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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nThe *heat kernel* of $\\R^n$ is the function $\\Phi \\colon \\R^n \\times (0, \\infty) \\to (0, \\infty)$ given by\n$$\\Phi(x, t) = \\frac{1}{(4 \\pi t)^{n/2}} \\exp\\left( -\\frac{\\abs{x}^2}{4t} \\right).$$\nIt solves the [[heat equation|VP2HN86VAG]] $\\partial_t \\Phi = \\Delta \\Phi$ on $\\R^n \\times (0, \\infty)$, satisfies $\\int_{\\R^n} \\Phi(x, t) \\, dx = 1$ for every $t \u003e 0$, and $\\Phi(\\cdot, t)$ tends to the Dirac measure at $0$ as $t \\to 0^+$.\n\n## Remarks\n\nCheck for $n = 1$: $\\partial_t \\Phi = \\Phi \\cdot \\big( -\\tfrac{1}{2t} + \\tfrac{x^2}{4t^2} \\big)$ and $\\partial_x \\Phi = -\\tfrac{x}{2t} \\Phi$, so $\\partial_x^2 \\Phi = \\Phi \\cdot \\big( \\tfrac{x^2}{4t^2} - \\tfrac{1}{2t} \\big)$, which is the same.\nFor bounded continuous $g$, the function $u(x, t) = \\int_{\\R^n} \\Phi(x - y, t) g(y) \\, dy$ solves the heat equation for $t \u003e 0$ and tends to $g(x)$ as $t \\to 0^+$.\nFor each $t$, $\\Phi(\\cdot, t)$ is the density of a [[normal distribution|05AQE0QR0D]] with mean $0$ and covariance $2t$ times the identity.\nNon-example: $p(x, t) = (2 \\pi t)^{-n/2} e^{-\\abs{x}^2/(2t)}$ is not the heat kernel of $\\partial_t = \\Delta$; it solves $\\partial_t p = \\tfrac12 \\Delta p$ and is the transition density of Brownian motion.\nMore generally, a domain with a boundary condition, or a Riemannian manifold, has a heat kernel $K(x, y, t)$.\nIt is the function such that $u(x, t) = \\int K(x, y, t) g(y) \\, dy$ solves the heat equation with that boundary condition and initial value $g$.\n",
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