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We consider a nonlinear stochastic partial differential equation (SPDE) indivergence form where the forcing term is a Gaussian noise, that is white intime and colored in space such that the gradient of the solution isH\"older-continuous, but not differentiable. Then, we prove a generalizedTaylor expansion of the difference between the solution to the SPDE and thesolution to its linearization around a fixed basepoint. The result isreminiscent of the theory of (controlled) rough paths and agrees with thegeneral observation, that, in settings with a rough driver, subtracting thesolution to the linearized equation yields a more regular object.

A first order description of a nonlinear SPDE in the spirit of rough paths