In this work we present examples of the effects of noise on the solution of a partial differential equation (PDE) in three different settings. We will first consider random initial conditions for two nonlinear dispersive PDEs, the nonlinear Schrödinger equation and the Korteweg - de Vries equation, and analyze their effects on some special solutions, the soliton solutions. The second case considered is a linear PDE, the wave equation, with random initial condi- tions. We will show that special random initial conditions allow to substantially decrease the computational and data storage costs of an algorithm to solve the inverse problem based on the boundary measurements of the solution of this equation. Finally, the third example considered is that of the linear transport equation with a singular drift term, where we will show that the addition of a multiplicative noise term forbids the blow up of solutions, under very weak hypothesis for which we have finite-time blow up of solutions in the deterministic case.
