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In this article we consider existence and uniqueness of the solutions to alarge class of stochastic partial differential of form $\partial_t u = L_x u +b(t,u)+\sigma(t,u)\dot{W}$, driven by a Gaussian noise $\dot{W}$, white in timeand spatial correlations given by a generic covariance $\gamma$. We providenatural conditions under which classical Picard iteration procedure provides aunique solution. We illustrate the applicability of our general result byproviding several interesting particular choices for the operator $L_x$ underwhich our existence and uniqueness result hold. In particular, we show thatDalang condition given in Dalang (1999) is sufficient in the case of manyparabolic and hypoelliptic operators $L_x$.

Remarks on existence and uniqueness of the solution for stochastic partial differential equations