  We consider a \emph{family} $(P_\omega)_{\omega \in \Omega}$ of elliptic
second order differential operators on a domain $U_0 \subset \mathbb{R}^m$
whose coefficients depend on the space variable $x \in U_0$ and on $\omega \in
\Omega,$ a probability space. We allow the coefficients $a_{ij}$ of $P_\omega$
to have jumps over a fixed interface $\Gamma \subset U_0$ (independent of
$\omega \in \Omega$). We obtain polynomial in the norms of the coefficients
estimates on the norm of the solution $u_\omega$ to the equation $P_\omega
u_\omega = f$ with transmission and mixed boundary conditions (we consider
``sign-changing'' problems as well). In particular, we show that, if $f$ and
the coefficients $a_{ij}$ are smooth enough and follow a log-normal-type
distribution, then the map $\Omega \ni \omega \to \|u_\omega\|_{H^{k+1}(U_0)}$
is in $L^p(\Omega)$, for all $1 \le p < \infty$. The same is true for the norms
of the inverses of the resulting operators. We expect our estimates to be
useful in Uncertainty Quantification.
