The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, and Roman Urban. "The evolution and Poisson kernels on nilpotent meta-abelian groups." Studia Mathematica 219.1 (2013): 69-96. <http://eudml.org/doc/285719>.

@article{RichardPenney2013,
abstract = {Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^\{k\}$, k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_\{α\} = L^\{a\} + Δ_\{α\}$, where $α ∈ ℝ^\{k\}$, and for each $a ∈ ℝ^\{k\}, L^a$ is left-invariant second order differential operator on N and $Δ_\{α\} = Δ - ⟨α,∇⟩$, where Δ is the usual Laplacian on $ℝ^\{k\}$. 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 $L^\{σ(t)\}$, where σ is a continuous function from [0,∞) to $ℝ^\{k\}$. We also give an upper bound for the Poisson kernel for $ℒ_\{α\}$.},
author = {Richard Penney, Roman Urban},
journal = {Studia Mathematica},
keywords = {Poisson kernel; evolution kernel; left-invariant differential operators; meta-abelian nilpotent Lie groups; solvable Lie groups; Brownian motion},
language = {eng},
number = {1},
pages = {69-96},
title = {The evolution and Poisson kernels on nilpotent meta-abelian groups},
url = {http://eudml.org/doc/285719},
volume = {219},
year = {2013},
}

TY - JOUR
AU - Richard Penney
AU - Roman Urban
TI - The evolution and Poisson kernels on nilpotent meta-abelian groups
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 1
SP - 69
EP - 96
AB - Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$, k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$, where $α ∈ ℝ^{k}$, and for each $a ∈ ℝ^{k}, L^a$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨α,∇⟩$, where Δ is the usual Laplacian on $ℝ^{k}$. 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 $L^{σ(t)}$, where σ is a continuous function from [0,∞) to $ℝ^{k}$. We also give an upper bound for the Poisson kernel for $ℒ_{α}$.
LA - eng
KW - Poisson kernel; evolution kernel; left-invariant differential operators; meta-abelian nilpotent Lie groups; solvable Lie groups; Brownian motion
UR - http://eudml.org/doc/285719
ER -