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

Abstract

Does a minimal harmonic function h 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 ⊂ R d D\subset \mathbb {R}^{d} of variable width and minimal harmonic functions h h corresponding to the boundary point of D D “at infinity.” Suppose f ( u ) f(u) is the width of the tube u u units away from its endpoint and f f is a Lipschitz function. The answer to the question is affirmative if and only if ∫ ∞ f 3 ( u ) d u = ∞ \int ^{\infty }f^{3}(u)du = \infty . If the test fails, there exist parabolic h h -transforms of space-time Brownian motion in D D with infinite lifetime which are not time-homogenous.