Convergence problems for singular stochastic dynamics

Over the last twenty years there has been significant progress in the well-posedness study of singular stochastic PDEs in both parabolic and dispersive settings. In this talk, I will discuss some convergence problems for singular stochastic nonlinear PDEs. In a seminal work, Da Prato and Debussche (2003) established well-posedness of the stochastic quantization equation, also known as the parabolic Φ^k+1 _2 -model in the two-dimensional case. More recently, Gubinelli, Koch, Oh, and Tolomeo proved the corresponding well-posedness for the canonical stochastic quantization equation, also known as the hyperbolic Φ^k+1 _2 -model in the two-dimensional case. In the first part of this talk, I will describe convergence of the hyperbolic Φ^k+1 _2 -model to the parabolic Φ^k+1 _2 -model. In the dispersive setting, Bourgain (1996) established well-posedness for the dispersive Φ^4 _2 -model (=deterministic cubic nonlinear Schrödinger equation) on the two-dimensional torus with Gibbsian initial data. In the second part of the talk, I will discuss the convergence of the stochastic complex Ginzburg-Landau equation (= complex-valued version of the parabolic Φ^4 _2 -model) to the dispersive Φ^4 _2-model at statistical equilibrium.