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  "body": "\n## Definition\n\nLet $\\nu \u003e 0$, $\\lambda \\in \\R$ and $D \u003e 0$ be constants.\nLet $\\xi$ be a Gaussian space-time white noise on $\\R \\times [0, \\infty)$, that is, a centred Gaussian random distribution with formal covariance $\\mathbb{E}[\\xi(x, t)\\xi(y, s)] = \\delta(x - y)\\delta(t - s)$.\nThe *Kardar–Parisi–Zhang equation* is the [[stochastic partial differential equation|PRCQG82QTK]] for a random function $h(x, t)$:\n$$\\partial_t h = \\nu\\, \\partial_x^2 h + \\frac{\\lambda}{2} \\big(\\partial_x h\\big)^2 + \\sqrt{D}\\, \\xi.$$\n\n## Remarks\n\nThe equation is formal: a solution $h(\\cdot, t)$ is only Hölder continuous of every order less than $\\frac{1}{2}$ in $x$, so $(\\partial_x h)^2$ is not defined.\nFor $\\lambda \\ne 0$, a standard meaning is the Cole–Hopf solution $h = \\frac{2\\nu}{\\lambda}\\log Z$, where $Z \u003e 0$ solves the stochastic heat equation $\\partial_t Z = \\nu\\, \\partial_x^2 Z + \\frac{\\lambda\\sqrt{D}}{2\\nu} Z \\xi$ in the Itô sense; see [[Cole–Hopf transformation|VBMQ9SJ3DH]].\nHairer's theory of regularity structures gives a solution theory that agrees with the Cole–Hopf solution, up to an infinite renormalization constant.\nExample: with $\\lambda = 0$ the equation is linear, the additive stochastic heat equation, and its solution is a Gaussian process given by a stochastic convolution with the heat kernel.\nNon-example: the deterministic equation $h_t = \\nu h_{xx} + \\frac{\\lambda}{2}(h_x)^2$ has no noise term, so it is not the KPZ equation; it is a viscous Hamilton–Jacobi equation.\n",
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