Stochastic PDEs and weakly interacting particle systems

We study two problems relating to weakly interacting particle systems in the presence of exogenous noise. Part I presents a pathwise regularisation by noise result for McKean–Vlasov equations with singular interaction kernels. Using ideas from the theory of averaged fields and non-linear Young integrals we obtain well-posedness of the McKean–Vlasov systems, particle approximations and mean field convergence. In Part II we study a family of semi-linear, convection-diffusion SPDEs that are closely related to PDEs coming from the theory of collision-less kinetics. We study these equations in the presence of additive spacetime white noise. In one dimension we show global well-posedness and exponential ergodicity for an equation with a cubic non-linearity and repulsive sign choice. In two dimensions we show local well-posedness for a renormalised equation.