Some Discussions on Restrictions of Gaussian Free Fields and Massive Gaussian Free Fields

We are interested in two kinds of GFF. The rst one, mainly referred as GFF in the forth comings, is indexed by H(D), where D ⊆ R is some bounded domain, and H (D) is the closure of C∞ c (D) under the Dirichlet inner product (·, ·)∇ on D. The other one is the massive GFF. which is indexed by the standard Sobolev space H (R). As we can see from [3] and [4], when n = 2, the circle "average" of the GFF has the same law as a multiple of Brownian motion. Inspired by this, we discussed the sphere "average" of the GFF for n ≥ 3, and also the restrictions of the GFF on general curves (n = 2) and hypersurfaces (n ≥ 3). We also studied the massive GFF, mainly with the approach of Fourier analysis. By examining the decay rate of the Fourier transform of the restriction distribution, we got the conclusion that the massive GFF can be restricted on any smooth compact hypersurface with non-zero Gaussian curvature everywhere.