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Examples of non-local time dependent or parabolic Dirichlet spaces

Niels Jacob — 1993

Colloquium Mathematicae

In [23] M. Pierre introduced parabolic Dirichlet spaces. Such spaces are obtained by considering certain families ( E ( τ ) ) τ of Dirichlet forms. He developed a rather far-reaching and general potential theory for these spaces. In particular, he introduced associated capacities and investigated the notion of related quasi-continuous functions. However, the only examples given by M. Pierre in [23] (see also [22]) are Dirichlet forms arising from strongly parabolic differential operators of second order. To...

Further pseudodifferential operators generating Feller semigroups and Dirichlet forms.

Niels Jacob — 1993

Revista Matemática Iberoamericana

We prove for a large class of symmetric pseudo differential operators that they generate a Feller semigroup and therefore a Dirichlet form. Our construction uses the Yoshida-Hille-Ray Theorem and a priori estimates in anisotropic Sobolev spaces. Using these a priori estimates it is possible to obtain further information about the stochastic process associated with the Dirichlet form under consideration.

On some translation invariant balayage spaces

Walter HohNiels Jacob — 1991

Commentationes Mathematicae Universitatis Carolinae

It is well known that strong Feller semigroups generate balayage spaces provided the set of their excessive functions contains sufficiently many elements. In this note, we give explicit examples of strong Feller semigroups which do generate balayage spaces. Further we want to point out the role of the generator of the semigroup in the related potential theory.

Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces

We introduce and systematically investigate Bessel potential spaces associated with a real-valued continuous negative definite function. These spaces can be regarded as (higher order) L p -variants of translation invariant Dirichlet spaces and in general they are not covered by known scales of function spaces. We give equivalent norm characterizations, determine the dual spaces and prove embedding theorems. Furthermore, complex interpolation spaces are calculated. Capacities are introduced and the...

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