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Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space

  • Martin Heida
  • Robert I. A. Patterson
  • D. R. Michiel Renger
Article
  • 27 Downloads

Abstract

We study functions of bounded variation with values in a Banach or in a metric space. In finite dimensions, there are three well-known topologies; we argue that in infinite dimensions there is a natural fourth topology. We provide some insight into the structure of these four topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin–Lions theorem. After this we study the Borel \(\sigma \)-algebras induced by these topologies, and we provide some results about probability measures on the space of functions of bounded variation, which can be used to study stochastic processes of bounded variation.

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Notes

Acknowledgements

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 “Scaling Cascades in Complex Systems”, Projects C08 “Stochastic Spatial Coagulation Particle Processes” and C05 “Effective Models for Materials and Interfaces with Multiple Scales”.

References

  1. 1.
    L. Ambrosio. Metric space valued functions of bounded variation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, \(4^e\) série, 17(3):439–478, 1990.Google Scholar
  2. 2.
    L. Ambrosio and S. Di Marino. Equivalent definitions of BV space and of total variation on metric measure spaces. Journal of Functional Analysis, 266(7):4150–4188, 2014.MathSciNetCrossRefGoogle Scholar
  3. 3.
    L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, NY, USA, 2006.zbMATHGoogle Scholar
  4. 4.
    L. Ambrosio and R. Ghezzi. Sobolev and bounded variation functions on metric measure spaces (lecture notes). http://cvgmt.sns.it/paper/2738/, 2016.
  5. 5.
    L. Ambrosio, R. Ghezzi, and V. Magnani. BV functions and sets of finite perimeter in sub-Riemannian manifolds. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire, 32(3):489–517, 2015.MathSciNetCrossRefGoogle Scholar
  6. 6.
    L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Birkhauser, Basel, Switzerland, 2nd edition, 2008.zbMATHGoogle Scholar
  7. 7.
    L. Bertini, A. Faggionato, and D. Gabrielli. Large deviations of the empirical flow for continuous time markov chains. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, 51(3):867–900, 2015.MathSciNetCrossRefGoogle Scholar
  8. 8.
    V. I. Bogachev. Measure theory. Vol. I & II. Springer-Verlag, Berlin, Germany, 2007.CrossRefGoogle Scholar
  9. 9.
    H. Brézis. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973.zbMATHGoogle Scholar
  10. 10.
    D. Chiron. On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Communications in Contemporary Mathematics, 9(4):473, 2007.MathSciNetCrossRefGoogle Scholar
  11. 11.
    V. V. Chistyakov. Selections of bounded variation. Journal of Applied Analysis, 10(1):1–82, 2004.MathSciNetCrossRefGoogle Scholar
  12. 12.
    J.B. Conway. A course in functional analysis. Springer, New York, NY, USA, 2nd edition, 2007.CrossRefGoogle Scholar
  13. 13.
    S. Di Marino. Sobolev and BV spaces on metric measure spaces via derivations and integration by parts. http://cvgmt.sns.it/paper/2521, 2014.
  14. 14.
    J. Diestel and J. J. Jr. Uhl. Vector Measures, volume 95. American Mathematical Society, Providence, RI, 1967.Google Scholar
  15. 15.
    N. Dinculeanu. Vector Measures. Pergamon press / Deutscher Verlag der Wissenschaften, Berlin, Germany, 1967.CrossRefGoogle Scholar
  16. 16.
    N. Dunford and J.T. Schwartz. Linear operators, part one: general theory. Interscience, New York, NY, USA, 1957.Google Scholar
  17. 17.
    L.C. Evans and R.F. Gariepy. Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, USA, 1992.zbMATHGoogle Scholar
  18. 18.
    H. Federer. Geometric measure theory. Springer, Berlin, Germany, 1969.zbMATHGoogle Scholar
  19. 19.
    A. Jakubowski. A non-Skorokhod topology on the Skorokhod space. Electronic Journal of Probability, 2(4):1–21, 1997.MathSciNetzbMATHGoogle Scholar
  20. 20.
    P. Krejčí. Regulated evolution quasivariational inequalities. Notes to Lectures held at the University of Pavia, 2003.Google Scholar
  21. 21.
    P. Krejčí. The Kurzweil integral and hysteresis. In Journal of Physics: Conference Series, volume 55, page 144. IOP Publishing, 2006.Google Scholar
  22. 22.
    P. Logaritsch and E. Spadaro. A representation formula for the p-energy of metric space-valued Sobolev maps. Communications in Contemporary Mathematics, 14(6):1250043, 2012.MathSciNetCrossRefGoogle Scholar
  23. 23.
    T.-W. Ma. Banach-Hilbert Spaces, Vector Measures and Group Representations. World Scientific, Singapore, 2002.Google Scholar
  24. 24.
    A. Mainik and A. Mielke. Existence results for energetic models for rate-independent systems. Calculus of Variations and Partial Differential Equations, 22(1):73–99, 2005.MathSciNetCrossRefGoogle Scholar
  25. 25.
    P.A. Meyer and W.A. Zheng. Tightness criteria for laws of semimartingales. Annales de l’I.H.P., section B, 20(4):353–372, 1984.MathSciNetzbMATHGoogle Scholar
  26. 26.
    A. Mielke and T. Roubicek. Rate-Independent Systems. Springer, Berlin, Germany, 2015.CrossRefGoogle Scholar
  27. 27.
    A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rational Mechanics and Analysis, 162(2):137–177, 2012.MathSciNetCrossRefGoogle Scholar
  28. 28.
    J. J. Moreau, P. D. Panagiotopoulos, and G. Strang. Topics in nonsmooth mechanics. Birkhäuser, Basel, Switzerland, 1988.zbMATHGoogle Scholar
  29. 29.
    R. I. A. Patterson and D. R. M. Renger. Large deviations of reaction fluxes. arXiv:1802.02512, 2018.
  30. 30.
    V. Recupero. BV solutions of rate independent variational inequalities. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 10(2):269, 2011.MathSciNetzbMATHGoogle Scholar
  31. 31.
    V. Recupero. Hysteresis operators in metric spaces. Discrete Contin. Dyn. Syst. Ser. S, 8:773–792, 2015.MathSciNetCrossRefGoogle Scholar
  32. 32.
    W. Rudin. Functional Analysis. McGraw-Hill, New York, NY, 1973.zbMATHGoogle Scholar
  33. 33.
    J. Simon. Compact sets in the space \(L^p(0,T;B)\). Annali di Matematica pura ed applicata, 146(1):65–96, 1986.CrossRefGoogle Scholar
  34. 34.
    A. Visintin. Differential models of hysteresis. Springer Science & Business Media, Berlin, Germany, 2013.zbMATHGoogle Scholar
  35. 35.
    A. Wiweger. Linear spaces with mixed topology. Studia Mathematica, 20(1):47–68, 1961.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Weierstrass InstituteBerlinGermany

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