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We introduce a new notion of G-expectation-weighted Sobolev spaces, or inshort, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownianmotion are in fact path dependent PDEs in the corresponding Sobolev spacesunder G-norms. For the linear case of G corresponding the classical Wienerprobability space with Wiener measure P, we have established a 1-1correspondence between BSDE and such new type of quasilinear PDE in thecorresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1correspondence between a fully nonlinear PDE in the corresponding G-Sobolevspace and BSDE driven by G-Brownian. Consequently, the existence and uniquenessof such type of fully nonlinear path-dependence PDE in G-Sobolev space havebeen obtained via a recent results of BSDE driven by G-Brownian motion.

G-Expectation Weighted Sobolev Spaces, Backward SDE and Path Dependent PDE