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Concentration for multidimensional diffusions and their boundary local times
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  • Published: 07 June 2011

Concentration for multidimensional diffusions and their boundary local times

  • Soumik Pal1 

Probability Theory and Related Fields volume 154, pages 225–254 (2012)Cite this article

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Abstract

We prove that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy quadratic transportation cost inequality under the uniform metric. From this we derive concentration properties of Lipschitz functions of process paths that depend on the entire history. In particular, we estimate concentration of boundary local time of reflected Brownian motions on a polyhedral domain. We work out explicit applications of consequences of measure concentration for the case of Brownian motion with rank-based drifts.

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References

  1. Arguin L.-P., Aizenman M.: On the structure of quasi-stationary competing particles systems. Ann. Probab. 37, 1080–1113 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banner A., Fernholz R., Karatzas I.: Atlas models of equity markets. Ann. Appl. Probab. 15(4), 2296–2330 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banner A., Ghomrasni R.: Local times of ranked continuous semimartingales. Stoch. Process. Appl. 118, 1244–1253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bobkov S., Gentil I., Ledoux M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80, 669–696 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bobkov S., Götze F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chatterjee S., Pal S.: A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Relat. Fields 147(1–2), 123–159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chatterjee, S., Pal, S.: A combinatorial analysis of interacting diffusions. J. Theor. Probab. (2008) (in press)

  8. Dembo A.: Information inequalities and concentration of measures. Ann. Probab. 25, 927–939 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dembo A., Zeitouni O.: Transportation approach to some concentration inequalities in product spaces. Electron. Commun. Probab. 1, 83–90 (1996)

    MathSciNet  MATH  Google Scholar 

  10. Djellout H., Guillin A., Wu L.: Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32(3B), 2702–2732 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dupuis P., Ishii H.: On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics 35, 31–62 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Dupuis P., Ramanan K.: Convex duality and the Skorokhod problem. I. Probab. Theory Relat. Fields 115, 153–195 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dupuis P., Ramanan K.: Convex duality and the Skorokhod problem. II. Probab. Theory Relat. Fields 115, 197–236 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fang S., Shao J.: Transportation cost inequalities on path and loop groups. J. Funct. Anal. 218(2), 293–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fang S., Shao J.: Optimal transport maps for Monge-Kantorovich problem on loop groups. J. Funct. Anal. 248(1), 225–257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fang S., Wang F.-Y., Wu B.: Transportation-cost inequality on path spaces with uniform distance. Stoch. Process. Appl. 118(12), 2181–2197 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fernholz, R., Karatzas, I.: Stochastic portfolio theory: a survey. In: Handbook of Numerical Analysis: Mathematical Modeling and Numerical Methods in Finance, pp. 89–168. Elsevier, Amsterdam (2009)

  18. Feyel D., Ustunel A.S.: The Monge-Kantorovich problem and Monge-Ampère equation on Wiener space. Probab. Theory Relat. Fields 128(3), 347–385 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gourcy M., Wu L.: Logarithmic Sobolev inequalities of diffusions for the L 2 metric. Potential Anal. 25(1), 77–102 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gozlan N.: Characterization of Talagrand’s like transportation-cost inequalities on the real line. J. Funct. Anal. 250, 400–425 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gozlan N.: A characterization for dimension free concentration in terms of transportation inequalities. Ann. Probab. 37(6), 2480–2498 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gozlan N., Roberto C., Samson P.-M.: A new characterization of Talagrand’s transport-entropy inequalities and applications. Ann. Probab. 39(3), 857–880 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Harrison J.M., Reiman M.I.: Reflected Brownian motion on an orthant. Ann. Probab. 9, 302–308 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Houdré C., Privault N.: Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8(6), 697–720 (2002)

    MathSciNet  MATH  Google Scholar 

  25. Ichiba, T., Karatzas, I.: Collisions of Brownian particles. Ann. Appl. Probab. (2010) (in press)

  26. Ichiba, T., Papathanakos, V., Banner, A., Karatzas, I., Fernholz, R.: Hybrid atlas models. Ann. Appl. Probab. Preprint (2010). arXiv:0909.0065

