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

Abstract

Does a minimal harmonic function hh remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes DβŠ‚RdD\subset \R^d of variable width and minimal harmonic functions hh corresponding to the boundary point of DD ``at infinity.'' Suppose f(u)f(u) is the width of the tube uu units away from its endpoint and ff is a Lipschitz function. The answer to the question is affirmative if and only if ∫∞f3(u)du=∞\int^\infty f^3(u)du = \infty. If the test fails, there exist parabolic hh-transforms of space-time Brownian motion in DD with infinite lifetime which are not time-homogenous

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