FINITE DIMENSIONAL GAUSSIAN MEASURES :PROBABILITY LAW APPROACH AND FUNCTIONAL APPROACH

Abstract

In this thesis work, several notions from Functional analysis, Topology, Measure theory, and integration are used in the study of Gaussian measures in fi nite dimensions. We discussed this using the probability law approach and functional analysis approach. For the probability law frame, we began by introducing the Random vectors where we vividly talked about the Gaussian random vectors in which the Orthogonal matrices were a very important tool for the discourse. Here, we discussed enough content and results of the fi nite dimensional Gaussian measures which aids us to move to functional frame. For the functional frame, as the name implies, we re-frame the study and properties of Gaussian measures using notions of functional analysis where the Hermite polynomials both in one dimension and multi-dimension were introduced. We fi nally introduced the Ornstein Uhlencbeck Semi-group and one of its applications in Integro-differential equations.