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  "body": "\n## Definition\n\nLet $\\mathbb{T}^d = \\R^d / \\Z^d$ be the torus, let $T \u003e 0$, and let $\\nu \u003e 0$ be a constant.\nLet $\\mathcal{P}$ be the set of probability densities on $\\mathbb{T}^d$.\nLet $H \\colon \\mathbb{T}^d \\times \\R^d \\to \\R$ be a smooth function, convex in its second variable, the Hamiltonian.\nLet $F, G \\colon \\mathbb{T}^d \\times \\mathcal{P} \\to \\R$ be given coupling functions, and let $m_0 \\in \\mathcal{P}$ be the initial distribution.\nThe *mean-field game system* is the system\n$$\\begin{aligned} -\\partial_t u - \\nu \\Delta u + H(x, \\nabla u) \u0026= F(x, m(t)), \\\\ \\partial_t m - \\nu \\Delta m - \\nabla \\cdot \\big( m \\, \\partial_p H(x, \\nabla u) \\big) \u0026= 0, \\\\ m(0, x) = m_0(x), \\qquad u(T, x) \u0026= G(x, m(T)), \\end{aligned}$$\nfor unknown functions $u, m \\colon [0, T] \\times \\mathbb{T}^d \\to \\R$.\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]], $\\nabla$ the [[gradient|0ABK9V9Y64]] and $\\nabla \\cdot$ the [[divergence|0MWV6ZTK6S]] in $x$, and $\\partial_p H$ is the gradient of $H$ in its second variable.\nA solution $(u, m)$ is a *mean-field game equilibrium*.\nIn it, $u$ is the value function of a typical player and $m(t)$ is the distribution of the players at time $t$.\n\n## Remarks\n\nThe first equation is a [[Hamilton–Jacobi–Bellman equation|SJP9HW82VX]] solved backward from time $T$.\nThe second is a [[Fokker–Planck equation|FVQWR7NS00]] solved forward from time $0$, for players who use the optimal feedback $-\\partial_p H(x, \\nabla u)$ with noise of intensity $\\nu$.\n\nExample: let $H(x, p) = \\tfrac{1}{2} \\abs{p}^2$ and $F = G = 0$.\nThen $u = 0$ and $m$ equal to the solution of the heat equation $\\partial_t m = \\nu \\Delta m$ with $m(0) = m_0$ form a solution.\nThe first equation and the terminal condition hold since $u = 0$.\nThe second equation reduces to the heat equation because $\\partial_p H(x, 0) = 0$.\n\nNon-example: in the same setting, if $m_0$ is not constant, then $u = 0$ and $m(t, x) = m_0(x)$ for all $t$ is not a solution, because $\\partial_t m - \\nu \\Delta m = -\\nu \\Delta m_0 \\ne 0$.\n\nLasry and Lions proved existence of solutions under regularity assumptions, and uniqueness when $F$ and $G$ are monotone.\n",
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