An invariance principle for the stochastic heat equation
Stochastics and Partial Differential Equations: Analysis and Computations
We approximate the white-noise driven stochastic heat equation by replacing the fractional Laplacian by the generator of a discrete time random walk on the one dimensional lattice, and approximating white noise by a collection of i.i.d. mean zero random variables. As a consequence, we give an alternative proof of the weak convergence of the scaled partition function of directed polymers in the intermediate disorder regime, to the stochastic heat equation; an advantage of the proof is that it gives the convergence of all moments.
Joseph, Mathew, "An invariance principle for the stochastic heat equation" (2018). Journal Articles. 1112.