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We consider the following stochastic partial differential equation, $$\begin{array}{@{}rcl@{}} &&dY_{t} = L^{\ast} Y_{t} dt + A^{\ast} Y_{t} \cdot dB_{t}\\ &&Y_{0} = \psi, \end{array} $$ associated with a stochastic flow {X(t,x)}, for t ≥ 0, $x \in \mathbb {R}^{d}$ , as in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162, 2008). We show that the strong solutions constructed there are ‘locally of compact support’. Using this notion,we define the mild solutions of the above equation and show the equivalence between strong and mild solutions in the multi Hilbertian space $\mathscr{S}^{\prime }$ . We show uniqueness of solutions in the case when ψ is smooth via the ‘monotonicity inequality’ for (L∗,A∗), which is a known criterion for uniqueness.

This paper contains a list of corrections to the paper ‘ Solutions of SPDEs associated with a stochastic flow’.

Let \((M, \mathcal {F})\) be a foliated compact Riemannian manifold. We consider a family of compatible Feller semigroups in C(Mn) associated to laws of the n-point motion. Under some assumptions (Le Jan and Raimond, Ann. Probab. 32:1247–1315, 2004) there exists a stochastic flow of measurable mappings in M. We study the degeneracy of these semigroups such that the flow of mappings is foliated, i.e. each trajectory lays in a single leaf of the foliation a.s, hence creating a geometrical obstruction for coalescence of trajectories in different leaves. As an application, an averaging principle is proved for a first order perturbation transversal to the leaves. Estimates for the rate of convergence are calculated.