Wick Calculus For Nonlinear Gaussian Functionals
نویسندگان
چکیده
This paper surveys some results on Wick product and Wick renormalization. The framework is the abstract Wiener space. Some known results on Wick product and Wick renormalization in the white noise analysis framework are presented for classical random variables. Some conditions are described for random variables whose Wick product or whose renormalization are integrable random variables. Relevant results on multiple Wiener integrals, second quantization operator, Malliavin calculus and their relations with the Wick product and Wick renormalization are also briefly presented. A useful tool for Wick product is the S-transform which is also described without the introduction of generalized random variables. Keyword: Malliavin calculus, Multiple integral, Chaos decomposition, Wick product, Wick renormalization 2000 MR Subject Classification: 60G15, 60H05, 60H07, 60H40
منابع مشابه
Convergence of densities of some functionals of Gaussian processes
We study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functionals of a Gaussian process. The tool we are using is the Malliavin calculus, in particular, the integration by parts formula and the Stein’s method. Applications to the convergence of densities of the least square estimator for the drift parameter in ...
متن کاملWick calculus for regular generalized functions
Wick calculus in Gaussian analysis is investigated. It is shown that this calculus can be developed in a space of regular generalized functions. The results are applied to the discussion of solutions for Wick type stochastic (partial) diierential equations. In particular, the viscous Burgers equation with a stochastic source is studied. Its solution is shown to be a regular generalized process ...
متن کاملApplication of Malliavin calculus to stochastic partial differential equations
The Malliavin calculus is an infinite dimensional calculus on a Gaussian space, which is mainly applied to establish the regularity of the law of nonlinear functionals of the underlying Gaussian process. Suppose that H is a real separable Hilbert space with scalar product denoted by 〈·, ·〉H . The norm of an element h ∈ H will be denoted by ‖h‖H . Consider a Gaussian family of random variables W...
متن کاملAlmost sure central limit theorems on the Wiener space
In this paper, we study almost sure central limit theorems for sequences of functionals of general Gaussian elds. We apply our result to non-linear functions of stationary Gaussian sequences. We obtain almost sure central limit theorems for these non-linear functions when they converge in law to a normal distribution.
متن کاملSecond order Poincaré inequalities and CLTs on Wiener space
We prove in nite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian elds, Stein's method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a xed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated elds.
متن کامل