Skip to main content
SpringerLink
Log in

On Skorokhod Differentiable Measures

  • Published:
Ukrainian Mathematical Journal Aims and scope

We present a survey of the Skorohod differentiability of measures on linear spaces, which also gives new proofs of some key results in this area parallel with some new observations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Airault and P. Malliavin, “Intégration géometrique sur l’espace de Wiener,” Bull. Sci. Math., 112, No. 1, 3–52 (1988).

    MathSciNet  MATH  Google Scholar 

  2. A. D. Alexandroff, “Additive set functions in abstract spaces,” Rec. Math. [Mat. Sbornik] N. S., 8, 307–348 (1940); 9, 563–628 (1941); 13, 169–238 (1943).

  3. L. Ambrosio and S. Di Marino, “Equivalent definitions of BV space and of total variation on metric measure spaces,” J. Funct. Anal., 266, 4150–4188 (2014).

    Article  MathSciNet  Google Scholar 

  4. L. Ambrosio, M. Miranda (jr.), S. Maniglia, and D. Pallara, “BV functions in abstract Wiener spaces,” J. Funct. Anal., 258, No. 3, 785–813 (2010).

  5. V. I. Averbuh, O. G. Smolyanov, and S. V. Fomin, “Generalized functions and differential equations in linear spaces. I, Differentiable measures,” Trudy Moskov. Mat. Obshch., 24, 133–174 (1971); English translation: Trans. Moscow Math. Soc., 24, 140–184 (1971).

  6. V. I. Bogachev, “Negligible sets in locally convex spaces,” Mat. Zametki, 36, No. 1, 51–64 (1984); English translation: Math. Notes, 36, 519–526 (1984).

  7. V. I. Bogachev, “Some results on differentiable measures,” Mat. Sb., 127, 336–351 (1985); English translation: Math. USSR Sb., 55, No. 2, 335–349 (1986).

  8. V. I. Bogachev, “On Skorokhod differentiability of measures,” Teor. Veroyatn. Primen., 33, No. 2, 349–354 (1988); English translation: Theory Probab. Appl., 33, 330–334 (1988).

  9. V. I. Bogachev, “Subspaces of differentiability of smooth measures on infinite-dimensional spaces,” Dokl. Akad. Nauk SSSR, 299, No. 1, 18–22 (1988); English translation: Sov. Math. Dokl., 37, No. 2, 304–308 (1988).

  10. V. I. Bogachev, “Gaussian measures,” Amer. Math. Soc., Providence, RI (1998).

    Book  Google Scholar 

  11. V. I. Bogachev, Measure Theory, Vols. 1, 2, Springer, Berlin (2007).

  12. V. I. Bogachev, “Differentiable measures and the Malliavin calculus,” Amer. Math. Soc., Providence, RI (2010).

    Book  Google Scholar 

  13. V. I. Bogachev, “Sobolev classes on infinite-dimensional spaces,” Geom. Meas. Theory Real Anal., 17, 1–56 (2014).

    MathSciNet  MATH  Google Scholar 

  14. V. I. Bogachev, “Distributions of polynomials on multidimensional and infinite-dimensional spaces with measures,” Uspekhi Mat. Nauk, 71, No. 4, 107–154 (2016); English translation: Russian Math. Surveys, 71, No. 4, 703–749 (2016).

  15. V. I. Bogachev, “Surface measures in infinite-dimensional spaces,” in: Measure Theory in Non-Smooth Spaces, De Gruyter Open, Warsaw (2017), pp. 52–97.

  16. V. I. Bogachev, “Weak convergence of measures,” Amer. Math. Soc., Providence, RI (2018).

    Book  Google Scholar 

  17. V. I. Bogachev, E. D. Kosov, and S. N. Popova, “A characterization of Nikolskii–Besov classes via integration by parts,” Dokl. Akad. Nauk, 476, No. 3, 251–255 (2017); English translation: Dokl. Math., 96, No. 2, 449–453 (2017).

