• Singular stochastic partial differential equations

The aim of the thesis is to review in detail the concept of Energy Solutions developed by Jara, Goncalves and Gubinelli in order to tackle the problem of KPZ equation
well-posedness and the related weak universality conjecture.
This equation has been introduced in a celebrated Physics paper in 1986 to give a universal description of growing interfaces fluctuations in one dimension.
The KPZ equation can not be understood by means of classical SPDEs techniques,
since nonlinearity and the presence of white noise make it impossible to give a meaning to the quadratic term.
The first rigorous results on KPZ, by Bertini and Giacomin, was the convergence of an interacting particle system density field to the solution of the Stochastic Burgers Equation (SBE),
which is the derivative of KPZ, and which shares the ill-posed nature of the latter.
In fact this convergence result was obtained by means of an exponential transformation at the microscopic level,
peculiar of the considered model, and a parallel exponential change of variables in SBE, known as the Hopf-Cole transformation: this strategy could hardly lead to more general results.

One major breakthrough were the results obtained by Hairer (and later with Quastel),
where existence and uniqueness, along with an invariance principle were derived for KPZ equation by means of the theory of Regularity Structures,
for which Hairer was awarded the Fields Medal in 2014.

We will be concerned with a different approach to KPZ, introduced by Goncalves and Jara and later
refined by Gubinelli and Jara: the concept of Energy Solutions will provide a notion of solution to the
stationary KPZ equation by means of a martingale problem. Our setting will thus be somewhat more familiar than Hairer's Regularity Structures,
however it will require stationarity, which is the main drawback of the theory.
Nonetheless, Energy Solutions have already proved to be a useful tool in the study of KPZ universality,
and convergence of particle models to these solutions has been rigorously established,
whereas Regularity Structures look difficult to apply to particle problems, the only universality result in the latter case considering SDEs not coming from particle systems.

The core idea of the energy solution approach is to understand the solutions of SBE as an antisymmetric perturbation of its linear part,
commonly known as the Ornstein-Uhlenbeck (OU) equation. More precisely, it is defined a class of processes controlled by OU, consisting in
reversible processes with space white noise invariant measure which are the sum of the OU process and a zero quadratic variation part.
Strong regularising estimates hold for such processes, just like in the linear case OU, and this allows to give a meaning to the quadratic nonlinearity
in SBE. An energy solution is thus defined as a process controlled by OU such that the aforementioned zero quadratic part coincides with Burgers'
drift.

It is a recent result that energy solutions to SBE are unique up to indistinguishability.
This fact has been established by means of the aforementioned Hopf-Cole transformation, which maps SBE to the Stochastic Heat Equation (SHE),
whose solution is known to be unique.
The formal proof is carried out regularising the energy solution, applying Ito's formula to the Hopf-Cole transform of the regularisation,
which produces an approximate version of SHE, and then checking that the remainder vanishes in the limit.
This last step contitutes the true difficulty of the proof.

Once uniqueness is established, we can combine it with invariance results to obtain well-posedness. Namely, it has been proved that the weak universality
conjecture holds for a large class of SPDEs if we consider solutions of KPZ to be defined as our energy solutions, which thus arise as the limit of well-posed
equations. The proof uses again the regularising estimates for controlled processes to gain the necessary tightness.
However, computations are in this case much harder, and (together with the uniqueness result) this is the subject of the most part of the thesis.