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An asymptotic result for brownian polymers

Thomas Mountford; Pierre Tarrès

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 1, page 29-46
  • ISSN: 0246-0203

Abstract

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We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields92 (1992) 337–349). We prove their conjecture about the asymptotic behavior of the underlying continuous process Xt (corresponding to the location of the end of the polymer at time t) for a particular type of repelling interaction function without compact support.

How to cite

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Mountford, Thomas, and Tarrès, Pierre. "An asymptotic result for brownian polymers." Annales de l'I.H.P. Probabilités et statistiques 44.1 (2008): 29-46. <http://eudml.org/doc/77963>.

@article{Mountford2008,
abstract = {We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields92 (1992) 337–349). We prove their conjecture about the asymptotic behavior of the underlying continuous process Xt (corresponding to the location of the end of the polymer at time t) for a particular type of repelling interaction function without compact support.},
author = {Mountford, Thomas, Tarrès, Pierre},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {self-interacting diffusions; repulsive interaction; superdiffusive process; almost sure law of large numbers},
language = {eng},
number = {1},
pages = {29-46},
publisher = {Gauthier-Villars},
title = {An asymptotic result for brownian polymers},
url = {http://eudml.org/doc/77963},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Mountford, Thomas
AU - Tarrès, Pierre
TI - An asymptotic result for brownian polymers
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 1
SP - 29
EP - 46
AB - We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields92 (1992) 337–349). We prove their conjecture about the asymptotic behavior of the underlying continuous process Xt (corresponding to the location of the end of the polymer at time t) for a particular type of repelling interaction function without compact support.
LA - eng
KW - self-interacting diffusions; repulsive interaction; superdiffusive process; almost sure law of large numbers
UR - http://eudml.org/doc/77963
ER -

