An approximation result for a class of stochastic heat equations with colored noise

We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations.As a consequence, we prove various comparison principles extending earlier works of [Stoch. Stoch. Rep. 37 (1991) 225.245] and [Ann. Probab. 45 (2017) 377.403] among others.Among other things, our results enable us to obtain sharp estimates on the moments of the solution.A main technical ingredient of our method is a local limit theorem which is of independent interest.