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XVI: 06, 95-132, LNM 920 (1982)
MEYER, Paul-André
Note sur les processus d'Ornstein-Uhlenbeck (Malliavin's calculus)
With every Gaussian measure $\mu$ one can associate an Ornstein-Uhlenbeck semigroup, for which $\mu$ is a reversible invariant measure. When $\mu$ is Wiener's measure on ${\cal C}(R)$, this semigroup is a fundamental tool in Malliavin's own approach to the Malliavin calculus''. See for instance Stroock's exposition of it in Math. Systems Theory, 13, 1981. With this semigroup one can associate its generator $L$ which plays the role of the classical Laplacian, and the positive bilinear functional $\Gamma(f,g)= L(fg)-fLg-gLf$---leaving aside domain problems for simplicity---sometimes called carré du champ'', which plays the role of the squared classical gradient. As in classical analysis, one can define it as $\sum_i \nabla_i f\nabla i g$, the derivatives being relative to an orthonormal basis of the Cameron-Martin space. We may define Sobolev-like spaces of order one in two ways: either by the fact that $Cf$ belongs to $L^p$, where $C=-\sqrt{-L}$ is the Cauchy generator'', or by the fact that $\sqrt{\Gamma(f,f)}$ belongs to $L^p$. A result which greatly simplifies the analytical part of the Malliavin calculus'' is the fact that both definitions are equivalent. This is the main topic of the paper, and its proof uses the Littlewood-Paley-Stein theory for semigroups as presented in 1010, 1510
Comment: An important problem is the extension to higher order Sobolev-like spaces. For instance, we could define the Sobolev space of order 2 either by the fact that $C^2f=-Lf$ belongs to $L^p$, and on the other hand define $\Gamma_2(f,g)=\sum_{ij} \nabla_i\nabla_j f \nabla_i\nabla_j g$ (derivatives of order 2) and ask that $\sqrt{\Gamma_2(f,f)}\in L^p$. For the equivalence of these two definitions and general higher order ones, see 1816, which anyhow contains many improvements over 1606. Also, proofs of these results have been given which do not involve Littlewood-Paley methods. For instance, Pisier has a proof which only uses the boundedness in $L^p$ of classical Riesz transforms.\par Another trend of research has been the correct definition of higher gradients'' within semigroup theory (the preceding definition of $\Gamma_2(f,g)$ makes use of the Gaussian structure). Bakry investigated the fundamental role of true'' $\Gamma_2$, the bilinear form $\Gamma_2(f,g)=L\Gamma(f,g)-\Gamma(Lf,g)-\Gamma(Lf,g)$, which is positive in the case of the Ornstein-Uhlenbeck semigroup but is not always so. See 1909, 1910, 1912
Keywords: Ornstein-Uhlenbeck process, Gaussian measures, Littlewood-Paley theory, Hypercontractivity, Hermite polynomials, Riesz transforms, Test functions