Estimates for the Poisson kernel and the evolution kernel on nilpotent meta-abelian groups

Abstract

Let $S$ be a semi direct product $S=N⋊A$ where $N$ is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is isomorphic with ${\mathbb{R}}^k,$ $k>1.$ We consider a class of second order left-invariant differential operators on $S$ of the form ${\mathcal{L}}_{\alpha }=L^a+{\Delta }_{\alpha },$ 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 ${\Delta }_{\alpha }=\Delta -<\alpha ,\nabla >,$ where $\Delta$ 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 bound for the Poisson kernel for ${\mathcal{L}}_{\alpha }.$ We also give 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 ${\mathbb{R}}^k.$

28 pages; this is a shorter version; some sections of the previous version (on skew-product formula) have already appeared in print in J. Evol. Equ. 12, No. 2 (2012), 327-351

References in corpus (1)

• Exponential functionals of Brownian motion, I: Probability laws at fixed time