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  "body": "\n## Statement\n\nLet $T \u003e 0$ and let $C_0 = C_0([0, T])$ be the [[classical Wiener space|5Z00K7K1HY]] of continuous functions $\\omega \\colon [0, T] \\to \\R$ with $\\omega(0) = 0$.\nLet $W$ be Wiener measure on $C_0$, the law of a [[Wiener process|22735FGJV6]] on $[0, T]$.\nLet $H$ be the Cameron–Martin space: the set of absolutely continuous $h \\in C_0$ with $h' \\in L^2([0, T])$.\nFor $h \\in C_0$, let $W_h$ be the [[pushforward|3N66TM3TSQ]] of $W$ under the translation $\\omega \\mapsto \\omega + h$.\n\n1. If $h \\in H$, then $W_h$ and $W$ are [[mutually absolutely continuous|P0P21KDV6E]], and for $W$-almost every $\\omega$\n$$\\frac{dW_h}{dW}(\\omega) = \\exp\\left( \\int_0^T h'(t) \\, d\\omega(t) - \\frac12 \\int_0^T h'(t)^2 \\, dt \\right).$$\n2. If $h \\notin H$, then $W_h$ and $W$ are [[mutually singular|EBGECZNCV5]].\n\n## Remarks\n\nHere $dW_h / dW$ is the [[Radon–Nikodym derivative|1NTKG64F7N]], and $\\int_0^T h'(t) \\, d\\omega(t)$ is the Wiener integral of the deterministic function $h'$, defined as an $L^2(W)$ limit.\nIn one dimension the formula is a direct computation: the density of $N(a, 1)$ with respect to $N(0, 1)$ is $e^{a x - a^2/2}$.\nThe general case reduces to finite-dimensional projections of the path, since the Wiener integral is a normal variable with variance $\\int_0^T h'(t)^2 \\, dt$.\n\nExample: for $h(t) = c t$, one has $h' = c$, and the density is $\\exp(c\\, \\omega(T) - c^2 T / 2)$.\nSo Brownian motion with drift $c$ has a law equivalent to that of Brownian motion on $[0, T]$.\n\nThe analogous result holds for every centered [[Gaussian measure|P0DV2TCH1N]] on a separable Banach space, with $H$ its Cameron–Martin space.\nIn infinite dimensions $H$ has measure zero, so most translations give singular measures.\n\n## Explanations\n\n### L1 human\n\nShifting a Brownian path by a fixed function changes its law.\nIf the shift is smooth enough, with square-integrable derivative, the new law has a density with respect to the old one, given by an explicit formula.\nIf the shift is rougher, the two laws live on disjoint sets of paths.\n\n### L2 human\n\nThe theorem is the starting point of Girsanov's theorem, where the deterministic shift is replaced by an adapted random drift.\nIt shows that Wiener measure is quasi-invariant only along the directions of $H$, which is a dense subspace of measure zero.\nIntegration by parts on Wiener space, the base of Malliavin calculus, comes from differentiating the density in $h$.\n",
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