Parametrix technique for a class of degenerate parabolic operators with measurable coefficients under the weak Hörmander condition

In this thesis we study a wide class of differential operators of Kolmogorov type characterized by two structural hypotheses: it is satisfied the weak Hörmander condition and the coefficients are merely measurable in time variable. The main purpose of this work is to adapt the classical Levi parametrix method to this framework, in order to construct a fundamental solution of the operator. In particular, we will use a time-dependent parametrix. This problem is strongly related to the study of stochastic differential equations (SDEs), since the backward Kolmogorov operators associated to some linear SDEs take the form of the operator in the class we have considered. Therefore, the obtained fundamental solution can be seen as the transition density of the solution of a SDE. Choosing coefficients that are merely measurable in time, these results may be applied in the study of stochastic partial differential equations (SPDEs), which naturally appear in applications with rough coefficients.