• cumulants
• flow approach
• renormalisation
• singular stochastic partial differential equations
• stochastic Burgers equation

The thesis aims to introduce the flow approach, originally developed by Paweł Duch, for the study of singular stochastic partial differential equations. We consider the explicit example of the stochastic Burgers equation on the one-dimensional torus. The method consists of obtaining an effective equation for a regularisation of the solution of the equation. The functional of the effective equation is expanded into a power series of the nonlinearity strength and chaoses of the (regularised) noise. The core of the argument is to prove a bound on the cumulants of the coefficients of the series. To do this, we introduce a flow equation for the cumulants, which allows us to prove these bounds inductively. A key step in the induction is to renormalise the coefficients by adding some artificial terms that are constants in the space variable. These don't change the equation because they disappear in one of its derivatives. We then improve the estimates by obtaining some pointwise inequalities and conclude the study of the equation with a fixed-point argument.