In this poster, we begin by defining the Fractional Gaussian Field (FGF) X_s on S1 with parameter s. For s>1/4, we index our Gaussian random variables by points on the circle S1, and for s\leq 1/4 we index these random variables by smooth, real-valued functions on S1. Using methods from Fourier analysis and probability, we prove that these definitions make sense (i.e. that the series defining the FGF pointwise converges almost surely). We then prove that the sample paths of X_s are almost surely Lipschitz when s>3/4, Holder with exponent 1/k for s>1/2k, and k times continuously differentiable for s>(k+1)/2. We then prove the analogous results on the d dimensional torus Td.
The theory of FGF’s on manifolds is currently a topic of interest to cosmologists studying cosmic microwave background radiation and particle physicists studying Liouville quantum gravity. We outline a procedure for defining the FGF on an arbitrary compact Riemannian manifold (M,g), using tools from functional analysis and probability. Our current interest is in regularity conditions for paths of FGF on the sphere S2, which are more difficult to study due to the sphere’s positive Ricci curvature.