  We develop a notion of nonlinear expectation --G-expectation-- generated by a
nonlinear heat equation with infinitesimal generator G. We first study
multi-dimensional G-normal distributions. With this nonlinear distribution we
can introduce our G-expectation under which the canonical process is a multi
dimensional 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 obtained
the existence and uniqueness of stochastic differential equation under our
G-expectation.
