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1
Content available remote

Noncoercive differential operators on homogeneous manifolds of negative curvature and their Green functions

100%
Urban R.
Colloquium Mathematicum
|
2001
|
tom 88
|
nr 1
121-134
EN
We obtain upper and lower estimates for the Green function for a second order noncoercive differential operator on a homogeneous manifold of negative curvature.
2
Content available remote

Corrigendum to "On density modulo 1 of some expressions containing algebraic integers" (Acta Arith. 127 (2007), 217-229)

100%
Urban R.
Acta Arithmetica
|
2010
|
tom 141
|
nr 2
209-210
3
Content available remote

On density modulo 1 of some expressions containing algebraic integers

100%
Urban R.
Acta Arithmetica
|
2007
|
tom 127
|
nr 3
217-229
4
Content available remote

Some remarks on the random walk on finite groups

100%
Urban R.
Colloquium Mathematicum
|
1997
|
tom 74
|
nr 2
287-298
5
Content available remote

Semigroup actions on tori and stationary measures on projective spaces

64%
Guivarc'h Y., Urban R.
Studia Mathematica
|
2005
|
tom 171
|
nr 1
33-66
EN
Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on $ℝ^{d}$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on $ℝ^{d}$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space $ℙ^{d-1}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits on $𝕋^{d} = ℝ^{d}/ℤ^{d}$ are finite or dense.
6
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The evolution and Poisson kernels on nilpotent meta-abelian groups

64%
Penney R., Urban R.
Studia Mathematica
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2013
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tom 219
|
nr 1
69-96
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 $ℒ_{α}$.
7
Content available remote

Corrigendum to the paper "Semigroup actions on tori and stationary measures on projective spaces" (Studia Math. 171 (2005), 33-66)

64%
Guivarc'h Y., Urban R.
Studia Mathematica
|
2007
|
tom 183
|
nr 2
195-196
8
Content available remote

Estimates for the Poisson kernel on higher rank NA groups

64%
Penney R., Urban R.
Colloquium Mathematicum
|
2010
|
tom 118
|
nr 1
259-281
EN
We obtain an estimate for the Poisson kernel for the class of second order left-invariant differential operators on higher rank NA groups.
9
Content available remote

Unbounded harmonic functions on homogeneous manifolds of negative curvature

64%
Penney R., Urban R.
Colloquium Mathematicum
|
2002
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tom 91
|
nr 1
99-121
EN
We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems answering partially the following question. Is a given harmonic function the Poisson integral of "something" from the boundary N?
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