  In this article, we will first introduce a class of Gaussian processes, and
prove the quasi-invariant theorem with respect to the Gaussian Wiener measure,
which is the law of the associated Gaussian process. In particular, it includes
the case of the fractional Brownian motion.
  As applications, we will establish the integration by parts formula and
Bismut-Elworthy-Li formula on the Gaussian path space, and by which some
logarithmic Sobolev inequalities will be presented. Moreover, we will also
provides some applications in the field of financial mathematics.
