Skew products, regular conditional probabilities and stochastic differential equations : a technical remark
On the existence of resolvents
Some remarks on Malliavin's comparison lemma and related topics
On Deny's characterization of the potential kernel for a convolution Feller semi-group
The convolution kernels on a homogeneous space , where is a compact sub-group of , that satisfy the complete maximum principle are characterized. Deny’s result for abelian groups , but with a stronger hypothesis, is a special case.
Duality and the Martin compactification
Let be a Bauer sheaf that admits a Green function. Then there exists a diffusion process corresponding to the sheaf whose resolvent possesses a Hunt-Kunita-Watanabe dual resolvent that comes from a diffusion process. If is a Brelot sheaf which possesses an adjoint sheaf the dual process corresponds to . The Martin compactification defined by a Brelot sheaf that admits a Green function coincides with a Kunita-Watanabe compactification defined by the dual resolvent.
On the Martin compactification of a bounded Lipschitz domain in a riemannian manifold
The Martin compactification of a bounded Lipschitz domain is shown to be for a large class of uniformly elliptic second order partial differential operators on . Let be an open Riemannian manifold and let be open relatively compact, connected, with Lipschitz boundary. Then is the Martin compactification of associated with the restriction to of the Laplace-Beltrami operator on . Consequently an open Riemannian manifold has at most one compactification which is a compact...
The Martin boundaries of equivalent sheaves
The Martin compactification of defined by a Brelot sheaf satisfying proportionality is shown to be the same as for if the sheaves agree outside a compact set. Minimal points coincide and hence and are isomorphic topological cones. Nakai’s result on the extension to of a function harmonic outside a compact set is extended to Bauer’s theory. The connected components of the Martin boundary correspond to the ends of which are related to direct decomposition of the cone .
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