اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSobolev regularity for the Monge-Ampere equation in the Wiener space
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Given the standard Gaussian measure $gamma$ on the countable product of lines $mathbb{R}^{infty}$ and a probability measure $g cdot gamma$ absolutely continuous with respect to $gamma$, we consider the optimal transportation $T(x) = x + abla varphi(x)$ of $g cdot gamma$ to $gamma$. Assume that the function $| abla g|^2/g$ is $gamma$-integrable. We prove that the function $varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {det}_2(I + D^2 varphi) exp bigl(mathcal{L} varphi - 1/2 | abla varphi|^2 bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $varphi$.
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