Markov field properties of solutions of white noise driven quasi-linear parabolic pdes

In this paper we study a one-dimensional quasi-linear parabolic stochastic partial differential equation driven by a space-time white noise. First the germ Markov field property is proved for the solution of a Cauchy problem for this equation. Secondly, we introduce periodic (in time) boundary conditions and we study the existence and uniqueness of a solution and its Markov properties. The main result is that for the periodic solution the Markov field property only holds in the linear case