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The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds
, 2008
"... The Lie group SU(2) endowed with its canonical subriemannian structure appears as a threedimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities. ..."
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The Lie group SU(2) endowed with its canonical subriemannian structure appears as a threedimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.
Teichmann: OrnsteinUhlenbeck processes on Lie groups
 J. Funct. Analysis
"... Abstract. We consider OrnsteinUhlenbeck processes (OUprocesses) associated to hypoelliptic diffusion processes on finitedimensional Lie groups: let L be a hypoelliptic, leftinvariant “sum of the squares”operator on a Lie group G with associated Markov process X, then we construct OUprocesses b ..."
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Abstract. We consider OrnsteinUhlenbeck processes (OUprocesses) associated to hypoelliptic diffusion processes on finitedimensional Lie groups: let L be a hypoelliptic, leftinvariant “sum of the squares”operator on a Lie group G with associated Markov process X, then we construct OUprocesses by adding negative horizontal gradient drifts of functions U. In the natural case U(x) = − log p(1, x), where p(1, x) is the density of the law of X starting at identity e at time t = 1 with respect to the rightinvariant Haar measure on G, we show the Poincaré inequality by applying the DriverMelcher inequality for “sum of the squares ” operators on Lie groups. The resulting Markov process is called the natural OUprocess associated to the hypoelliptic diffusion on G. We prove the global strong existence of these OUtype processes on G under an integrability assumption on U. The Poincaré inequality for a large class of potentials U is then shown by a perturbation technique. These results are applied to obtain a hypoelliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces M. 1.
MALLIAVIN CALCULUS FOR LIE GROUPVALUED WIENER FUNCTIONS
, 2005
"... Abstract. Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξ on Wiener space into G via the stochastic version of Cartan’s rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differenti ..."
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Cited by 1 (1 self)
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Abstract. Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξ on Wiener space into G via the stochastic version of Cartan’s rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differentiable to all orders and these derivatives belong to L p (µ) for all p> 1, where µ is Wiener measure.
Gradient Estimates for the Subelliptic Heat Kernel on Htype Groups
, 904
"... We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of Htype: ∇Ptf  ≤ KPt(∇f) where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of H.Q. Li [10] for ..."
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We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of Htype: ∇Ptf  ≤ KPt(∇f) where Pt is the heat semigroup corresponding to the sublaplacian on G, ∇ is the subelliptic gradient, and K is a constant. This extends a result of H.Q. Li [10] for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafaï [3].