References

top
  1. M. Benaïm. Vertex-reinforced random walks and a conjecture of Pemantle. Ann. Probab. 25 (1997) 361–392. Zbl0873.60044MR1428513
  2. M. Benaïm, M. Ledoux and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields 122 (2002) 1–41. Zbl1042.60060MR1883716
  3. M. Benaïm and O. Raimond. Self-interacting diffusions II: convergence in law. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 1043–1055. Zbl1064.60191MR2010396
  4. M. Benaïm and O. Raimond. Self-interacting diffusions III: symmetric interactions. Ann. Probab. 33 (2005) 1716–1759. Zbl1085.60073MR2165577
  5. A. Collevecchio. Limit theorems for Diaconis walk on certain trees. Probab. Theory Related Fields 136 (2006) 81–101. Zbl1109.60027MR2240783
  6. A. Collevecchio. On the transience of processes defined on Galton–Watson trees. Ann. Probab. 34 (2006) 870–878. Zbl1104.60048MR2243872
  7. D. Coppersmith and P. Diaconis. Random walks with reinforcement. Unpublished manuscript, 1986. 
  8. M. Cranston and Y. Le Jan. Self-attracting diffusions: two case studies. Math. Ann. 303 (1995) 87–93. Zbl0838.60052MR1348356
  9. M. Cranston and T. S. Mountford. The strong law of large numbers for a Brownian polymer. Ann. Probab. 2 (1996) 1300–1323. Zbl0873.60014MR1411496
  10. B. Davis. Reinforced random walk. Probab. Theory Related Fields 84 (1990) 203–229. Zbl0665.60077MR1030727
  11. B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501–518. Zbl0930.60041MR1717528
  12. B. Davis. Reinforced and perturbed random walks. Random Walks (Budapest, 1998) János Bolyai Math. Soc., Budapest 9 (1999) 113–126. Zbl0953.60028MR1752892
  13. P. Del Moral and L. Miclo. On convergence of chains with occupational self-interactions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004) 325–346. Zbl1061.60069MR2052266
  14. P. Del Moral and L. Miclo. Self-interacting Markov chains. Stoch. Anal. Appl. 24 (2006) 615–660. Zbl1093.60068MR2220075
  15. P. Diaconis. Recent progress on de Finetti’s notions of exchangeability. Bayesian Statistics, 3 (Valencia, 1987) 111–125. Oxford Sci. Publ., Oxford University Press, New York, 1988. Zbl0707.60033MR1008047
  16. P. Diaconis and S. W. W. Rolles. Bayesian analysis for reversible Markov chains. Ann. Statist. 34 (2006) 1270–1292. Zbl1118.62085MR2278358
  17. R. T. Durrett, H. Kesten and V. Limic. Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122 (2002) 567–592. Zbl0995.60042MR1902191
  18. R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 (1992) 337–349. Zbl0767.60080MR1165516
  19. I. Gihman and A. G. Skorohod. Theory of Stochastic Processes, volume 3. Springer, 1979. Zbl0404.60061
  20. S. Herrmann and B. Roynette. Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 (2003) 81–96. Zbl1010.60033MR1957265
  21. N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. Zbl0495.60005MR1011252
  22. I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. Zbl0638.60065MR917065
  23. M. S. Keane and S. W. W. Rolles. Edge-reinforced random walk on finite graphs. Infinite Dimensional Stochastic Analysis (Amsterdam, 1999), R. Neth. Acad. Arts. Sci. 217–234, 2000. Zbl0986.05092MR1832379
  24. M. S. Keane and S. W. W. Rolles. Tubular recurrence. Acta Math. Hungar. 97 (2002) 207–221. Zbl1026.60089MR1933730
  25. V. Limic. Attracting edge property for a class of reinforced random walks. Ann. Probab. 31 (2003) 1615–1654. Zbl1057.60048MR1989445
  26. V. Limic and P. Tarrès. Attracting edge and strongly edge reinforced walks. Ann. Probab. 35 (2007) 1783–1806. Zbl1131.60036MR2349575
  27. F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on a ladder. Ann. Probab. 33 (2005) 2051–2093. Zbl1102.82010MR2184091
  28. F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on one-dimensional periodic graphs. Preprint, 2006. Zbl1186.82039MR2529432
  29. F. Merkl and S. W. W. Rolles. Linearly edge-reinforced random walks. In Dynamics and Stochastics: Festschrift in the Honor of Michael Keane 66–77, Inst. Math. Statist., Beachvood, OH, 2006. Zbl1125.82014MR2306189
  30. F. Merkl and S. W. W. Rolles. Asymptotic behavior of edge-reinforced random walks. Ann. Probab. 35 (2007) 115–140. Zbl1206.82082MR2303945
  31. F. Merkl and S. W. W. Rolles. Recurrence of edge-reinforced random walk on a two-dimensional graph. Preprint, 2007. Zbl1180.82085
  32. J. R. Norris, L. C. G. Rogers and D. Williams. Self-avoiding walk: a Brownian motion model with local time drift. Probab. Theory Related Fields 74 (1987) 271–287. Zbl0611.60052MR871255
  33. H. G. Othmer and A. Stevens. Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57 (1997) 1044–1081. Zbl0990.35128MR1462051
  34. R. Pemantle. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988) 1229–1241. Zbl0648.60077MR942765
  35. R. Pemantle. Random processes with reinforcement. Massachussets Institute of Technology doctoral dissertation, 1988. 
  36. R. Pemantle. Vertex-reinforced random walk. Probab. Theory Related Fields 92 (1992) 117–136. Zbl0741.60029MR1156453
  37. R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1–79. Zbl1189.60138MR2282181
  38. O. Raimond. Self-attracting diffusions: case of the constant interaction. Probab. Theory Related Fields 107 (1997) 177–196. Zbl0881.60055MR1431218
  39. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, New York, 1991. Zbl0731.60002MR1083357
  40. L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, volume 2. Wiley, New York, 1987. Zbl0627.60001MR921238
  41. S. W. W. Rolles. How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 (2003) 243–260. Zbl1029.60089MR1990056
  42. S. W. W. Rolles. On the recurrence of edge-reinforced random walk on ℤ×G. Probab. Theory Related Fields 135 (2006) 216–264. Zbl1206.82045MR2218872
  43. T. Sellke. Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11 (2006) 301–310. Zbl1113.60048MR2217818
  44. T. Sellke. Reinforced random walks on the d-dimensional integer lattice. Technical Report 94-26, Purdue University, 1994. Zbl1154.82011
  45. M. Takeshima. Behavior of 1-dimensional reinforced random walk. Osaka J. Math. 37 (2000) 355–372. Zbl0962.60017MR1772837
  46. P. Tarrès. Vertex-reinforced random walk on ℤ eventually gets stuck on five points. Ann. Probab. 32 (2004) 2650–2701. Zbl1068.60072MR2078554
  47. B. Tóth. The ‘true’ self-avoiding walk with bond repulsion on ℤ: limit theorems. Ann. Probab. 23 (1995) 1523–1556. Zbl0852.60083MR1379158
  48. B. Tóth. Self-interacting random motions – a survey. Random Walks (Budapest, 1999), Bolyai Society Mathematical Studies 9 (1999) 349–384. Zbl0953.60027MR1752900
  49. B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375–452. Zbl0912.60056MR1640799
  50. M. Vervoort. Games, walks and grammars: Problems I’ve worked on. PhD thesis, Universiteit van Amsterdam, 2000. Zbl1193.00001
  51. S. Volkov. Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 (2001) 66–91. Zbl1031.60089MR1825142
  52. S. Volkov. Phase transition in vertex-reinforced random walks on ℤ with non-linear reinforcement. J. Theoret. Probab. 19 (2006) 691–700. Zbl1107.60068MR2280515

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