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Some Fine Properties of BV Functions on Wiener Spaces

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Języki publikacji
EN
Abstrakty
EN
In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.
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Twórcy
  • Scuola Normale Superiore Piazza dei Cavalieri,7, 56126 Pisa, Italy
  • Dip. di Matematica e Informatica, Università di Ferrara, via Machiavelli 30, 44121 Ferrara, Italy
  • Dip. di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, P.O.B. 193, 73100 Lecce, Italy
Bibliografia
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  • [2] L. Ambrosio, A. Figalli, Surface measure and convergence of the Ornstein-Uhlenbeck semigroup inWiener spaces, Ann. Fac. Sci. Toulouse Math., 20(2011) 407-438.
  • [3] L. Ambrosio, A. Figalli, E. Runa, On sets of finite perimeter in Wiener spaces: reduced boundary and convergence to halfspaces, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24(2013), 111-122. [WoS]
  • [4] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000.
  • [5] L. Ambrosio, S. Maniglia, M. Miranda Jr, D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258(2010), 785– 813.
  • [6] L. Ambrosio, M. Miranda Jr, D. Pallara, Sets with finite perimeter in Wiener spaces, perimeter measure and boundary recti- fiability, Discrete Contin. Dyn. Syst., 28(2010), 591–606.
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  • [9] V. Caselles, A. Lunardi, M. Miranda Jr, M. Novaga, Perimeter of sublevel sets in infinite dimensional spaces, Adv. Calc. Var., 5(2012), 59–76. [WoS]
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  • [11] E. De Giorgi, L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., (8) 82(1988), n.2, 199–210, English translation in: Ennio De Giorgi: Selected Papers, (L. Ambrosio, G. DalMaso, M. Forti, M. Miranda, S. Spagnolo eds.) Springer, 2006, 686–696.
  • [12] N. Dunford, J.T. Schwartz, Linear operators Part I: General theory, Wiley, 1958.
  • [13] D. Feyel, A. de la Pradelle, Hausdorff measures on the Wiener space, Potential Anal. 1(1992), 177-189.
  • [14] M. Fukushima, BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174(2000), 227-249.
  • [15] M. Fukushima, M. Hino, On the space of BV functions and a Related Stochastic Calculus in Infinite Dimensions, J. Funct. Anal., 183(2001), 245-268.
  • [16] L. Gross, Abstract Wiener spaces, in: Proc. Fifth Berkeley Symp. Math. Stat. Probab. (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 1, p. 31-42, Univ. California Press, Berkeley.
  • [17] M. Hino, Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space, J. Funct. Anal., 258(2010), 1656– 1681. [WoS]
  • [18] M. Hino, H. Uchida, Reflecting Ornstein-Uhlenbeck processes on pinned path spaces, Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 111–128, RIMS Kokyuroku Bessatsu, B6, Kyoto, 2008.
  • [19] M. Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math., 118(1994), 485–510.
  • [20] P. Malliavin, Stochastic analysis, Grundlehren der Mathematischen Wissenschaften 313, Springer, 1997.
  • [21] M.Miranda Jr, M. Novaga, D. Pallara, An introduction to BV functions in Wiener spaces, Advanced Studies in Pure Mathematics, 67, 245–293, Tokyo 2015.
  • [22] M. Miranda Jr, D. Pallara, F. Paronetto, M. Preunkert, Short–time heat flow and functions of bounded variation in RN, Ann. Fac. Sci. Toulouse, XVI(2007), 125–145.
  • [23] R. O’Donnell, Analysis of Boolean functions, Cambridge University Press, 2014.
  • [24] M. Röckner, R.C. Zhu, X.C. Zhu, The stochastic reflection problem on an infinite dimensional convex set and BV functions in a Gelfand triple, Ann. Probab., 40(2012), 1759-1794. [WoS]
  • [25] D. Trevisan:, BV-regularity for the Malliavin derivative of the maximum of the Wiener process, Electron. Commun. Probab., 18(2013), no.29. [WoS][Crossref]
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  • [28] L. Zambotti, Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Related Fields, 123(2002) 579-600.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.doi-10_1515_agms-2015-0013
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