Complex Isonormal Gaussian Process And The Closability Of Malliavin Derivative Operators For The Complex Isonormal Gaussian Process

With a real separable Hilbert space exists a real Isonormal Gaussian process as the foundation, we establish a complex Isonormal Gaussian process on a complex separable Hilbert space. From the complex Isonormal Gaussian process, we define Wiener-It? chaos with respect to the complex Isonormal Gaussian process by using the complex Hermite polynomials, and through the properties of orthogonality of the complex Hermite polynomials, gives Wiener-It? chaos decomposition of a square integrable space. Next, the function class of the smooth random variables is set up on the complex Isonormal Gaussian process, and then we define and research the complex Malliavin derivative operators in the function class of the smooth random variables. The core of this paper is to explain the complex Malliavin derivative operators in the function class of the smooth random variables can be closable. To do this, firstly, we prove that the real Malliavin derivative operators on the real Isonormal Gaussian process is closable. And in the process of the study, we find and get the general Leibniz rule with respect to Malliavin operators, and we also get the representation theorem of divergence operators and integration by parts formula by applying the Hermite polynomials as a tool. As an application, by the way of analogy,the closability of Malliavin derivative for the complex Isonormal Gaussian processes is obtained.