The evolution and Poisson kernels on nilpotent meta-abelian groups

• Autorzy: Richard Penney , Roman Urban
• Języki publikacji: EN

Abstrakty

EN

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 $ℒ_{α}$.

Kategorie tematyczne

• 22E25: Nilpotent and solvable Lie groups
• 60J60: Diffusion processes
• 31B05: Harmonic, subharmonic, superharmonic functions
• 60J25: Continuous-time Markov processes on general state spaces
• 22E30: Analysis on real and complex Lie groups
• 43A85: Analysis on homogeneous spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2013
• Tom: 219
• Numer: 1
• Strony: 69-96

Daty

wydano

2013

Twórcy

autor

Richard Penney

• Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, U.S.A.

autor

Roman Urban

• Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Identyfikatory

DOI

10.4064/sm219-1-4