This thesis studies irregular stochastic (partial) differential equations arising in fluctuating hydrodynamics or regularization by noise, and homogenization limits thereof. In the first part, we consider a model for particles on a biological membrane. The membrane is given by an ultra-violet cutoff of the quasi-planar Helfrich surface, that is subject to space-time fluctuations. We study the homogenization limits of the Itô and Stratonovich rough paths lifts of the diffusion in different scaling regimes. As an outlook on the construction of the diffusion on the Helfrich membrane without cutoff, we prove convergence of the rescaled surface measures. Moreover, we study nonlinear approximations of the Dean-Kawasaki SPDE, a model for the dynamics of the empirical density of independent Brownian particles. We approximate this highly irregular SPDE such that the physical constraints of the particle system are preserved and derive weak error estimates. We prove well-posedness and a comparison principle for a class of nonlinear regularized Dean-Kawasaki equations. The second part of this thesis deals with the weak well-posedness of multidimensional singular SDEs with Besov drift in the rough regularity regime and additive stable jump noise. We first solve the associated fractional parabolic Kolmogorov equation. To that end, we employ the paracontrolled ansatz and furthermore generalize to irregular terminal conditions, that are itself paracontrolled. We then prove existence and uniqueness of a solution to the martingale problem. Motivated by the equivalence between probabilistic weak solutions of SDEs with bounded, measurable drift and solutions of the martingale problem, we define a rough-path-type weak solution concept for singular Lévy diffusions, proving moreover equivalence to the martingale solution in the Young and rough regime. To this end, we construct a rough stochastic sewing integral. In particular, we show that canonical weak solutions are in general non-unique in the rough case. We apply our theory to construct the Brox diffusion with Lévy noise. Finally, we combine the theory of periodic homogenization with the solution theory for singular SDEs with stable noise. For the martingale solution projected onto the torus, we prove existence of a unique invariant probability measure. We solve the singular Fokker-Planck equation and prove a strict maximum principle. Furthermore, we solve the singular resolvent and Poisson equation. Using Kipnis-Varadhan methods, we prove a central limit theorem and obtain a Brownian motion with constant diffusion matrix. In the pure stable noise case, we rescale differently and encounter no diffusivity enhancement. We conclude on the periodic homogenization result for the singular parabolic PDE via Feynman-Kac formula.
