Abstract
LetG be a connected solvable Lie group with abelian derived group and μ a Borel measure onG. We define the μ-harmonic functions as the bounded Borel measurable solutions of the equation:h(g)=∫ G h(gy)μ(dy). We prove that if μ is a spread-out measure, bounded μ-harmonic functions are given by a Poisson formula, where the Poisson boundary is characterized in terms of the asymptotic behaviour of the right random walk of law μ onG. Moreover, we give a complete description of the Poisson boundary for certain groups. The Poisson formula remains essentially valid for adapted μ and left uniformly continuous bounded harmonic functions.
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Cuny, C. Sur les fonctions harmoniques bornées sur les groupes de Lie résolubles connexes et l'existence de mesure invariante. J. Anal. Math. 94, 91–124 (2004). https://doi.org/10.1007/BF02789043
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DOI: https://doi.org/10.1007/BF02789043