Dissipative perturbations of integrable PDE

In the introductory part of my talk I will discuss what is known and unknown about Hamiltonian and dissipative perturbations of an integrable PDE under periodic boundary conditions. Next I will present an averaging theory for such equation, perturbed by a small viscosity and small random force. It turns out that the averaged dynamics is described by a quasilinear stochastic heat equation with a non-local nonlinearity. The theory applies both to solutions of Cauchy problem and to stationary in time solutions.