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.