Maximum principle for quasilinear SPDE’s on a bounded domain without regularity assumptions

We prove a maximum principle for local solutions of quasi-linear parabolic stochastic PDEs, with non-homogeneous second order operator on a bounded domain and driven by a space–time white noise. Our method based on an approximation of the domain and the coefficients of the operator, does not require regularity assumptions. As in previous works by Denis et al. (2005, 2009) [5,6], the results are consequences of Itô’s formula and estimates for the positive part of local solutions which are non-positive on the lateral boundary.