Li Chen: Fractional Gaussian Fields and the Parabolic Anderson Model on the Sierpiński gasket

Abstract: We will first introduce fractional Gaussian fields and discuss their regularity properties on the Sierpiński gasket. These fields are naturally associated with the Laplacian and can be constructed via discrete graph approximations. Formally, they can be viewed as random distributions of the form X_{\alpha}=(-\Delta)^{-\alpha} W, where \( W \) is a Gaussian white noise.

We then consider the Parabolic Anderson Model driven by Gaussian noises which are white in time and colored in space, with spatial covariance induced by the fractional Gaussian field. We study the existence and uniqueness of solutions in the Itô sense and present moment estimates. The main analytic tools are heat kernel estimates and spectral properties of the Laplacian on the gasket.