We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.
"Strong invariance and noise-comparison principles for some parabolic stochastic PDEs." Ann. Probab. 45 (1) 377 - 403, January 2017. https://doi.org/10.1214/15-AOP1009