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1993 | 65 | 2 | 241-265
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Examples of non-local time dependent or parabolic Dirichlet spaces

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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.
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  • Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1A, D-8520 Erlangen, Germany
  • [1] C. Berg and G. Forst, Non-symmetric translation invariant Dirichlet forms, Invent. Math. 21 (1973), 199-212.
  • [2] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergeb. Math. Grenzgeb. (2) 87, Springer, Berlin 1975.
  • [3] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208-215.
  • [4] J. Bliedtner and W. Hansen, Potential Theory -- An Analytic and Probabilistic Approach to Balayage, Universitext, Springer, Berlin 1986.
  • [5] M. Brzezina, On a class of translation invariant balayage spaces, Exposition. Math. 11 (1993), 181-184.
  • [6] J. Deny, Méthodes Hilbertiennes et théorie du potentiel, in: Potential Theory, C.I.M.E., Edizioni Cremonese, 1970, 123-201.
  • [7] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Math. Library 23, North-Holland, Amsterdam 1980.
  • [8] W. Hoh, Some commutator estimates for pseudo differential operators with negative definite functions as symbol, Integral Equations Operator Theory, in press.
  • [9] W. Hoh and N. Jacob, Some Dirichlet forms generated by pseudo differential operators, Bull. Sci. Math. 116 (1992), 383-398.
  • [10] W. Hoh and N. Jacob, On some translation invariant balayage spaces, Comment. Math. Univ. Carolinae 32 (1991), 471-478.
  • [11] N. Jacob, Commutator estimates for pseudo differential operators with negative definite functions as symbol, Forum Math. 2 (1990), 155-162.
  • [12] N. Jacob, Feller semigroups, Dirichlet forms, and pseudo differential operators, ibid. 4 (1992), 433-446.
  • [13] N. Jacob, A class of elliptic pseudo differential operators generating symmetric Dirichlet forms, Potential Analysis 1 (1992), 221-232.
  • [14] N. Jacob, Further pseudo differential operators generating Feller semigroups and Dirichlet forms, Rev. Mat. Iberoamericana 9 (2) (1993), in press.
  • [15] N. Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z., in press.
  • [16] N. Jacob, A further application of (r,2)-capacities to pseudo-differential operators, submitted.
  • [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundlehren Math. Wissensch. 181, Springer, Berlin 1972.
  • [18] Z.-M. Ma and M. Röckner, An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, Berlin 1992.
  • [19] U. Mosco, Composite media and asymptotic Dirichlet forms, preprint.
  • [20] Y. Oshima, On a construction of Markov processes associated with time dependent Dirichlet spaces, Forum Math. 4 (1992), 395-415.
  • [21] Y. Oshima, Some properties of Markov processes associated with time dependent Dirichlet forms, Osaka J. Math. 29 (1992), 103-127.
  • [22] M. Pierre, Problèmes d'évolution avec contraintes unilatérales et potentiels paraboliques, Comm. Partial Differential Equations 4 (1979), 1149-1197.
  • [23] M. Pierre, Représentant précis d'un potentiel parabolique, in: Sém. Théorie du Potentiel, Lecture Notes in Math. 814, Springer, Berlin 1980, 186-228.
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