Convergence problems for singular stochastic dynamics

Abstract

In this thesis, we investigate convergence problems for some nonlinear dispersive and parabolic PDEs in the singular stochastic setting. In the first part of the thesis, we study the so-called Smoluchowski-Kramers approximation on convergence of stochastic nonlinear wave equations (SNLW) to stochastic nonlinear heat equations (SNLH), with a polynomial nonlinearity. In particular, we prove that, in the over-damped regime, solutions of SNLW converge to those of the corresponding SNLH. This convergence is established for deterministic initial data. In the second part of the work, we study the inviscid limit for the stochastic complex Ginzburg-Landau equation (SCGL) with the cubic nonlinearity. We prove that, for Gaussian free field initial data, the solution of SCGL converges to that of the cubic nonlinear Schrödinger equation.