Mean-field game

Definition

Let $\mathbb{T}^d = \R^d / \Z^d$ be the torus, let $T > 0$, and let $\nu > 0$ be a constant. Let $\mathcal{P}$ be the set of probability densities on $\mathbb{T}^d$. Let $H \colon \mathbb{T}^d \times \R^d \to \R$ be a smooth function, convex in its second variable, the Hamiltonian. Let $F, G \colon \mathbb{T}^d \times \mathcal{P} \to \R$ be given coupling functions, and let $m_0 \in \mathcal{P}$ be the initial distribution. The mean-field game system is the system $$\begin{aligned} -\partial_t u - \nu \Delta u + H(x, \nabla u) &= F(x, m(t)), \\ \partial_t m - \nu \Delta m - \nabla \cdot \big( m \, \partial_p H(x, \nabla u) \big) &= 0, \\ m(0, x) = m_0(x), \qquad u(T, x) &= G(x, m(T)), \end{aligned}$$ for unknown functions $u, m \colon [0, T] \times \mathbb{T}^d \to \R$. Here $\Delta$ is the Laplacian, $\nabla$ the gradient and $\nabla \cdot$ the divergence in $x$, and $\partial_p H$ is the gradient of $H$ in its second variable. A solution $(u, m)$ is a mean-field game equilibrium. In it, $u$ is the value function of a typical player and $m(t)$ is the distribution of the players at time $t$.