If a function $u:[0,T]\times\mathbb{T}^{d}\rightarrow\mathbb{R}$ is sufficiently regular, then we denote by $\partial_{t}u,\;D_{x_{i}}u$ the partial derivative with respect to $t$ and the partial derivative with respect to $x_{i}$ (the $i$-th coordinate) for $i=1,2,...,d$, respectively. For $p\in[1,+\infty]$, we use the symbol $L^{p}_{t,x}$ (resp. $L^{p}_{x})$ for the space of functions $f$ such that $\int|f(t,x)|^{p}dtdx<+\infty$ (resp. $\int|f(x)|^{p}dx<+\infty$). For $k\geq 1$, $p\in[1,+\infty]$ and $\beta\in(0,1)$, we denote by $W^{k,p}$, $C^{k,\beta}$ the Sobolev ($k$ weak derivatives in $L^{p}$) and Hölder spaces ($k$ derivatives which are $\beta$-Hölder continuous) on $\mathbb{T}^{d}$, respectively. Also, we write $C^{k,\beta}_{t,x}$ for the standard parabolic Hölder spaces on $[0,T]\times\mathbb{T}^{d}$, e.g. $C^{2,\beta}_{t,x}$ will be the space of functions $f(t,x)$ such that $\partial_{t}f$, $Df$, $D^{2}f$ exist, and are $\beta/2$-Hölder in $t$ and $\beta$-Hölder in $x$. Similarly, we will use $W^{k,p}_{t,x}$ for the standard parabolic Hölder spaces, e.g. $W^{2,p}_{t,x}$ will indicate the space of functions $f(t,x)$ such that $\partial_{t}f$, $Df$, $D^{2}f$ are in $L^{p}$.

We now recall the precise definition of the space $W^{-1,\infty}$.

As in [JW18], since $K$ (and possibly $\bm{\alpha}$) are not smooth, in order to get controlled entropy bounds between $\rho^{N}$ and $\overline{\rho}^{N}=\overline{\rho}^{\otimes N}$ we must work with entropy solutions.

For technical reasons, we at times need to work specifically with entropy solutions of (1.2) which arise via a suitable mollification procedure. In particular, we fix throughout the paper a standard mollifier $(\rho_{\delta})_{\delta>0}$ on $\mathbb{T}^{d}$, and we define $K_{\delta}=K*\rho_{\delta}$, $F_{\delta}=F*\rho_{\delta}$. Then we make the following definition.

In this subsection we state and prove some results that will be useful in the paper. The first is a refinement of a compactness result borrowed from [Dau23, Proposition 1.2] for solutions $(m,\alpha)$ to the Fokker-Planck equation:

where $m\in\mathcal{C}([0,T],\mathcal{P}_{2}(\mathbb{T}^{d}))$ and $\alpha\in L^{2}_{dt\otimes m(t)}\left([0,T]\times\mathbb{T}^{d},\mathbb{R}^{d}\right)$.

We are also going to need the following two elementary lemmas, the first of which can be inferred [DE11, Proposition 2.4.2] (notice that the boundness assumption on $f$ there is not necessary) and the second of which is standard.

To prove Theorem 1.1, we are going to need the following variational lower bound for $(0,K)$-entropy solutions.