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 of 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 integral(infinity) f(3) (u)du = infinity. If the test fails, there exist parabolic h-transforms of space-time Brownian motion in D with infinite lifetime which are not time-homogenous.