The Quadratic Functionals Of Gaussian Processes And Their Applications

Gaussian processes are very important stochastic processes, which have importantapplications in many other felds. We mainly study two types of quadratic functionals ofGaussian processes in this paper, which we divide into three parts.Firstly, we discuss the frst type of (positive) quadratic functional of the additiveBrownian motions and additive Brownian bridges from the KL expansions. Based on theKL expansions, we show the interesting Pythagorean type theorems. As applications, weshow the associated Laplace transform and the small deviation estimates.Secondly, we study the Laplace transform of the frst type of quadratic functional ofcontinuous Gaussian processes through the Cameron-Martin formula. We try to extendthe result of Klepstyna et al. We want to show that, from the analysis point of view, theCameron-Martin type formula holds for the largest suitable domain.At last, we study a special type of quadratic functional of Gaussian processes: Le′vy’sarea process. We want to show the stochastic representation through the elementary idea:to study the Characteristic function. Moreover, we obtain that, for the Le′vy’s area processassociated to the H-type group, the stochastic representation holds.