  We prove existence of solutions to continuity equations in a separable
Hilbert space. We look for solutions which are absolutely continuous with
respect to a reference measure \gamma which is Fomin-differentiable with
exponentially integrable partial logarithmic derivatives. We describe a class
of examples to which our result applies and for which we can prove also
uniqueness. Finally, we consider the case where \gamma is the invariant measure
of a reaction-diffusion equation and prove uniqueness of solutions in this
case. We exploit that the gradient operator D_x is closable with respect to
L^p(H,\gamma) and a recent formula for the commutator D_xP_t - P_tD_x where P_t
is the transition semigroup corresponding to the reaction-diffusion equation,
[DaDe14]. We stress that P_t is not necessarily symmetric in this case. This
uniqueness result is an extension to such \gamma of that in [DaFlRo14] where
\gamma was the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck
process.
