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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a probability space.\nA *Wiener process*, or standard Brownian motion, is a real [[stochastic process|0QH5ENKZM1]] $(W_t)_{t \\ge 0}$ such that:\n\n1. $W_0 = 0$ almost surely.\n2. For all $0 \\le t_0 \u003c t_1 \u003c \\dots \u003c t_n$, the increments $W_{t_1} - W_{t_0}, \\dots, W_{t_n} - W_{t_{n-1}}$ are [[independent|0GBJACNAP6]].\n3. For all $0 \\le s \u003c t$, the increment $W_t - W_s$ has the [[normal distribution|05AQE0QR0D]] with mean $0$ and variance $t - s$.\n4. Almost surely, the path $t \\mapsto W_t$ is continuous.\n\n## Remarks\n\nSuch a process exists; one construction takes the uniform limit of random series $\\sum_k \\xi_k \\phi_k(t)$, with independent standard normal $\\xi_k$ and suitable functions $\\phi_k$ such as the Schauder functions.\nThe process $X_t = 2 W_t$ satisfies clauses 1, 2 and 4 but fails clause 3, since $X_t - X_s$ has variance $4(t - s)$.\nEquivalently, $W$ is a centred [[Gaussian process|22WEM6RTW0]] with $E[W_s W_t] = \\min(s, t)$ and continuous paths.\nA process $W_t + \\mu t$, or $\\sigma W_t$, is called a Brownian motion with drift $\\mu$ or with variance parameter $\\sigma^2$.\nA $d$-dimensional Wiener process is a vector of $d$ independent real Wiener processes.\n",
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