Stokes formula on the Wiener space and \(n\)-dimensional Nourdin-Peccati analysis. (English) Zbl 1202.60081
The authors extend the results of I. Nourdin and G. Peccati [Probab. Theory Relat. Fields 145, No. 1-2, 75–118 (2009; Zbl 1175.60053)] to the \(n\)-dimensional case, by obtaining a system
\[
{\partial \over \partial x_1} (h^{j,1} (x) \rho (x) ) + \cdots + {\partial \over \partial x_n} (h^{j,n} (x) \rho (x) ) = - x_j \rho (x), \qquad x\in {\mathbb R}^n, \quad j=1,\ldots ,n,
\]
of partial differential equations for the density \(\rho (x)\) of a smooth \({\mathbb R}^n\)-valued random variable \(Z = ( Z^1 , \ldots , Z^n )\), based on the function
\[
h^{i,j} ( x ) = - E[ \langle D {\mathcal L}^{-1} Z^i , D Z^i \rangle \mid Z = x ], \qquad x \in {\mathbb R}^n, \quad 1 \leq i , j \leq n,
\]
where \(D\) and \({\mathcal L}\) are respectively the Malliavin derivative and the Ornstein-Uhlenbeck operator on the Wiener space. Applications to quasi-sure analysis and to the estimation of conditional probability densities of smooth random variables are given.
Reviewer: Nicolas Privault (Singapore)
MSC:
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60H30 | Applications of stochastic analysis (to PDEs, etc.) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
Wiener space; quasi-sure analysis; density formula; Nourdin-Peccati analysis; Stein’s Lemma; conditional probabilityCitations:
Zbl 1175.60053References:
| [1] | Airault, H., Projection of the infinitesimal generator of a diffusion, J. Funct. Anal., 85, 2, 353-391 (1989) · Zbl 0683.60055 |
| [2] | Airault, H.; Malliavin, P., Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math., 112, 3-52 (1988) · Zbl 0656.60046 |
| [3] | Breton, J.-C.; Nourdin, I.; Peccati, G., Exact confidence intervals for the Hurst parameter of a fractional Brownian motion, Electron. J. Stat., 3, 416-425 (2009) · Zbl 1326.62065 |
| [4] | Federer, H., Geometric Measure Theory (1996), Springer-Verlag · Zbl 0176.00801 |
| [5] | Fukushima, M., Basic properties of Brownian motion and a capacity on the Wiener space, J. Math. Soc. Japan, 36, 161-176 (1984) · Zbl 0522.60081 |
| [6] | Ito, K., On Malliavin tensor fields, Comm. Pure Appl. Math., 47, 377-403 (1994) · Zbl 0808.60052 |
| [7] | Malliavin, P., Stochastic Analysis (1997), Springer-Verlag · Zbl 0878.60001 |
| [8] | Nourdin, I.; Peccati, G., Stein’s method on Wiener chaos, Probab. Theory Related Fields, 145, 1, 75-118 (2008) · Zbl 1175.60053 |
| [9] | I. Nourdin, G. Peccati, Stein’s method and exact Berry-Esséen asymptotics for functionals of Gaussian fields, Ann. Probab., in press; I. Nourdin, G. Peccati, Stein’s method and exact Berry-Esséen asymptotics for functionals of Gaussian fields, Ann. Probab., in press · Zbl 1196.60034 |
| [10] | I. Nourdin, G. Peccati, Non-central convergence of multiple integrals, Ann. Probab., in press; I. Nourdin, G. Peccati, Non-central convergence of multiple integrals, Ann. Probab., in press · Zbl 1171.60323 |
| [11] | Nourdin, I.; Peccati, G.; Reinert, G., Second order Poincaré inequalities and CLTs on Wiener space, J. Funct. Anal., 257, 593-609 (2009) · Zbl 1186.60047 |
| [12] | I. Nourdin, F. Viens, Density formula and concentration inequalities with Malliavin calculus, submitted for publication; I. Nourdin, F. Viens, Density formula and concentration inequalities with Malliavin calculus, submitted for publication · Zbl 1192.60066 |
| [13] | Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer-Verlag · Zbl 1099.60003 |
| [14] | D. Nualart, Ll. Quer-Sardayons, Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations, preprint, 2009, http://arxiv.org/abs/0902.1849; D. Nualart, Ll. Quer-Sardayons, Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations, preprint, 2009, http://arxiv.org/abs/0902.1849 |
| [15] | Stein, Ch., A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, (Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability Theory (1972), University of California Press: University of California Press Berkeley), 583-602 · Zbl 0278.60026 |
| [16] | Sugita, H., Positive generalized functions and potential theory over an abstract Wiener space, Osaka J. Math., 25, 665-696 (1988) · Zbl 0737.46038 |
| [17] | F. Viens, Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent, Stochastic Process. Appl., published online http://dx.doi.org/10.1016/j.spa.2009.07.002; F. Viens, Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent, Stochastic Process. Appl., published online http://dx.doi.org/10.1016/j.spa.2009.07.002 · Zbl 1175.60056 |
| [18] | Watanabe, S., Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels., Ann. Probab., 15, 1, 1-39 (1987) · Zbl 0633.60077 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
