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  "body": "\n## Definition\n\nLet $d, m \\ge 1$, let $b \\colon \\R^d \\to \\R^d$ and $\\sigma \\colon \\R^d \\to \\R^{d \\times m}$ be Lipschitz continuous, and let $W$ be an $m$-dimensional [[Wiener process|22735FGJV6]].\nThe *diffusion process* (Itô diffusion) with drift $b$ and dispersion $\\sigma$ started at $x \\in \\R^d$ is the unique continuous adapted solution $(X_t)_{t \\ge 0}$ of\n$$dX_t = b(X_t)\\, dt + \\sigma(X_t)\\, dW_t, \\qquad X_0 = x.$$\nThe matrix $a = \\sigma \\sigma^{\\mathsf{T}}$ is its diffusion matrix.\n\n## Remarks\n\nThe process is a strong [[Markov process|0B77KZCQ51]], and by [[Itô's lemma|2J1RVQB0CM]] its [[generator|VPA41PWRR1]] acts on twice continuously differentiable functions $f$ with compact support by\n$$Lf(x) = \\frac12 \\sum_{i,j=1}^d a_{ij}(x) \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(x) + \\sum_{i=1}^d b_i(x) \\frac{\\partial f}{\\partial x_i}(x).$$\nExample: $d = m = 1$, $b(x) = -\\theta x$ with $\\theta \u003e 0$ and $\\sigma(x) = 1$ give the Ornstein–Uhlenbeck process; both coefficients are Lipschitz, and $Lf = \\tfrac12 f'' - \\theta x f'$.\nNon-example: the Poisson process is a Markov process but not a diffusion, because its paths jump.\nIn a wider sense, a diffusion is any strong Markov process with continuous paths; the Itô diffusions above form the main class.\n",
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