  18. V. I. Bogachev, E. D. Kosov, and S. N. Popova, “On Gaussian Nikolskii–Besov classes,” Dokl. Akad. Nauk, 476, No. 6, 609–613 (2017); English translation: Dokl. Math., 96, No. 2, 498–502 (2017).

  19. V. I. Bogachev, E. D. Kosov, and S. N. Popova, “A new approach to Nikolskii–Besov classes,” Moscow Math. J., 19, No. 4, 619–654 (2019).

    Article  MathSciNet  Google Scholar 

  20. V. I. Bogachev, E. D. Kosov, and G. I. Zelenov, “Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality,” Trans. Amer. Math. Soc., 370, No. 6, 4401–4432 (2018).

    Article  MathSciNet  Google Scholar 

  21. V. I. Bogachev and I. I. Malofeev, “Surface measures generated by differentiable measures,” Potential Anal., 44, No. 4, 767–792 (2016).

    Article  MathSciNet  Google Scholar 

  22. V. I. Bogachev, A.Yu. Pilipenko, and E. A. Rebrova, “Classes of functions of bounded variation on infinite-dimensional domains,” Dokl. Akad. Nauk, 451, No. 2, 127–131 (2013); English translation: Dokl. Math., 88, No. 1, 391–395 (2013).

  23. V. I. Bogachev, A.Yu. Pilipenko, and A.V. Shaposhnikov, “Sobolev functions on infinite-dimensional domains,” J. Math. Anal. Appl., 419, 1023–1044 (2014).

    Article  MathSciNet  Google Scholar 

  24. V. I. Bogachev and E. A. Rebrova, “Functions of bounded variation on infinite-dimensional spaces with measures,” Dokl. Akad. Nauk, 449, No. 2, 131–135 (2013); English translation: Dokl. Math., 87, No. 2, 144–147 (2013).

  25. V. I. Bogachev and O. G. Smolyanov, “Analytic properties of infinite dimensional distributions,” Russian Math. Surveys, 45, No. 3, 1–104 (1990).

    Article  MathSciNet  Google Scholar 

  26. V. I. Bogachev and O. G. Smolyanov, Topological Vector Spaces and Their Applications, Springer, Cham (2017).

    Book  Google Scholar 

  27. V. I. Bogachev and O. G. Smolyanov, Real and Functional Analysis, Springer, Cham (2020).

    Book  Google Scholar 

  28. V. Caselles, A. Lunardi, M. Miranda (jr.), and M. Novaga, “Perimeter of sublevel sets in infinite dimensional spaces,” Adv. Calc. Var., 5, No. 1, 59–76 (2012).

  29. P. Celada and A. Lunardi, “Traces of Sobolev functions on regular surfaces in infinite dimensions,” J. Funct. Anal., 266, 1948–1987 (2014).

    Article  MathSciNet  Google Scholar 

  30. G. Da Prato, A. Lunardi, and L. Tubaro, “Surface measures in infinite dimension,” Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 25, No. 3, 309–330 (2014).

    Article  MathSciNet  Google Scholar 

  31. G. Da Prato, A. Lunardi, and L. Tubaro, “Malliavin calculus for non-Gaussian differentiable measures and surface measures in Hilbert spaces,” Trans. Amer. Math. Soc., 370, No. 8, 5795–5842 (2018).

    Article  MathSciNet  Google Scholar 

  32. Yu. L. Dalecky and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Space, Kluwer AP, Dordrecht (1991).

    Book  Google Scholar 

  33. G. M. Fichtenholz, “On absolutely continuous functions,” Mat. Sb., 31, No. 2, 286–295 (1923).

    Google Scholar 

  34. S. V. Fomin, “Differentiable measures in linear spaces,” Proc. Internat. Congr. Math., Sec. 5, 78–79 (1966).

  35. S. V. Fomin, “Differentiable measures in linear spaces,” Uspekhi Mat. Nauk, 23, No. 1, 221–222 (1968).

    MathSciNet  Google Scholar 

  36. M. Fukushima, “BV functions and distorted Ornstein–Uhlenbeck processes over the abstract Wiener space,” J. Funct. Anal., 174, 227–249 (2000).

