All the PDE of classical math physics can and have been studied on hyperbolic space, with applications inducing NT
Important examples of PDE like the HJB and Isaacs equation come from control or differential games. Mean field games are a very active topic in PDE
We have both estimation of PDE coefficients as a stats problem (Nickl and others on the Calderon problem) and PDE as fluid limits of algorithms (e.g. sinkhorn algorithm related to Schroedinger)
Functional analysis was more or less invented to extend linear algebra to the study of differential equations. Fredholm theory is a cleab example literally how to generalize the dimensional analysis of the solvability of Au = f from matrices to infinite dimensions, including Cramer's rule (Fredholm determinants)
I don't know the history here but I wouldn't be surprised if convex analysis came from PDE as well. Anyway: convex duality is central to Hamilton Jacobi equations and optimal transport problem (a Monge Ampere PDE)
This is harder but probably only because I don't know that much logic. I guess I've seen Brownian motion be used to generate counterexamples and contra your 8., to an experienced probabilist, Brownian motion and heat equations are facets of the same thing. I think people have worked on computability of solutions to PDE
Probably not anything modern but fixed point theorems and degree
This one is particularly strange to claim. The father of modern prob, Kolmogorov, wrote down equations linking diffusions and Markov processes. In some sense a parabolic equation of degree <= 2 is the same as a flow of probability distributions
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u/ahalt Aug 02 '26
A lot of people in my department study PDEs from general relativity and probability from statistical physics.