Exponential convergence for a convexifying equation

Guillaume Carlier; Alfred Galichon

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 611-620
  • ISSN: 1292-8119

Abstract

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We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.

How to cite

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Carlier, Guillaume, and Galichon, Alfred. "Exponential convergence for a convexifying equation." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 611-620. <http://eudml.org/doc/277828>.

@article{Carlier2012,
abstract = {We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. },
author = {Carlier, Guillaume, Galichon, Alfred},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Convex envelope; viscosity solutions; stochastic control representation; nonautonomous gradient flows; convex envelope},
language = {eng},
month = {11},
number = {3},
pages = {611-620},
publisher = {EDP Sciences},
title = {Exponential convergence for a convexifying equation},
url = {http://eudml.org/doc/277828},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Carlier, Guillaume
AU - Galichon, Alfred
TI - Exponential convergence for a convexifying equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 611
EP - 620
AB - We consider an evolution equation similar to that introduced by Vese in [Comm. Partial Diff. Eq. 24 (1999) 1573–1591] and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time.
LA - eng
KW - Convex envelope; viscosity solutions; stochastic control representation; nonautonomous gradient flows; convex envelope
UR - http://eudml.org/doc/277828
ER -

References

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  1. O. Alvarez, J.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints. J. Math. Pures Appl.76 (1997) 265–288.  Zbl0890.49013
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  7. A. Oberman, The convex envelope is the solution of a nonlinear obstacle problem. Proc. Amer. Math. Soc.135 (2007) 1689–1694.  Zbl1190.35107
  8. A. Oberman, Computing the convex envelope using a nonlinear partial differential equation. Math. Mod. Methods Appl. Sci.18 (2008) 759–780.  Zbl1154.35056
  9. A. Oberman and L. Silvestre, The Dirichlet Problem for the Convex Envelope. Trans. Amer. Math. Soc. (to appear).  Zbl1244.35056
  10. H.M. Soner and N. Touzi, Stochastic representation of mean curvature type geometric flows. Ann. Probab.31 (2003) 1145–1165.  Zbl1080.60076
  11. N. Touzi, Stochastic control and application to Finance. Lecture Notes available at .  URIhttp://www.cmap.polytechnique.fr/˜touzi/
  12. L. Vese, A method to convexify functions via curve evolution. Comm. Partial Diff. Eq.24 (1999) 1573–1591. Zbl0935.35087

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