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
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topMountford, 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- M. Benaïm. Vertex-reinforced random walks and a conjecture of Pemantle. Ann. Probab. 25 (1997) 361–392. Zbl0873.60044MR1428513
- M. Benaïm, M. Ledoux and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields 122 (2002) 1–41. Zbl1042.60060MR1883716
- M. Benaïm and O. Raimond. Self-interacting diffusions II: convergence in law. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 1043–1055. Zbl1064.60191MR2010396
- M. Benaïm and O. Raimond. Self-interacting diffusions III: symmetric interactions. Ann. Probab. 33 (2005) 1716–1759. Zbl1085.60073MR2165577
- A. Collevecchio. Limit theorems for Diaconis walk on certain trees. Probab. Theory Related Fields 136 (2006) 81–101. Zbl1109.60027MR2240783
- A. Collevecchio. On the transience of processes defined on Galton–Watson trees. Ann. Probab. 34 (2006) 870–878. Zbl1104.60048MR2243872
- D. Coppersmith and P. Diaconis. Random walks with reinforcement. Unpublished manuscript, 1986.
- M. Cranston and Y. Le Jan. Self-attracting diffusions: two case studies. Math. Ann. 303 (1995) 87–93. Zbl0838.60052MR1348356
- M. Cranston and T. S. Mountford. The strong law of large numbers for a Brownian polymer. Ann. Probab. 2 (1996) 1300–1323. Zbl0873.60014MR1411496
- B. Davis. Reinforced random walk. Probab. Theory Related Fields 84 (1990) 203–229. Zbl0665.60077MR1030727
- B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501–518. Zbl0930.60041MR1717528
- B. Davis. Reinforced and perturbed random walks. Random Walks (Budapest, 1998) János Bolyai Math. Soc., Budapest 9 (1999) 113–126. Zbl0953.60028MR1752892
- 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
- P. Del Moral and L. Miclo. Self-interacting Markov chains. Stoch. Anal. Appl. 24 (2006) 615–660. Zbl1093.60068MR2220075
- 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
- P. Diaconis and S. W. W. Rolles. Bayesian analysis for reversible Markov chains. Ann. Statist. 34 (2006) 1270–1292. Zbl1118.62085MR2278358
- 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
- R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 (1992) 337–349. Zbl0767.60080MR1165516
- I. Gihman and A. G. Skorohod. Theory of Stochastic Processes, volume 3. Springer, 1979. Zbl0404.60061
- S. Herrmann and B. Roynette. Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 (2003) 81–96. Zbl1010.60033MR1957265
- N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. Zbl0495.60005MR1011252
- I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. Zbl0638.60065MR917065
- 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
- M. S. Keane and S. W. W. Rolles. Tubular recurrence. Acta Math. Hungar. 97 (2002) 207–221. Zbl1026.60089MR1933730
- V. Limic. Attracting edge property for a class of reinforced random walks. Ann. Probab. 31 (2003) 1615–1654. Zbl1057.60048MR1989445
- V. Limic and P. Tarrès. Attracting edge and strongly edge reinforced walks. Ann. Probab. 35 (2007) 1783–1806. Zbl1131.60036MR2349575
- F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on a ladder. Ann. Probab. 33 (2005) 2051–2093. Zbl1102.82010MR2184091
- F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on one-dimensional periodic graphs. Preprint, 2006. Zbl1186.82039MR2529432
- 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
- F. Merkl and S. W. W. Rolles. Asymptotic behavior of edge-reinforced random walks. Ann. Probab. 35 (2007) 115–140. Zbl1206.82082MR2303945
- F. Merkl and S. W. W. Rolles. Recurrence of edge-reinforced random walk on a two-dimensional graph. Preprint, 2007. Zbl1180.82085
- 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
- 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
- R. Pemantle. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988) 1229–1241. Zbl0648.60077MR942765
- R. Pemantle. Random processes with reinforcement. Massachussets Institute of Technology doctoral dissertation, 1988.
- R. Pemantle. Vertex-reinforced random walk. Probab. Theory Related Fields 92 (1992) 117–136. Zbl0741.60029MR1156453
- R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1–79. Zbl1189.60138MR2282181
- O. Raimond. Self-attracting diffusions: case of the constant interaction. Probab. Theory Related Fields 107 (1997) 177–196. Zbl0881.60055MR1431218
- D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, New York, 1991. Zbl0731.60002MR1083357
- L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, volume 2. Wiley, New York, 1987. Zbl0627.60001MR921238
- S. W. W. Rolles. How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 (2003) 243–260. Zbl1029.60089MR1990056
- 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
- T. Sellke. Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11 (2006) 301–310. Zbl1113.60048MR2217818
- T. Sellke. Reinforced random walks on the d-dimensional integer lattice. Technical Report 94-26, Purdue University, 1994. Zbl1154.82011
- M. Takeshima. Behavior of 1-dimensional reinforced random walk. Osaka J. Math. 37 (2000) 355–372. Zbl0962.60017MR1772837
- P. Tarrès. Vertex-reinforced random walk on ℤ eventually gets stuck on five points. Ann. Probab. 32 (2004) 2650–2701. Zbl1068.60072MR2078554
- B. Tóth. The ‘true’ self-avoiding walk with bond repulsion on ℤ: limit theorems. Ann. Probab. 23 (1995) 1523–1556. Zbl0852.60083MR1379158
- B. Tóth. Self-interacting random motions – a survey. Random Walks (Budapest, 1999), Bolyai Society Mathematical Studies 9 (1999) 349–384. Zbl0953.60027MR1752900
- B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375–452. Zbl0912.60056MR1640799
- M. Vervoort. Games, walks and grammars: Problems I’ve worked on. PhD thesis, Universiteit van Amsterdam, 2000. Zbl1193.00001
- S. Volkov. Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 (2001) 66–91. Zbl1031.60089MR1825142
- 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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