Homogenization for a Class of SDEs in Hilbert Spaces with Applications to Anomalous Diffusion
In this talk, we will discuss homogenization problems for a class of generalized Langevin equations (GLEs), which are commonly employed as models to study anomalous diffusion. The GLEs are stochastic integro-differential equations that contain a memory kernel and are driven by a colored Gaussian noise. We will start with a brief overview of various representations of a stationary Gaussian process, focusing in particular on the stochastic realization of its covariance function on a state-space. We will then introduce and discuss various homogenization problems for the GLEs. In particular, we will see how an understanding of the representations of the Gaussian noise motivates us to lift the SDEs to Hilbert spaces. We will end the talk by presenting some preliminary results that we have obtained for these homogenization problems.