  27. Jourdain B., Malrieu F.: Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf. Ann. Appl. Probab. 18(5), 1706–1736 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Karatzas I., Shreve S.: Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. 2nd edn. Springer, Berlin (1991)

    Book  Google Scholar 

  29. Ledoux M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001)

    Google Scholar 

  30. Marton K.: Bounding \({\bar d}\) -distance by information divergence: a method to prove measure concentration. Ann. Probab. 24, 857–866 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Marton K.: A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6, 556–571 (1997)

    Article  MathSciNet  Google Scholar 

  32. Marton K.: Mesure concentration for a class of random processes. Probab. Theory Relat. Fields 110, 427–439 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. McKean, H.P., Shepp, L.: The advantage of capitalism vs. Socialism depends on the criterion (2005). http://www.emis.de/journals/ZPOMI/v328/p160.ps.gz

  34. Neveu, J.: Discrete Parameter Martingales. North-Holland Mathematical Library. North-Holland/Elsevier Science, Amsterdam (1975)

  35. Nourdin I., Viens F.G.: Density formula and concentration inequalities with Malliavin calculus. Electron. J. Probab. 14, 2287–2309 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pal S., Pitman J.: One-dimensional Brownian particle systems with rank dependent drifts. Ann. Appl. Probab. 18(6), 2179–2207 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pal, S., Shkolnikov, M.: Concentration of measure for systems of Brownian particles interacting through their ranks. Preprint (2010)

  39. Ruzmaikina A., Aizenman M.: Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33(1), 82–113 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Revuz D., Yor M.: Continuous Martingales and Brownian Motion. A Series of Comprehensive Studies in Mathematics, vol. 293. 3rd edn. Springer, Berlin (1999)

    Google Scholar 

  41. Shkolnikov M.: Competing particle systems evolving by I.I.D. increments. Electron. J. Probab. 14, 728–751 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Shkolnikov, M.: Competing particle systems evolving by interacting Levy processes. Preprint (2010)

  43. Talagrand, M.: A New Isoperimetric Inequality for Product Measure, and the Concentration of Measure Phenomenon. Israel Seminar (GAFA), Lecture Notes in Mathematics, vol. 1469, pp. 91–124. Springer, Berlin (1991)

  44. Talagrand M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 2876 (1994)

    Google Scholar 

  45. Talagrand M.: Concentration of measure and isoperimetric inequalities in product spaces. Publications Mathématiques de l’I.H.E.S. 81, 73205 (1995)

    Google Scholar 

  46. Talagrand M.: A new look at independence. Ann. Probab. 24, 134 (1996)

    Article  Google Scholar 

  47. Talagrand M.: Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. Üstünel, A.S.: Transportation cost inequalities for diffusions under uniform distance. Preprint (2010). arxiv.org/abs/1009.5251

  49. Wang F.-Y.: Transportation cost inequalities on path spaces over Riemannian manifolds. Ill. J. Math. 46(4), 1197–1206 (2002)

    MATH  Google Scholar 

  50. Wang F.-Y.: Generalized transportation-cost inequalities and applications. Potential Anal. 28(4), 321–334 (2008)

    Article  MathSciNet  Google Scholar 

  51. Wu L., Zhang Z.: Talagrand’s T 2-transportation inequality w.r.t. a uniform metric for diffusions. Acta Math. Appl. Sin. Engl. Ser. 20(3), 357–364 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Soumik Pal

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  1. Soumik Pal
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Corresponding author

Correspondence to Soumik Pal.

Additional information

This research is partially supported by NSF grant DMS-1007563.

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Cite this article

Pal, S. Concentration for multidimensional diffusions and their boundary local times. Probab. Theory Relat. Fields 154, 225–254 (2012). https://doi.org/10.1007/s00440-011-0368-1

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  • Received: 28 May 2010

  • Revised: 20 March 2011

  • Published: 07 June 2011

  • Issue Date: October 2012

  • DOI: https://doi.org/10.1007/s00440-011-0368-1

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Keywords

  • Concentration of diffusions
  • Concentration of local times
  • Transportation cost inequality

Mathematics Subject Classification (2000)

  • 60G17
  • 60G60
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