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On the relation between elliptic and parabolic Harnack inequalities
, 2001
"... We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in que ..."
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Cited by 66 (6 self)
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We show that, if a certain Sobolev inequality holds, then a scaleinvariant elliptic Harnack inequality suces to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suces to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on M , (i.e., for @ t + ) and elliptic Harnack inequality for @ 2 t + on R M . 1
Heat kernel on connected sums of Riemannian manifolds
 MR1713132 (2001b:58041), Zbl 0957.58023
, 1999
"... This note is about the heat kernel on a connected sum M of noncompact manifolds M1,M2,...,Mk assuming that one knows enough about the heat kernels for each Mi individually (which is the case when Mi are complete manifolds of nonnegative Ricci curvature). We announce here matching uniform upper an ..."
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This note is about the heat kernel on a connected sum M of noncompact manifolds M1,M2,...,Mk assuming that one knows enough about the heat kernels for each Mi individually (which is the case when Mi are complete manifolds of nonnegative Ricci curvature). We announce here matching uniform upper and
Asymptotic separation for independent trajectories of Markov processes
"... Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ffprocess : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 7.3 Random walk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 8 Proofs 24 8.1 ..."
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Cited by 1 (0 self)
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Markov processes 15 7 Examples 18 7.1 Diffusions on manifolds : : : : : : : : : : : : : : : : : : : : : : : : : : 19 7.2 The ffprocess : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 7.3 Random walk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 23 8 Proofs 24 8.1 Intersections of trajectories with covering balls : : : : : : : : : : : : : 24 8.2 Hitting probability and Green kernel : : : : : : : : : : : : : : : : : : 27 8.3 Asymptotic separation in terms of the Green kernel : : : : : : : : : : 28 8.4 Asymptotic separation for two trajectories : : : : : : : : : : : : : : : 31 Supported by the EPSRC Research Fellowship B/94/AF/1782 y Partially supported by the EPSRC Visiting Fellowship GR/M61573 1 8.5 Asymptotic separation for n trajectories : : : : : : : : : : : : : : : : 33 1
Dedicated to E.M.Landis
"... This paper is a survey of some recent results on the heat kernel of a noncompact complete Riemannian manifold. The fast progress during the last 1015 years has turned this field into a welldeveloped theory which on the one hand has its own traditions and stimulations for further investigations an ..."
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This paper is a survey of some recent results on the heat kernel of a noncompact complete Riemannian manifold. The fast progress during the last 1015 years has turned this field into a welldeveloped theory which on the one hand has its own traditions and stimulations for further investigations and, on the other hand, is connected heavily with adjacent spheres of geometry, potential theory, partial differential equations, theory