In the first three sections of this article I give a new proof of a theorem of Jack Lee which says that if $M$ is a compact strictly pseudoconvex domain with a real-analytic boundary, one can find a defining function on the boundary which satisfies the homogeneous complex Monge-Ampere equation. The proof involves complexifying a solution of a related real Monge-Ampere equation. The rest of this article is devoted to a generalization of a theorem of L. Boutet de Monvel. Boutet's theorem says that if $X$ is a compact manifold equipped with a real-analytic Riemannian metric and $f$ is a real-analytic function of $M$ then the following are equivalent (1) $f$ can be extended holomorphically to a Grauert of radius $r$, about $X$. (2) The diffusion equation, $\frac {\partial u}{\partial t} = \Delta ^{\frac 12}u$, can be solved backwards in time over the interval $-r\leq t\leq 0$ with initial data : $u(0, x) = f (x).$ In the second half of this article I show that this theorem has a generalization in which Grauert tubes are replaced by a family, $\phi = r$, of strictly pseudoconvex domains, $\phi$ satisfying homogeneous Monge-Ampere.