Abstract
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.
References
[1] Alberti, C. Mantegazza, A note on the theory of SBV functions, Boll. Un. Mat. Ital. B (7) 11 (1997), n.2, 375–382. Search in Google Scholar
[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. 10.5802/afst.1297Search in Google Scholar
[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. 10.4171/RLM/647Search in Google Scholar
[4] L. Ambrosio, N. Fusco, D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000. Search in Google Scholar
[5] L. Ambrosio, S. Maniglia, M. Miranda Jr, D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258(2010), 785– 813. 10.1016/j.jfa.2009.09.008Search in Google Scholar
[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. 10.3934/dcds.2010.28.591Search in Google Scholar
[7] V. I. Bogachev, Gaussian Measures, American Mathematical Society, Providence R.I., 1998. 10.1090/surv/062Search in Google Scholar
[8] V. I. Bogachev, A.Yu. Pilipenko, A.V. Shaposhnikov, Sobolev functions on infinite-dimensional domains, J.Math. Anal. Appl., 419(2014), 1023–1044. 10.1016/j.jmaa.2014.05.020Search in Google Scholar
[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. 10.1515/acv.2011.010Search in Google Scholar
[10] P. Celada, A. Lunardi, Traces of Sobolev functions on regular surfaces in infinite dimensions, J. Funct. Anal., 266(2014), 1948–1987. 10.1016/j.jfa.2013.11.013Search in Google Scholar
[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. Search in Google Scholar
[12] N. Dunford, J.T. Schwartz, Linear operators Part I: General theory, Wiley, 1958. Search in Google Scholar
[13] D. Feyel, A. de la Pradelle, Hausdorff measures on the Wiener space, Potential Anal. 1(1992), 177-189. 10.1007/BF01789239Search in Google Scholar
[14] M. Fukushima, BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174(2000), 227-249. 10.1006/jfan.2000.3576Search in Google Scholar
[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. 10.1006/jfan.2000.3738Search in Google Scholar
[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. Search in Google Scholar
[17] M. Hino, Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space, J. Funct. Anal., 258(2010), 1656– 1681. 10.1016/j.jfa.2009.06.033Search in Google Scholar
[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. Search in Google Scholar
[19] M. Ledoux, Semigroup proofs of the isoperimetric inequality in Euclidean and Gauss space, Bull. Sci. Math., 118(1994), 485–510. Search in Google Scholar
[20] P. Malliavin, Stochastic analysis, Grundlehren der Mathematischen Wissenschaften 313, Springer, 1997. 10.1007/978-3-642-15074-6Search in Google Scholar
[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. Search in Google Scholar
[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. 10.5802/afst.1142Search in Google Scholar
[23] R. O’Donnell, Analysis of Boolean functions, Cambridge University Press, 2014. Search in Google Scholar
[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. 10.1214/11-AOP661Search in Google Scholar
[25] D. Trevisan:, BV-regularity for the Malliavin derivative of the maximum of the Wiener process, Electron. Commun. Probab., 18(2013), no.29. 10.1214/ECP.v18-2314Search in Google Scholar
[26] D. Trevisan, Lagrangian flows driven by BV fields in Wiener spaces, Probab. Theory Related Fields, DOI 10.1007/s00440- 014-0589-1. Search in Google Scholar
[27] A.I. Vol’pert, S.I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Martinus Nijhoff Publishers, Dordrecht, NL, 1985. Search in Google Scholar
[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. 10.1007/s004400200203Search in Google Scholar
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