Abstract
We study functions of bounded variation with values in a Banach or in a metric space. In finite dimensions, there are three well-known topologies; we argue that in infinite dimensions there is a natural fourth topology. We provide some insight into the structure of these four topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin–Lions theorem. After this we study the Borel \(\sigma \)-algebras induced by these topologies, and we provide some results about probability measures on the space of functions of bounded variation, which can be used to study stochastic processes of bounded variation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Ambrosio. Metric space valued functions of bounded variation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, \(4^e\) série, 17(3):439–478, 1990.
L. Ambrosio and S. Di Marino. Equivalent definitions of BV space and of total variation on metric measure spaces. Journal of Functional Analysis, 266(7):4150–4188, 2014.
L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, NY, USA, 2006.
L. Ambrosio and R. Ghezzi. Sobolev and bounded variation functions on metric measure spaces (lecture notes). http://cvgmt.sns.it/paper/2738/, 2016.
L. Ambrosio, R. Ghezzi, and V. Magnani. BV functions and sets of finite perimeter in sub-Riemannian manifolds. Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire, 32(3):489–517, 2015.
L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Birkhauser, Basel, Switzerland, 2nd edition, 2008.
L. Bertini, A. Faggionato, and D. Gabrielli. Large deviations of the empirical flow for continuous time markov chains. Annales de l’Institut Henri Poincaré - Probabilités et Statistiques, 51(3):867–900, 2015.
V. I. Bogachev. Measure theory. Vol. I & II. Springer-Verlag, Berlin, Germany, 2007.
H. Brézis. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973.
D. Chiron. On the definitions of Sobolev and BV spaces into singular spaces and the trace problem. Communications in Contemporary Mathematics, 9(4):473, 2007.
V. V. Chistyakov. Selections of bounded variation. Journal of Applied Analysis, 10(1):1–82, 2004.
J.B. Conway. A course in functional analysis. Springer, New York, NY, USA, 2nd edition, 2007.
S. Di Marino. Sobolev and BV spaces on metric measure spaces via derivations and integration by parts. http://cvgmt.sns.it/paper/2521, 2014.
J. Diestel and J. J. Jr. Uhl. Vector Measures, volume 95. American Mathematical Society, Providence, RI, 1967.
N. Dinculeanu. Vector Measures. Pergamon press / Deutscher Verlag der Wissenschaften, Berlin, Germany, 1967.
N. Dunford and J.T. Schwartz. Linear operators, part one: general theory. Interscience, New York, NY, USA, 1957.
L.C. Evans and R.F. Gariepy. Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, USA, 1992.
H. Federer. Geometric measure theory. Springer, Berlin, Germany, 1969.
A. Jakubowski. A non-Skorokhod topology on the Skorokhod space. Electronic Journal of Probability, 2(4):1–21, 1997.
P. Krejčí. Regulated evolution quasivariational inequalities. Notes to Lectures held at the University of Pavia, 2003.
P. Krejčí. The Kurzweil integral and hysteresis. In Journal of Physics: Conference Series, volume 55, page 144. IOP Publishing, 2006.
P. Logaritsch and E. Spadaro. A representation formula for the p-energy of metric space-valued Sobolev maps. Communications in Contemporary Mathematics, 14(6):1250043, 2012.
T.-W. Ma. Banach-Hilbert Spaces, Vector Measures and Group Representations. World Scientific, Singapore, 2002.
A. Mainik and A. Mielke. Existence results for energetic models for rate-independent systems. Calculus of Variations and Partial Differential Equations, 22(1):73–99, 2005.
P.A. Meyer and W.A. Zheng. Tightness criteria for laws of semimartingales. Annales de l’I.H.P., section B, 20(4):353–372, 1984.
A. Mielke and T. Roubicek. Rate-Independent Systems. Springer, Berlin, Germany, 2015.
A. Mielke, F. Theil, and V. I. Levitas. A variational formulation of rate-independent phase transformations using an extremum principle. Archive for Rational Mechanics and Analysis, 162(2):137–177, 2012.
J. J. Moreau, P. D. Panagiotopoulos, and G. Strang. Topics in nonsmooth mechanics. Birkhäuser, Basel, Switzerland, 1988.
R. I. A. Patterson and D. R. M. Renger. Large deviations of reaction fluxes. arXiv:1802.02512, 2018.
V. Recupero. BV solutions of rate independent variational inequalities. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie V, 10(2):269, 2011.
V. Recupero. Hysteresis operators in metric spaces. Discrete Contin. Dyn. Syst. Ser. S, 8:773–792, 2015.
W. Rudin. Functional Analysis. McGraw-Hill, New York, NY, 1973.
J. Simon. Compact sets in the space \(L^p(0,T;B)\). Annali di Matematica pura ed applicata, 146(1):65–96, 1986.
A. Visintin. Differential models of hysteresis. Springer Science & Business Media, Berlin, Germany, 2013.
A. Wiweger. Linear spaces with mixed topology. Studia Mathematica, 20(1):47–68, 1961.
Acknowledgements
This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through Grant CRC 1114 “Scaling Cascades in Complex Systems”, Projects C08 “Stochastic Spatial Coagulation Particle Processes” and C05 “Effective Models for Materials and Interfaces with Multiple Scales”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Heida, M., Patterson, R.I.A. & Renger, D.R.M. Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space. J. Evol. Equ. 19, 111–152 (2019). https://doi.org/10.1007/s00028-018-0471-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-018-0471-1