Quasilinear SPDEs in divergence-form

We develop a solution theory in H\"older spaces for a quasilinear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE Lemmas which establish a priori H\"older bounds for an equation with irregular right hand side written in divergence form. We apply these deterministic bounds to the case of a noise term which is white in time and trace class in space to obtain stretched exponential bounds for the H\"older semi-norms of the solution for the stochastic equation.