Finite Dimensional Gaussian Measure

This article mainly concerns about the finite dimensional Gaussian measure and its properties. The first chapter is devoted to one-dimension Gaussian measure and n-dimension Gaussian measure, as well as their Fourier transforms and properties. The second chapter is devoted to the Hermite polynomials and Ornstein-Uhlenbeck semi-group with respect to Gaussian measure. Moreover, by using Hermite polynomials and the integral representation of the Hermite polynomials, we can prove the Mehler’s for-mula. The third chapter is devoted to Sobolev class and Sobolev space in Lp(γn).