In three vignettes, I will explore situations where the presence of randomness leads to new phenomena and requires a change of perspective and a different tool box.

I shall start with a drift-diffusion equation where the stream function is given by the Gaussian free-field, producing a borderline anomalous diffusive behavior in two dimensions.

Then, I will consider the optimal matching problem between random point clouds, which amounts to a Monge–Ampère equation with a rough and random right-hand-side; again the two-dimensional case is critical and does not admit a stationary matching.

Finally, I will present a differential geometric view upon regularity structures for quasi-linear classes of singular stochastic PDEs, and use Malliavin calculus to obtain estimates on a parameterization of the renormalized solution manifold.