  We introduce a notion of nonlinear expectation --G--expectation-- generated
by a nonlinear heat equation with infinitesimal generator G. We first discuss
the notion of G-standard normal distribution. With this nonlinear distribution
we can introduce our G-expectation under which the canonical process is a
G--Brownian motion. We then establish the related stochastic calculus,
especially stochastic integrals of Ito's type with respect to our G--Brownian
motion and derive the related Ito's formula. We have also give the existence
and uniqueness of stochastic differential equation under our G-expectation. As
compared with our previous framework of g-expectations, the theory of
G-expectation is intrinsic in the sense that it is not based on a given
(linear) probability space.
