  We consider the following stochastic partial differential equation,
\begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=\psi,
\end{align*} associated with a stochastic flow $\{X(t,x)\}$, for $t \geq 0$, $x
\in \mathbb{R}^d$, as in [Rajeev \& Thangavelu, \emph{{Probabilistic
representations of solutions of the forward
  equations}}, Potential Anal. \textbf{28} (2008), no.~2, 139--162]. 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
$\mathcal{S}^\prime$. We show uniqueness of solutions in the case when $\psi$
is smooth via the `monotonicity inequality' for $(L^\ast,A^\ast)$, which is a
known criterion for uniqueness.
