Jinqiao Duan Illinois Institute of Technology IPAM Associate Director The Laplace operator appears in the Navier-Stokes equations, the geophysical flow equations, and the equation for substance transport in fluids. It comes out so often that we at times forget its origin, let alone its connection with modern stochastic analysis, stochastic dynamical systems, especially non-Gaussian stochastic dynamical systems. Where does the Laplace operator come from? How is it related to normal diffusion or Brownian motion? How does it link to a nonlocal Laplace operator, anomalous diffusion, or Levy motion? The speaker will present an overview of these issues, with simple derivations. Related topics include (Gaussian) Brownian motion, (non-Gaussian) Levy motions, generators of stochastic processes, microscopic and macroscopic descriptions of diffusion processes, Fokker-Planck equations, among others.