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

Burdzy, Krzysztof; Salisbury, Thomas

doi:10.1090/S0002-9947-99-02471-X

On minimal parabolic functions and time-homogeneous parabolic $h$-transforms
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by Krzysztof Burdzy and Thomas S. Salisbury PDF

Trans. Amer. Math. Soc. 351 (1999), 3499-3531 Request permission

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 $D\subset \mathbb {R}^{d}$ 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 $\int ^{\infty }f^{3}(u)du = \infty$. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous.

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