S(P)DEs and the study of random fields

This thesis presents three contributions to stochastic functional equations and various Gaussian fields. In the first work, we establish a Fokker-Planck type of relationship between a certain class of McKean-Vlasov SDEs and the Keller-Segel PDE model. In the second contribution, we show that the Hausdorff dimension of the zeros of the Brownian sheet is preserved by all linear projections if and only if a certain condition on the dimensions is satisfied. Finally, in the third project, we prove a new characterization theorem for the Gaussian free field by developing an approach via resampling dynamics. This approach allows us to generalize previous results and applies to the more general class of fractional Gaussian fields.