Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

Examples of non-local time dependent or parabolic Dirichlet spaces

100%

Jacob N.

Colloquium Mathematicum

1993

tom 65

nr 2

241-265

In [23] M. Pierre introduced parabolic Dirichlet spaces. Such spaces are obtained by considering certain families $(E^{(τ)})_{τ ∈ ℝ}$ 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 our knowledge, only very recently, when Y. Oshima in [20] was able to construct a Markov process associated with a time dependent or parabolic Dirichlet space, these parabolic Dirichlet spaces attracted the attention of probabilists. The proof of the existence of such a Markov process depends much on the potential theory developed by M. Pierre. Moreover, in [21] Y. Oshima proved that a lot of results valid for symmetric Dirichlet spaces (see [7] as a standard reference) also hold for time dependent Dirichlet spaces. The purpose of this note is to give some concrete examples of time dependent Dirichlet spaces which are generated by pseudo-differential operators and therefore are non-local. In Section 1 we recall the basic definition of a time dependent Dirichlet space and in Section 2 we give some auxiliary results. Sections 3-5 are devoted to examples. In Section 3 we discuss some spatially translation invariant operators. We do not really give there any surprising examples, but we emphasize the relation to the theory of balayage spaces. In Section 4 we consider time dependent Dirichlet spaces constructed from a special class of symmetric pseudo-differential operators analogous to those handled in our joint paper [9] with W. Hoh. Finally, in Section 5 we construct time dependent generators of (symmetric) Feller semigroups following [15]. The estimates used in this construction already ensure that we get non-local time dependent Dirichlet spaces. We would like to mention that non-local Dirichlet forms have recently been investigated by U. Mosco [19] in his study of composite media.

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

55%

Farkas W., Jacob N., Schilling R.

Instytut Matematyczny Polskiej Akademii Nauk

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 existence of quasi-continuous modifications is shown.

Ograniczanie wyników

1 Colloquium Mathematicum

2 Jacob N.

1 Farkas W.

1 Schilling R.

1 2001

1 1993

Wyniki wyszukiwania

Examples of non-local time dependent or parabolic Dirichlet spaces

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