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Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...

One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE $$\partial_t u=\Delta u-u^3+\xi,$$ where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...