In this paper we extend a previous result of the author [S. Lisini, Calc. Var. Partial Differ. Eq. 28 (2007) 85–120.] on the characterization of absolutely continuous curves in Wasserstein spaces to a more general class of spaces: the spaces of probability measures endowed with the Wasserstein−Orlicz distance constructed on extended Polish spaces (in general non separable), recently considered in [L. Ambrosio, N. Gigli and G. Savaré, Invent. Math. 195 (2014) 289–391.] An application to the geodesics of this Wasserstein−Orlicz space is also given.
DOI : 10.1051/cocv/2015020
Mots clés : Spaces of probability measures, Wasserstein−Orlicz distance, absolutely continuous curves, superposition principle, geodesic in spaces of probability measures
@article{COCV_2016__22_3_670_0,
author = {Lisini, Stefano},
title = {Absolutely continuous curves in extended {Wasserstein\ensuremath{-}Orlicz} spaces},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {670--687},
publisher = {EDP-Sciences},
volume = {22},
number = {3},
year = {2016},
doi = {10.1051/cocv/2015020},
zbl = {1348.49048},
mrnumber = {3527938},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2015020/}
}
TY - JOUR AU - Lisini, Stefano TI - Absolutely continuous curves in extended Wasserstein−Orlicz spaces JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 670 EP - 687 VL - 22 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2015020/ DO - 10.1051/cocv/2015020 LA - en ID - COCV_2016__22_3_670_0 ER -
%0 Journal Article %A Lisini, Stefano %T Absolutely continuous curves in extended Wasserstein−Orlicz spaces %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 670-687 %V 22 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2015020/ %R 10.1051/cocv/2015020 %G en %F COCV_2016__22_3_670_0
Lisini, Stefano. Absolutely continuous curves in extended Wasserstein−Orlicz spaces. ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 3, pp. 670-687. doi : 10.1051/cocv/2015020. http://www.numdam.org/articles/10.1051/cocv/2015020/
and , Equivalent definitions of BV space and of total variation on metric measure spaces. J. Funct. Anal. 266 (2014) 4150–4188. | DOI | MR | Zbl
L. Ambrosio, N. Gigli and G. Savarè, Gradient Flows in Metric Spaces and in the Wasserstein Spaces of Probability Measures. Birkhäuser (2005). | MR | Zbl
, and , Density of lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoamericana 29 (2013) 969–986. | DOI | MR | Zbl
, and , Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195 (2014) 289–391. | DOI | MR | Zbl
and , An equivalent path functional formulation of branched transportation problems. Discrete Contin. Dyn. Syst. 29 (2011) 845–871. | DOI | MR | Zbl
, and , A variational method for a class of parabolic PDEs. Ann. Sc. Norm. Super. Pisa Cl. Sci. 10 (2011) 207–252. | Numdam | MR | Zbl
, Gradient estimate for Markov kernels, Wasserstein control and Hopf-Lax formula. RIMS Kôkyûroku Bessatsu, B43 (2013) 61–68. | MR | Zbl
, Characterization of absolutely continuous curves in Wasserstein spaces. Calc. Var. Partial Differ. Eq. 28 (2007) 85–120. | DOI | MR | Zbl
M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces. Marcel Dekker Inc. (1991). | MR | Zbl
and , Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. 2 (2003) 395–431. | Numdam | Zbl
D.W. Stroock, Probability Theory. 2nd edition. Cambridge University Press (2011). | Zbl
, Generalized Orlicz spaces and Wasserstein distances for convex-concave scale functions. Bull. Sci. Math. 135 (2011) 795–802. | DOI | Zbl
Cité par Sources :
