May  2020, 19(5): 2679-2711. doi: 10.3934/cpaa.2020117

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

1. 

Department of Mathematics and applications, University of Milano Bicocca, via Cozzi 55, 20125 Milano, Italy

2. 

Department of Mathematics and Computer Science, University of Ferrara, via Machiavelli 30, 44121 Ferrara, Italy

* Corresponding author

Received  February 2019 Revised  October 2019 Published  March 2020

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 $.

Citation: Davide Addona, Giorgio Menegatti, Michele Miranda jr.. $ BV $ functions on open domains: the Wiener case and a Fomin differentiable case. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2679-2711. doi: 10.3934/cpaa.2020117
References:
[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, Vol. 62, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[9]

V. I. Bogachev, Measure Theory, Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Da Prato and A. Lunardi, $BV$ functions in Hilbert spaces, preprint, arXiv: 1801.03344v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Ferrari, Sobolev spaces with respect to weighted gaussian measures in infinite dimension, preprint, arXiv: 1510.08283. Google Scholar

[18]

G. Fichera, Lezioni sulle trasformazioni lineari, Introduzione all'analisi lineare, Vol. I, Istituto Matematico, Università, Trieste, 1954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. Jordan, Sur la serie de Fourier, Comptes Rendus de l'Academie des Sciences Paris, 2, 1881. Google Scholar

[24]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Vol. 132, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

M. Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Super. Pisa, 18 (1964), 27-56.   Google Scholar

[29]

M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Super. Pisa, 18 (1964), 515-542.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4] L. AmbrosioN. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. I. Bogachev, Gaussian Measures, Mathematical Surveys and Monographs, Vol. 62, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[9]

V. I. Bogachev, Measure Theory, Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Da Prato and A. Lunardi, $BV$ functions in Hilbert spaces, preprint, arXiv: 1801.03344v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

S. Ferrari, Sobolev spaces with respect to weighted gaussian measures in infinite dimension, preprint, arXiv: 1510.08283. Google Scholar

[18]

G. Fichera, Lezioni sulle trasformazioni lineari, Introduzione all'analisi lineare, Vol. I, Istituto Matematico, Università, Trieste, 1954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

C. Jordan, Sur la serie de Fourier, Comptes Rendus de l'Academie des Sciences Paris, 2, 1881. Google Scholar

[24]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Vol. 132, Springer-Verlag, Berlin, 1995.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[28]

M. Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Super. Pisa, 18 (1964), 27-56.   Google Scholar

[29]

M. Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Super. Pisa, 18 (1964), 515-542.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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