Complex analysis and Brownian motion

Here there have a been a variety of results eg. conformal invariance of Brownian motion, various proofs of known complex analysis results. It has been used for intuition purposes eg. see answers by B.Thurston Does Riemann map depend continuously on the domain?. The connections are also still being explored in the study of SLE curves (defined in terms of a 1d-Brownian motion) in various statistical models.

A priori this bridge is surprising even if one just starts with the original construction using the heat equation.

Random matrices and Partial differential equations

Here the main connections come through integrability theory. The main equation that stands out here is the KPZ equation and the KPZ universality class. The height function has beautiful formulas in terms of Airy processes and Fredholm determinant quantities that show up in random matrix theory. In particular, when dealing with boundary conditions, we find different types of random matrix families. So the connections run deep.

eg. see survey by PL Ferrari "From interacting particle systems to random matrices".

This bridge is surprising given the fact that KPZ originates from the study of interfaces and the Burgers equation which in turn was studied as a model for turbulence.

Renormalization, Regularity structures and SPDEs

This is more of a recent development. But I personally think it is quite surprising that so many of the tools/concepts developed in the specific setting of quantum fields such as Feynman-diagrams, BPHZ renormalization turn out to have analogues that apply to large families of stochastic differential equations. And that the generalization of rough paths was the natural framework to formalize those notions.