Nonlinear Brownian motion and nonlinear Feynman-Kac formula of path-functions

We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated nonlinear i.i.d random variables of order \(1/n\) tend to a maximal distribution according a new law of large number, whereas, with a new central limit theorem, the accumulation of order \(1/\sqrt{n}\) tends to a nonlinear normal distribution. The continuous time uncertainty accumulation derives a nonlinear Brownian motion as well as the corresponding OU-process driven by this nonlinear Brownian motion which converges to a nonlinear invariant measure of Gaussian type. The related stochastic calculus provides us a powerful tools to introduce time-space derivatives for functional of paths. The corresponding Feynman-Kac formula for gives one to one correspondence between fully nonlinear parabolic partial differential equations and backward stochastic differential equations driven by the nonlinear Brownian motion.