The evolution and Poisson kernels on nilpotent meta-abelian groups
Volume 219 / 2013
Abstract
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 $\mathcal L_\alpha.$
