Weak solutions to granular medium equations via stochastic particle approximations

Introduction

The granular medium equation with porous medium diffusion has been studied for many years. However, a detailed proof of existence of weak solutions to the associated McKean–Vlasov stochastic differential equation does not appear to be readily available in the literature. The aim of this work is to fill this gap. Specifically, we establish the existence of weak solutions to a stochastic differential equation (SDE) whose time-marginal densities solve the following partial differential equation (PDE):

(PDE)

Here, $m > 1$, $V$ and $W$ are the confinement and interaction potentials, respectively, and $*$ denotes convolution in the spatial variable. For each $t \in[0,T)$, the function $u_t = u(t,\cdot)$ is a probability density on $\mathbb{R}^d$.

More precisely, we construct a process $(Y_t)_{t\in[0,T)}$ that solves, in the weak sense, the McKean–Vlasov SDE

(SDE)

where $(B_t)_{t\in[0,T)}$ is a standard $d$-dimensional Brownian motion.

In the one-dimensional case, equation PDE was introduced to model granular media in [3, 4]. Consider many infinitesimal particles colliding inelastically, then $u(t,x)$ represents a density of the velocity of a representative particle (among infinitely many). The potential $V$ represents the friction and $W$ the inelastic collisions between particles with different velocities. In a different context, equations of this type were also considered in the recent survey [20] where it is used to describe such phenomena as aggregation, diffusion, and self-propulsion mechanisms among particles.

Several works study the long-time behaviour of solutions to this equation, including the existence of stationary measures and rates of convergence to equilibrium [11, 12, 26], or develop numerical schemes for its approximation [5, 27]. The question of existence, however, is not always treated in full detail in these works. In particular, [11] provides a detailed existence proof on bounded domains and outlines the argument for $\mathbb{R}^d$. Under stronger assumptions on $V$ and $W$, existence in the convex regime can also be established using gradient flow theory [2, 18], whereas our approach applies to a broader class of potentials than those covered by these results.

The problem of existence for linear diffusion ($m=1$) was rigorously solved in [13] using a probabilistic technique called propagation of chaos that has been known since [23, 28, 32] and surveyed in [14, 15]. The core of this approach is to consider a system of $N$ particles $X_t^{1,N},\ldots,X_t^{N,N}$ that are governed by $N$ linear SDEs and consider their empirical measure. If it converges to a deterministic limit as $N \to\infty$, we call it propagation of chaos. The name comes from the observation that, under natural assumptions, this property is equivalent to the asymptotic independence of the particles. More precisely, if we fix $k \in\mathbb{N}$, then $\operatorname{Law}(X_t^{1,N},\ldots,X_t^{k,N}) \xrightarrow[N\to\infty]{} f_t \otimes\cdots\otimes f_t$ in the weak sense, where $f_t$ is the limit of the empirical measures.

In [13], the authors considered the following system of SDEs for $i=1,\ldots,N$ and $t>0$:

where $B_t^1,\ldots,B_t^N$ are independent $d$-dimensional standard Brownian motions. They showed that this problem is well-posed, the sequence of distributions of their empirical measures, i.e,

is tight and the limit is $\delta_Q\in\mathcal{P}\left(\mathcal{P}\left(C([0,T],\mathbb{R}^{d})\right)\right)$, where $Q$ is the law of a solution to SDE for $m=1$. For $t>0$, the time marginals of $Q$ have densities $u_t$ solving PDE. Moreover $X_{\cdot}^{1,N}\to Y_{\cdot}$ in law, where $Y_{\cdot}$ solves SDE for linear diffusion case, i.e. with $m=1$.

Our goal is to adapt this approach to obtain an analogous result for $m>1$. A key difficulty in constructing a particle approximation of this SDE is that the diffusion coefficient depends on pointwise values of the probability density. Since empirical measures are atomic, they cannot be used directly to evaluate this coefficient. We address this difficulty by regularizing the empirical measure and using the resulting density approximation to define the diffusion coefficients of the particle system.

When $V=W=0$, equation (PDE) reduces to the classical porous medium equation

Stochastic particle approximations of this equation have been studied in several settings. In one dimension, Jourdain [21] established a propagation of chaos result for a formulation in which the solution represents a cumulative distribution function. This result was extended to broader classes of functions in [6]. Philipowski [30] constructed an interacting diffusion approximation for the quadratic porous medium equation, $m = 2$, in arbitrary spatial dimension. Figalli and Philipowski [19] subsequently extended this approach to general exponents $m > 1$, using an additional approximation of the nonlinearity when $1 < m < 2$. Their construction represents nonlinear diffusion through a regularized interaction drift and includes auxiliary Brownian noise. In our construction, nonlinear diffusion is instead approximated through a diffusion coefficient depending on a regularized empirical density.

Alongside these stochastic approaches, deterministic particle approximations of the porous medium equation have also been studied. In [1, 10], convergence of such approximations to solutions of (PME) was established using a gradient flow approach. Related results in [8, 9] concern the convergence of solutions to non-local diffusion equations, whose diffusion operators involve an interaction kernel, to solutions of (PDE). These results cover the cases $m \le1$ and $m = 2$, under the assumption that $V$ and $W$ are semiconvex and grow at most quadratically. In particular, the authors of [10] used the JKO scheme (see, e.g., [2]) to construct solutions to the following non-local equation:

where $\varphi_\varepsilon(x) = \frac{1}{\varepsilon^d}\varphi\left(\frac{x}{\varepsilon}\right)$ is a standard mollifier, and showed the convergence of $u^\varepsilon$ with $\varepsilon\to0$ to the solution of PME.

Most of the approximations discussed above can be viewed as variants of the following particle system, which is deterministic for $\delta= 0$ and stochastic for $\delta> 0$:

In most of the particle approximations discussed above, nonlinear diffusion is represented through a regularized interaction drift, while any additional Brownian noise serves as an auxiliary regularization that vanishes in the limit. Exceptions include [21], which is restricted to one spatial dimension, and [22], which assumes a Lipschitz diffusion coefficient.

The key feature of our construction is that nonlinear diffusion is generated entirely by a Brownian term whose coefficient depends on a regularized empirical density. Our approach applies in arbitrary spatial dimension and does not require the limiting diffusion coefficient to be Lipschitz. Crucially, the density-dependent stochastic term remains present in the limiting SDE: letting $\delta$ tend to zero removes only the auxiliary viscosity. More precisely, for $i=1,\ldots,N$, we consider the particle system

(PSDE)

For fixed $\epsilon$ and $\delta$, we establish convergence of particle system, as $N\to\infty$. The limiting one-particle distributions are the martingale solution to the following non-local SDE: