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  "body": "\n## Definition\n\nLet $d \\ge 1$.\nA *Lévy process* in $\\R^d$ is a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \\ge 0}$ with values in $\\R^d$ such that:\n\n1. $X_0 = 0$ almost surely;\n2. for all $0 \\le t_0 \u003c t_1 \u003c \\cdots \u003c t_n$, the increments $X_{t_1} - X_{t_0}, \\ldots, X_{t_n} - X_{t_{n-1}}$ are [[independent|0GQ938GDE0]];\n3. for all $0 \\le s \\le t$, the increment $X_t - X_s$ has the same [[distribution|0DVDCMA2DK]] as $X_{t-s}$;\n4. for every $t \\ge 0$ and $\\varepsilon \u003e 0$, $\\lim_{h \\to 0} P(\\abs{X_{t+h} - X_t} \u003e \\varepsilon) = 0$.\n\n## Remarks\n\nExample: the Poisson process $N$ of rate $\\lambda$ satisfies the four clauses; $N_t - N_s$ is Poisson with mean $\\lambda(t - s)$, and $P(N_{t+h} \\ne N_t) = 1 - e^{-\\lambda \\abs{h}} \\to 0$.\nNon-example: the deterministic process $X_t = t^2$ satisfies clauses 1, 2 and 4, but $X_{s+1} - X_s = 2s + 1$ depends on $s$, so clause 3 fails.\nEvery Lévy process has a modification whose paths are right-continuous with left limits, and one usually works with it.\nThere is a function $\\psi$, the characteristic exponent, with $\\mathbb{E}[e^{i \\langle u, X_t \\rangle}] = e^{t \\psi(u)}$ for all $u \\in \\R^d$ and $t \\ge 0$.\n",
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