models of self-organized criticality which can be described by stochastic partial differential equations with noncoercive mono- tone diffusivity function and multiplicative wiener forcing term (as e.g. the bak/tang/wiesenfeld- and zhang-models) are analyzed. existence and uniqueness of nonnegative strong solutions are proved. previously numerically predicted transition to the critical state in 1-d is confirmed by a rigorous proof that this indeed happens in finite time with high probability.