On minimal parabolic functions and time-homogeneous parabolic h-transforms

Does a minimal harmonic function $h$ remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes $D\subset \R^d$ of variable width and minimal harmonic functions $h$ corresponding to the boundary point of $D$ ``at infinity.'' Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and $f$ is a Lipschitz function. The answer to the question is affirmative if and only if $\int^\infty f^3(u)du = \infty$. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous.