Infinite-dimensional Wiener processes with drift

A countable-dimensional stochastic differential equation (*) dX(t) = a(t, X) dt + dW(t) is considered. Here a is a vector function on (C is the set of continuous functions on [0, 1]) which, for each t, depends only on the past of X up to time t and W symbolizes a Wiener process with future increments independent of the past of X. The existence of a weak (distribution sense) solution of (*) is proved by a partial absolute continuous change of measures. It is assumed that the components of the vector a(·, X) depend asymptotically only on finitely many components of X without the restriction that the norm of a in l2 is square-t-integrable. (The latter would allow one to apply directly the Girsanov theorem.) There are also no regularity restrictions on a in X.