The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney ¹   ; Roman Urban ²

¹ Department of Mathematics Purdue University 150 N. University St. West Lafayette, IN 47907, U.S.A.
² Institute of Mathematics Wrocław University Pl. Grunwaldzki 2/4 50-384 Wrocław, Poland

Studia Mathematica, Tome 219 (2013) no. 1, pp. 69-96

Résumé

Let $S$ be a semidirect product $S=N\rtimes A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic to $\mathbb{R}^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form $\mathcal L_\alpha=L^a+\varDelta_\alpha,$ where $\alpha\in\mathbb{R}^k,$ and for each $a\in\mathbb{R}^k,$ $L^a$ is left-invariant second order differential operator on $N$ and $\varDelta_\alpha=\varDelta-\langle\alpha,\nabla\rangle,$ where $\varDelta$ is the usual Laplacian on $\mathbb{R}^k.$ Using some probabilistic techniques (e.g., skew-product formulas for diffusions on $S$ and $N$ respectively) we obtain an upper estimate for the transition probabilities of the evolution on $N$ generated by $L^{\sigma(t)},$ where $\sigma$ is a continuous function from $[0,\infty)$ to $\mathbb R^k.$ We also give an upper bound for the Poisson kernel for $\mathcal L_\alpha.$

Détail

DOI : 10.4064/sm219-1-4

Keywords: semidirect product rtimes where connected simply connected non abelian nilpotent meta abelian lie group isomorphic mathbb consider class second order left invariant differential operators form mathcal alpha vardelta alpha where alpha mathbb each mathbb left invariant second order differential operator vardelta alpha vardelta langle alpha nabla rangle where vardelta usual laplacian mathbb using probabilistic techniques skew product formulas diffusions respectively obtain upper estimate transition probabilities evolution generated sigma where sigma continuous function infty mathbb upper bound poisson kernel mathcal alpha