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 DβRd 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 β«βf3(u)du=β. If the test fails, there exist
parabolic h-transforms of space-time Brownian motion in D with infinite
lifetime which are not time-homogenous