    Article  MathSciNet  Google Scholar 

  37. M. Fukushima and M. Hino, “On the space of BV functions and a related stochastic calculus in infinite dimensions,” J. Funct. Anal., 183, No. 1, 245–268 (2001).

    Article  MathSciNet  Google Scholar 

  38. M. Hino, “Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space,” J. Funct. Anal., 258, No. 5, 1656–1681 (2010).

    Article  MathSciNet  Google Scholar 

  39. M. Hino, “Dirichlet spaces on H-convex sets in Wiener space,” Bull. Sci. Math., 135, 667–683 (2011); Erratum: Ibid., 137, 688–689 (2013).

  40. E. D. Kosov, “Besov classes on finite and infinite-dimensional spaces,” Mat. Sb., 210, No. 5, 41–71 (2019); English translation: Sb. Math., 210, No. 5, 663–692 (2019).

  41. E. P. Krugova, “On differentiability of convex measures,” Mat. Zametki, 57, No. 6, 51–61 (1995); English translation: Math. Notes, 58, No. 6, 1294–1301 (1995).

  42. A. M. Kulik, “Log-Sobolev inequality, exponential integrability, and large deviation estimates for C(α, β) log-concave measures,” Random Oper. Stochast. Equat., 10, No. 2, 105–122 (2002).

    MathSciNet  MATH  Google Scholar 

  43. A. Lunardi, M. Miranda (jr.), and D. Pallara, “BV functions on convex domains inWiener spaces,” Potential Anal., 43, 23–48 (2015).

  44. T. S. Pitcher, “Likelihood ratios for diffusion processes with shifted mean value,” Trans. Amer. Math. Soc., 101, 168–176 (1961).

    MathSciNet  MATH  Google Scholar 

  45. M. Röckner, R.-Ch. Zhu, and X.-Ch. Zhu, “The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple,” Ann. Probab., 40, No. 4, 1759–1794 (2012).

  46. M. Röckner, R. Zhu, and X. Zhu, “BV functions in a Gelfand triple for differentiable measures and its applications,” Forum Math., 27, No. 3, 1657–1687 (2015).

  47. V. A. Romanov, “On continuous and totally discontinuous measures in linear spaces,” Dokl. Akad. Nauk SSSR, 227, No. 3, 569–570 (1976); English translation: Sov. Math. Dokl., 17, 472–474 (1976).

  48. A.V. Shaposhnikov, “Differentiability of measures in the sense of Skorokhod and related properties,” Dokl. Akad. Nauk, 429, No. 2, 163–167 (2009); English translation: Dokl. Math., 80, 818–822 (2009).

  49. A.V. Skorohod, Integration in Hilbert Space, Springer-Verlag, New-York–Heidelberg (1974).

    Book  Google Scholar 

  50. O. G. Smolyanov, Analysis on Topological Vector Spaces and Its Applications [in Russian], Izd. MGU, Moscow (1979).

    Google Scholar 

  51. O. G. Smolyanov and H. von Weizsäcker, “Differentiable families of measures,” J. Funct. Anal., 118, No. 2, 454–476 (1993).

  52. A.V. Uglanov, Integration on Infinite-Dimensional Surfaces and Its Applications, Kluwer AP, Dordrecht (2000).

    Book  Google Scholar 

  53. W. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York (1989).

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    V. I. Bogachev

  2. National Research University, Higher School of Economics, Moscow, Russia

    V. I. Bogachev

Authors
  1. V. I. Bogachev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Bogachev.

Additional information

(Editor’s remark.) We preserve author’s spelling “A.V. Skorohod” throughout the paper. This spelling was chosen for the English translation of the book “Integration in Hilbert space”, where the notation of (Skorohod) differentiability of measures was introduced (see [49]).

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 9, pp. 1159–1178, September, 2020. Ukrainian DOI: 10.37863/umzh.v72i9.6277.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogachev, V.I. On Skorokhod Differentiable Measures. Ukr Math J 72, 1335–1357 (2021). https://doi.org/10.1007/s11253-021-01861-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-021-01861-x

Search

Navigation