The Infinitesimal Generator Under G-expectation, The G-supermartingale And G-superharmonic Function

In this paper, we study several problems about the application of BSDE and the related g-expectation in nonlinear analysis. In §1 and §3 we give two new results: The stochastic Infinitesimal Generator under g-expectation and the Corresponding relation between g-supermartingale and g-superharmonic function. Particularly, in §3.4 we expand the range of the g-superharmonic functions by applicating the concept of viscosity solution, however the correspondence still hold. In §2 we use BSDE to construct a nonlinear semi-group and discuss its infitesimal generator and its differential operator, and we expect more new development in this direction.