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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, \\mathbb{P})$ be a probability space, let $D \\subseteq \\R^d$ be open and let $T \u003e 0$.\nLet $L$ be a linear differential operator in the variable $x \\in D$, and let $F$ and $G$ be functions.\nLet $\\xi$ be a random noise on $[0, T] \\times D$, for example space-time white noise, the formal time derivative of a cylindrical [[Wiener process|22735FGJV6]] $W$.\nA *stochastic partial differential equation* is an equation for a random field $u(t, x, \\omega)$ of the form\n$$\\partial_t u = L u + F(u) + G(u) \\, \\xi, \\qquad t \\in (0, T], \\ x \\in D,$$\nwith initial and boundary conditions.\nIn Itô form it reads $du = (L u + F(u)) \\, dt + G(u) \\, dW$.\nA solution is a [[stochastic process|0QH5ENKZM1]], adapted to the filtration of $W$, that satisfies the equation in a [[weak|SJZX81A783]] or [[mild|3VBRS94V1G]] sense.\nClassical derivatives are not used, because $\\xi$ is too irregular.\n\n## Remarks\n\nExample: the stochastic heat equation $\\partial_t u = \\partial_x^2 u + \\xi$ on $(0, 1)$ with $u = 0$ at $x = 0, 1$ and $u(0, \\cdot) = 0$, where $\\xi$ is space-time white noise.\nIts mild solution is $u(t, x) = \\int_0^t \\int_0^1 p_{t-s}(x, y) \\, W(ds, dy)$, with $p$ the Dirichlet heat kernel of $(0, 1)$.\nThis is a well-defined Gaussian random field, continuous in $(t, x)$.\nNon-example: the stochastic differential equation $dX = -X \\, dt + dB$, with $B$ a Brownian motion, is driven by noise.\nBut it involves no derivatives in a space variable, so it is not a partial differential equation.\nThe same stochastic heat equation in dimension $d \\ge 2$ still has a mild solution, but it is a random distribution rather than a function.\n",
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