On some stochastic parabolic differential equations in a Hilbert space

We consider some stochastic difference partial differential equations of the form d u ( x , t , c ) = L ( x , t , D ) u ( x , t , c ) d t + M ( x , t , D ) u ( x , t − a , c ) d w ( t ) , where L ( x , t , D ) is a linear uniformly elliptic partial differential operator of the second order, M ( x , t , D ) is a linear partial differential operator of the first order, and w ( t ) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.