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$ BV $ functions on open domains: the Wiener case and a Fomin differentiable case

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  • We provide three different characterizations of the space $ BV(O, \gamma) $ of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure $ \gamma $ on open domains $ O $ in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to $ BV(O, \gamma) $ by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures $ \nu $ on a Hilbert space $ X $, and we infer a characterization of the space $ BV(O, \nu) $ of the functions of bounded variation with respect to $ \nu $ on open domains $ O\subseteq X $.

    Mathematics Subject Classification: Primary: 49Q20, 58E99, 28C20; Secondary: 26B30, 60H07.


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  • [1] D. Addona, G. Menegatti and M. Miranda, Jr., On integration by parts formula on open convex sets in Wiener spaces, preprint, arXiv: 1808.06825.
    [2] S. AlbeverioZ. Ma and M. Röckner, Partitions of unity in Sobolev spaces over infinite-dimensional state spaces, J. Funct. Anal., 143 (1997), 247-268.  doi: 10.1006/jfan.1996.2968.
    [3] L. Ambrosio, G. Da Prato and A. C. G. Mennucci, Introduction to Measure Theory and Integration, Lecture Notes Scuola Normale Superiore di Pisa (New Series), Vol. 10, Edizioni della Normale, Pisa, 2011. doi: 10.1007/978-88-7642-386-4.
    [4] L. AmbrosioN. Fusco and  D. PallaraFunctions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. 
    [5] L. AmbrosioM. MirandaJr. S. Maniglia and D. Pallara, BV functions in abstract Wiener spaces, J. Funct. Anal., 258 (2010), 785-813.  doi: 10.1016/j.jfa.2009.09.008.
    [6] L. Ambrosio and S. Di Marino, Equivalent definitions of $BV$ space and of total variation on metric measure spaces, J. Funct. Anal., 266 (2014), 4150-4188.  doi: 10.1016/j.jfa.2014.02.002.
    [7] J. Assaad and J. van Neerven, $L^{2}$-theory for non-symmetric ornstein–uhlenbeck semigroups on domains, J. Evol. Equ., 13 (2013), 107-134.  doi: 10.1007/s00028-012-0171-1.
    [8] V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, Vol. 62, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.
    [9] V. I. Bogachev, Measure Theory, Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.
    [10] V. I. BogachevA. Y. Pilipenko and A. V. Shaposhnikov, Sobolev functions on infinite-dimensional domains, J. Math. Anal. Appl., 419 (2014), 1023-1044.  doi: 10.1016/j.jmaa.2014.05.020.
    [11] G. Cappa, On the Ornstein-Uhlenbeck operator in convex sets of Banach spaces, Studia Mathematica, 247 (2019), 217-239.  doi: 10.4064/sm8229-3-2018.
    [12] G. Da Prato and A. Lunardi, $BV$ functions in Hilbert spaces, preprint, arXiv: 1801.03344v1.
    [13] G. Da PratoA. Lunardi and L. Tubaro, Malliavin calculus for non-Gaussian differentiable measures and surface measures in Hilbert spaces, Trans. Amer. Math. Soc., 370 (2018), 5795-5842.  doi: 10.1090/tran/7195.
    [14] E. De Giorgi, Su una teoria generale della misura $(r-1)$-dimensionale in uno spazio ad $r$ dimensioni, Ann. Mat. Pura Appl., 36 (1954), 191-213.  doi: 10.1007/BF02412838.
    [15] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993.
    [16] J. Diestel and J. J. Uhl, Jr., Vector Measures, with a Foreword by B. J. Pettis, Mathematical Surveys, Vol. 15, American Mathematical Society, Providence, R.I., 1977.
    [17] S. Ferrari, Sobolev spaces with respect to weighted gaussian measures in infinite dimension, preprint, arXiv: 1510.08283.
    [18] G. Fichera, Lezioni sulle trasformazioni lineari, Introduzione all'analisi lineare, Vol. I, Istituto Matematico, Università, Trieste, 1954.
    [19] M. Fukushima, $BV$ functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174 (2000), 227-249.  doi: 10.1006/jfan.2000.3576.
    [20] M. Fukushima and M. Hino, On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal., 183 (2001), 245-268.  doi: 10.1006/jfan.2000.3738.
    [21] N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc., 251 (2018), v+161. doi: 10.1090/memo/1196.
    [22] M. Hino, On Dirichlet spaces over convex sets in infinite dimensions, In Finite and infinite dimensional analysis in honor of Leonard Gross (New Orleans, LA, 2001), Contemporary Mathematics, Vol. 317, American Mathematical Society, Providence, RI, (2003), 143–156. doi: 10.1090/conm/317/05525.
    [23] C. Jordan, Sur la serie de Fourier, Comptes Rendus de l'Academie des Sciences Paris, 2, 1881.
    [24] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Vol. 132, Springer-Verlag, Berlin, 1995.
    [25] A. LunardiM. Miranda, Jr. and D. Pallara, $BV$ functions on convex domains in Wiener spaces, Potential Anal., 43 (2015), 23-48.  doi: 10.1007/s11118-015-9462-9.
    [26] G. Menegatti, Sobolev Classes and Bounded Variation Functions on Domains of Wiener Spaces, and Applications, Ph.D thesis, Università degli studi di Ferrara, 2018.
    [27] N.G. Meyers and J. Serrin, $H = W$, Proc. Natl. Acad. Sci. U. S. A., 51 (1964), 1055-1056.  doi: 10.1073/pnas.98.20.11836/a.
    [28] M. Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Super. Pisa, 18 (1964), 27-56. 
    [29] M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Super. Pisa, 18 (1964), 515-542. 
    [30] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol 146, Marcel Dekker, Inc., New York, 1991.
    [31] I. Shigekawa, Stochastic Analysis, Translations of Mathematical Monographs, Vol. 224, Translated from the 1998 Japanese original by the author, Iwanami Series in Modern Mathematics, American Mathematical Society, Providence, RI, 2004.
    [32] N. N. Vakhania, V. I. Tarieladze and S. A. Chobanyan, Probability Distributions on Banach Spaces, Mathematics and its Applications (Soviet Series), Vol. 14, Translated from the Russian and with a preface by Wojbor A. Woyczynski, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3873-1.
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