Examples of non-local time dependent or parabolic Dirichlet spaces

Niels Jacob

Colloquium Mathematicae (1993)

  • Volume: 65, Issue: 2, page 241-265
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Jacob, Niels. "Examples of non-local time dependent or parabolic Dirichlet spaces." Colloquium Mathematicae 65.2 (1993): 241-265. <http://eudml.org/doc/210218>.

@article{Jacob1993,
abstract = {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.},
author = {Jacob, Niels},
journal = {Colloquium Mathematicae},
keywords = {non-local Dirichlet form; time dependent Dirichlet spaces; Markov process; time dependent Dirichlet forms; pseudodifferential operators},
language = {eng},
number = {2},
pages = {241-265},
title = {Examples of non-local time dependent or parabolic Dirichlet spaces},
url = {http://eudml.org/doc/210218},
volume = {65},
year = {1993},
}

TY - JOUR
AU - Jacob, Niels
TI - Examples of non-local time dependent or parabolic Dirichlet spaces
JO - Colloquium Mathematicae
PY - 1993
VL - 65
IS - 2
SP - 241
EP - 265
AB - 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.
LA - eng
KW - non-local Dirichlet form; time dependent Dirichlet spaces; Markov process; time dependent Dirichlet forms; pseudodifferential operators
UR - http://eudml.org/doc/210218
ER -

References

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  1. [1] C. Berg and G. Forst, Non-symmetric translation invariant Dirichlet forms, Invent. Math. 21 (1973), 199-212. Zbl0263.31010
  2. [2] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergeb. Math. Grenzgeb. (2) 87, Springer, Berlin 1975. Zbl0308.31001
  3. [3] A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 208-215. Zbl0089.08201
  4. [4] J. Bliedtner and W. Hansen, Potential Theory -- An Analytic and Probabilistic Approach to Balayage, Universitext, Springer, Berlin 1986. Zbl0706.31001
  5. [5] M. Brzezina, On a class of translation invariant balayage spaces, Exposition. Math. 11 (1993), 181-184. Zbl0779.31005
  6. [6] J. Deny, Méthodes Hilbertiennes et théorie du potentiel, in: Potential Theory, C.I.M.E., Edizioni Cremonese, 1970, 123-201. 
  7. [7] M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland Math. Library 23, North-Holland, Amsterdam 1980. Zbl0422.31007
  8. [8] W. Hoh, Some commutator estimates for pseudo differential operators with negative definite functions as symbol, Integral Equations Operator Theory, in press. Zbl0795.47033
  9. [9] W. Hoh and N. Jacob, Some Dirichlet forms generated by pseudo differential operators, Bull. Sci. Math. 116 (1992), 383-398. Zbl0769.31007
  10. [10] W. Hoh and N. Jacob, On some translation invariant balayage spaces, Comment. Math. Univ. Carolinae 32 (1991), 471-478. Zbl0755.31012
  11. [11] N. Jacob, Commutator estimates for pseudo differential operators with negative definite functions as symbol, Forum Math. 2 (1990), 155-162. Zbl0693.35168
  12. [12] N. Jacob, Feller semigroups, Dirichlet forms, and pseudo differential operators, ibid. 4 (1992), 433-446. Zbl0759.60078
  13. [13] N. Jacob, A class of elliptic pseudo differential operators generating symmetric Dirichlet forms, Potential Analysis 1 (1992), 221-232. Zbl0784.31007
  14. [14] N. Jacob, Further pseudo differential operators generating Feller semigroups and Dirichlet forms, Rev. Mat. Iberoamericana 9 (2) (1993), in press. Zbl0780.31007
  15. [15] N. Jacob, A class of Feller semigroups generated by pseudo-differential operators, Math. Z., in press. Zbl0795.35154
  16. [16] N. Jacob, A further application of (r,2)-capacities to pseudo-differential operators, submitted. Zbl0831.60084
  17. [17] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundlehren Math. Wissensch. 181, Springer, Berlin 1972. Zbl0223.35039
  18. [18] Z.-M. Ma and M. Röckner, An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Universitext, Springer, Berlin 1992. Zbl0826.31001
  19. [19] U. Mosco, Composite media and asymptotic Dirichlet forms, preprint. 
  20. [20] Y. Oshima, On a construction of Markov processes associated with time dependent Dirichlet spaces, Forum Math. 4 (1992), 395-415. Zbl0759.60081
  21. [21] Y. Oshima, Some properties of Markov processes associated with time dependent Dirichlet forms, Osaka J. Math. 29 (1992), 103-127. Zbl0759.60082
  22. [22] M. Pierre, Problèmes d'évolution avec contraintes unilatérales et potentiels paraboliques, Comm. Partial Differential Equations 4 (1979), 1149-1197. Zbl0426.31005
  23. [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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