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Heat kernel estimates for the ¯ ∂Neumann problem on Gmanifolds
, 2011
"... We prove heat kernel estimates for the ¯ ∂Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated LaplaceBeltrami operator on functions. ..."
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Cited by 3 (3 self)
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We prove heat kernel estimates for the ¯ ∂Neumann Laplacian □ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated LaplaceBeltrami operator on functions.
Essential selfadjointness, generalized eigenforms, and . . .
, 2011
"... www.elsevier.com/locate/jfa ..."
On the heat kernel and the Dirichlet form of Liouville Brownian Motion
"... In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville qua ..."
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In [14], a Feller process called Liouville Brownian motion on R2 has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field eγ X and is the right diffusion process to consider regarding 2dLiouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially [13] and the techniques introduced in [14]. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a nontrivial theorem of Fukushima and al. One of the motivations which led to introduce the Liouville Brownian motion in [14] was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. In particular, the theory developed for example in [28, 29, 30], whose aim is to capture the “geometry ” of the underlying space out of the Dirichlet form of a process living on that space, suggests a notion of distance associated to a Dirichlet form. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide a distance in the wide sense, called intrinsic metric, which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.
Geometry and Analysis of Dirichlet forms (II)
"... Abstract Given a regular, strongly local Dirichlet form E, under the lower bound of the Ricci curvature of BakryEmery, doubling and Poincaré, we obtain that (i) the intrinsic differential and distance structures of E coincide; (ii) Cheeger energy functional ChdE is a quadratic norm. This shows tha ..."
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Abstract Given a regular, strongly local Dirichlet form E, under the lower bound of the Ricci curvature of BakryEmery, doubling and Poincaré, we obtain that (i) the intrinsic differential and distance structures of E coincide; (ii) Cheeger energy functional ChdE is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature by AmbrosioGigliSavaré. Dirichlet forms are constructed to show that doubling and Poincare ́ are not enough to obtain either (i) or (ii) above. 1
L∞Variational Problem Associated to Dirichlet forms
"... Abstract We study the L∞variational problem associated to a general regular, strongly local Dirichlet form. We show that the intrinsic distance determines the absolute minimizer (infinite harmonic function) of the corresponding L∞functional. This leads to the existence and uniqueness of the absol ..."
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Abstract We study the L∞variational problem associated to a general regular, strongly local Dirichlet form. We show that the intrinsic distance determines the absolute minimizer (infinite harmonic function) of the corresponding L∞functional. This leads to the existence and uniqueness of the absolute minimizer on a bounded domain, given a continuous boundary data. Applying this, we also obtain that an infinity harmonic function on Rn may be the minimizer for several different variational problems. Finally, we apply our results to CarnotCarathéodory spaces. 1
Essential . . . EIGENFORMS, AND SPECTRA FOR THE ¯∂NEUMANN PROBLEM ON GMANIFOLDS.
, 2011
"... Let M be a complex manifold with boundary, satisfying a subelliptic estimate, which is also the total space of a principal G–bundle with G a Lie group and compact orbit space M — /G. Here we investigate the ¯ ∂Neumann Laplacian □ on M. We show that it is essentially selfadjoint on its restrictio ..."
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Let M be a complex manifold with boundary, satisfying a subelliptic estimate, which is also the total space of a principal G–bundle with G a Lie group and compact orbit space M — /G. Here we investigate the ¯ ∂Neumann Laplacian □ on M. We show that it is essentially selfadjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to σ(□) if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well–behaved eigenforms so that, spectrally, almost every energy comes with such a generalized eigenform.