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

Abstract

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 [is an element of the subset of real numbers to the power of] 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 [definite integral to the power of infinity] f [to the power of] 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.