We consider the following interacting particle system in $\mathbb{R}^d$:

$$
\begin{aligned}
\dot{x}_i(t) &= -\frac{1}{N}\sum _{j=1}^{N}\nabla K(x_i(t)-x_j(t)) -\frac{1}{N} \sum _{j=1} ^{N}K\big(x_i(t)-x_j(t)\bigr)\nabla V (x_j(t)), \\\
x_i(0) &= x_i ^0 \in \mathbb R^d, \qquad i=1,\cdots,N.
\end{aligned} \tag{1}
$$

We refer to each of the $N$ functions $x_i(\cdot) \in \mathbb{R}^d$ as a particle. The function $K : \mathbb{R}^d \mapsto \mathbb{R}$ is a smooth, symmetric, and positive definite kernel. The function $V : \mathbb{R}^d \to \mathbb{R}$ is a smooth potential such that $e^{-V(x)}$ is integrable. More specific assumptions about $K$ and $V$ are given below.

We are interested in the macroscopic behavior of the particle system $(1)$ as $N \to \infty$ in the framework of mean field limit. Formally this mean field limit is described by the following non-local, nonlinear partial differential equation (PDE):
$$
\begin{aligned}
&\partial_t \rho = \nabla \cdot \left(\rho\left(K \ast (\nabla \rho+\nabla V\rho)\right)\right), \\\
&\rho(0,\cdot) = \rho_0(\cdot).
\end{aligned} \tag{2}
$$

In this article, we will prove the well-posedness of (1) and (2).

The Stein Variational Gradient Descent (SVGD) algorithm was first introduced by Liu and Wang [LW16], whose idea is to transport a set of $N$ particles $\{x_i\}_{i=1}^N$ in $\mathbb R^d$ so that their empirical measure

$$
\mu^N:=\frac{1}{N}\sum_{i=1}^N\delta_{x_i}
$$

approximates the target probability measure

$$
\rho_\infty (x)\ dx=Z^{-1} e^{-V(x)}\ dx
$$

with an unknown normalization factor $Z$.

In this article, we will introduce the gradient flows in Wasserstein space.

In this article, we will introduce the general theory in metric spaces.

In this article, we will introduce the gradient flows in the metric setting.

In this article, we will introduce some basic results of gradient flows in the Euclidean space.

Throughout this article $(X,d)$ will be a given complete metric space.