Well-posedness of stochastic PDEs

In this talk, we first discuss the second iteration argument introduced by Bourgain to establish LWP of KdV with measures as initial data. Then, we establish LWP of the stochastic KdV (SKdV) with additive space-time white noise by estimating the stochastic convolution via Ito calculus and showing its continuity via the factorization method. Next, we discuss well-posedness of SKdV with multiplicative noise in $L^2$. In order to treat the non-zero mean case, we derive a coupled system of a SDE and a SPDE. Lastly, as a toy model to study KPZ equation and stochastic Burgers equation, we study stochastic KdV-Burgers equation (SKdVB). We discuss how Fourier analytic technique can be applied to show LWP. If time permits, we discuss how one can obtain global well-posedness of these equations via (1) analogue of conservation laws, (2) Applying Bourgain's argument for invariant measures (for deterministic PDEs) to SPDEs.