Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 23 (2013), No. 3, 655--668Copyright Heldermann Verlag 2013 Upper Bound for the Heat Kernel on Higher-Rank NA Groups Richard Penney Dept. of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, U.S.A. rcp@math.purdue.edu Roman Urban Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland urban@math.uni.wroc.pl [Abstract-pdf] \def\R{{\Bbb R}} Let $S$ be a semi-direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with $\R^k,$ $k>1$. We consider a class of second order left-invariant differential operators ${\cal L}_\alpha$, $\alpha\in\R^k$, on $S$. We obtain an upper bound for the heat kernel for ${\cal L}_\alpha$. Keywords: Heat kernel, left invariant differential operators, meta-abelian nilpotent Lie groups, solvable Lie groups, homogeneous groups, higher rank $NA$ groups, Brownian motion, exponential functionals of Brownian motion. MSC: 43A85, 31B05, 22E25, 22E30, 60J25, 60J60 [ Fulltext-pdf  (327  KB)] for subscribers only.