  In this paper, we develop a new general approach to the existence and
uniqueness theory of infinite dimensional stochastic equations of the form
dX+A(t)Xdt = XdW in (0;T)xH, where A(t) is a nonlinear monotone and
demicontinuous operator from V to V', coercive and with polynomial growth.
Here, V is a reflexive Banach space continuously and densely embedded in a
Hilbert space H of (generalized) functions on a domain $O\subset \mathbb{R}^d$
and V' is the dual of V in the duality induced by H as pivot space.
Furthermore, W is a Wiener process in H. The new approach is based on an
operatorial reformulation of the stochastic equation which is quite robust
under perturbation of A(t). This leads to new existence and uniqueness results
of a larger class of equations with linear multiplicative noise than the one
treatable by the known approaches. In addition, we obtain regularity results
for the solutions with respect to both the time and spatial variable which are
sharper than the classical ones. New applications include stochastic partial
differential equations, as e.g. stochastic transport equations.
