Examples of non-local time dependent or parabolic Dirichlet spaces

In [23] M. Pierre introduced parabolic Dirichlet spaces. Such spaces are obtained by considering certain families ${\displaystyle {\left(E^{(\tau )}\right)}_{\tau \in ℝ}}$ 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.