In order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are... Show moreIn order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are introduced in stochastic dynamical systems. In this thesis we investigate stochastic differential equations with non-Gaussian Lévy noise, and the singular two-dimensional Anderson model equation with spatial white noise potential. This thesis consists of the following three main parts. In the first part, we establish a linear response theory for stochastic differential equations driven by an α-stable Lévy noise (1<α<2). We first prove the ergodic property of the stochastic differential equation and the regularity of the corresponding stationary Fokker-Planck equation. Then we establish the linear response theory. This result is a general fluctuation-dissipation relation between the response of the system to the external perturbations and the Lévy type fluctuations at a steady state.In the second part, we study the global well-posedness of the singular nonlinear parabolic Anderson model equation on a two-dimensional torus. This equation can be viewed as the nonlinear heat equation with a random potential. The method is based on paracontrolled distribution and renormalization. After splitting the original nonlinear parabolic Anderson model equation into two simpler equations, we prove the global existence by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by classical energy estimates. This work improves the local well-posedness results in earlier works.In the third part, we investigate the variation problem associated with the elliptic Anderson model equation in a two-dimensional torus in the paracontrolled distribution framework. The energy functional in this variation problem is arising from the Anderson localization. We obtain the existence of minimizers by a direct method in the calculus of variations, and show that the Euler-Lagrange equation of the energy functional is an elliptic singular stochastic partial differential equation with the Anderson Hamiltonian. We further establish the L2 estimates and Schauder estimates for the minimizer as weak solution of the elliptic singular stochastic partial differential equation. Therefore, we uncover the natural connection between the variation problem and the singular stochastic partial differential equation in the paracontrolled distribution framework.Finally, we summarize our results and outline some research topics for future investigation. Show less
