An asymptotic result for brownian polymers

Mountford, Thomas; Tarrès, Pierre

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 29-46 / Harvested from Numdam

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Abstract

Nous considérons un modèle de formation de polymères introduit par Durrett et Rogers (Probab. Theory Related Fields 92 (1992) 337-349). Nous prouvons leur conjecture sur le comportement asymptotique du processus continu associé $X_{t}$ (correspondant à l’emplacement de l’extrémité du polymère au temps $t$ ) pour un type particulier de fonction d’interaction répulsive à support non compact.

We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337-349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_{t}$ (corresponding to the location of the end of the polymer at time $t$ ) for a particular type of repelling interaction function without compact support.

Publié le : 2008-01-01

MR 2451570 | Zbl 1175.60084 | 2 citations dans Numdam

DOI : https://doi.org/10.1214/07-AIHP113
Classification: 60F15, 60K35

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     author = {Mountford, Thomas and Tarr\`es, Pierre},
     title = {An asymptotic result for brownian polymers},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
     pages = {29-46},
     doi = {10.1214/07-AIHP113},
     mrnumber = {2451570},
     zbl = {1175.60084},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_1_29_0}
}

Mountford, Thomas; Tarrès, Pierre. An asymptotic result for brownian polymers. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 29-46. doi : 10.1214/07-AIHP113. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_1_29_0/

Bibliographie

M. Benaïm. Vertex-reinforced random walks and a conjecture of Pemantle. Ann. Probab. 25 (1997) 361-392. | MR 1428513 | Zbl 0873.60044

M. Benaïm, M. Ledoux and O. Raimond. Self-interacting diffusions. Probab. Theory Related Fields 122 (2002) 1-41. | MR 1883716 | Zbl 1042.60060

M. Benaïm and O. Raimond. Self-interacting diffusions II: convergence in law. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 1043-1055. | Numdam | MR 2010396 | Zbl 1064.60191

M. Benaïm and O. Raimond. Self-interacting diffusions III: symmetric interactions. Ann. Probab. 33 (2005) 1716-1759. | MR 2165577 | Zbl 1085.60073

A. Collevecchio. Limit theorems for Diaconis walk on certain trees. Probab. Theory Related Fields 136 (2006) 81-101. | MR 2240783 | Zbl 1109.60027

A. Collevecchio. On the transience of processes defined on Galton-Watson trees. Ann. Probab. 34 (2006) 870-878. | MR 2243872 | Zbl 1104.60048

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. | MR 1348356 | Zbl 0838.60052

M. Cranston and T. S. Mountford. The strong law of large numbers for a Brownian polymer. Ann. Probab. 2 (1996) 1300-1323. | MR 1411496 | Zbl 0873.60014

B. Davis. Reinforced random walk. Probab. Theory Related Fields 84 (1990) 203-229. | MR 1030727 | Zbl 0665.60077

B. Davis. Brownian motion and random walk perturbed at extrema. Probab. Theory Related Fields 113 (1999) 501-518. | MR 1717528 | Zbl 0930.60041

B. Davis. Reinforced and perturbed random walks. Random Walks (Budapest, 1998) János Bolyai Math. Soc., Budapest 9 (1999) 113-126. | MR 1752892 | Zbl 0953.60028

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. | MR 2052266 | Zbl 1061.60069

P. Del Moral and L. Miclo. Self-interacting Markov chains. Stoch. Anal. Appl. 24 (2006) 615-660. | MR 2220075 | Zbl 1093.60068

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. | MR 1008047 | Zbl 0707.60033

P. Diaconis and S. W. W. Rolles. Bayesian analysis for reversible Markov chains. Ann. Statist. 34 (2006) 1270-1292. | MR 2278358 | Zbl 1118.62085

R. T. Durrett, H. Kesten and V. Limic. Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122 (2002) 567-592. | MR 1902191 | Zbl 0995.60042

R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields 92 (1992) 337-349. | MR 1165516 | Zbl 0767.60080

I. Gihman and A. G. Skorohod. Theory of Stochastic Processes, volume 3. Springer, 1979. | Zbl 0404.60061

S. Herrmann and B. Roynette. Boundedness and convergence of some self-attracting diffusions. Math. Ann. 325 (2003) 81-96. | MR 1957265 | Zbl 1010.60033

N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981. | MR 1011252 | Zbl 0495.60005

I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, New York, 1988. | MR 917065 | Zbl 0638.60065

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. | MR 1832379 | Zbl 0986.05092

M. S. Keane and S. W. W. Rolles. Tubular recurrence. Acta Math. Hungar. 97 (2002) 207-221. | MR 1933730 | Zbl 1026.60089

V. Limic. Attracting edge property for a class of reinforced random walks. Ann. Probab. 31 (2003) 1615-1654. | MR 1989445 | Zbl 1057.60048

V. Limic and P. Tarrès. Attracting edge and strongly edge reinforced walks. Ann. Probab. 35 (2007) 1783-1806. | MR 2349575 | Zbl 1131.60036

F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on a ladder. Ann. Probab. 33 (2005) 2051-2093. | MR 2184091 | Zbl 1102.82010

F. Merkl and S. W. W. Rolles. Edge-reinforced random walk on one-dimensional periodic graphs. Preprint, 2006. | MR 2529432 | Zbl pre05607256

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. | MR 2306189 | Zbl 1125.82014

F. Merkl and S. W. W. Rolles. Asymptotic behavior of edge-reinforced random walks. Ann. Probab. 35 (2007) 115-140. | MR 2303945 | Zbl pre05150500

F. Merkl and S. W. W. Rolles. Recurrence of edge-reinforced random walk on a two-dimensional graph. Preprint, 2007. | Zbl pre05625050

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. | MR 871255 | Zbl 0611.60052

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. | MR 1462051 | Zbl 0990.35128

R. Pemantle. Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16 (1988) 1229-1241. | MR 942765 | Zbl 0648.60077

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. | MR 1156453 | Zbl 0741.60029

R. Pemantle. A survey of random processes with reinforcement. Probab. Surv. 4 (2007) 1-79. | MR 2282181

O. Raimond. Self-attracting diffusions: case of the constant interaction. Probab. Theory Related Fields 107 (1997) 177-196. | MR 1431218 | Zbl 0881.60055

D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 2nd edition. Springer, New York, 1991. | MR 1083357 | Zbl 0731.60002

L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, volume 2. Wiley, New York, 1987. | MR 921238 | Zbl 0627.60001

S. W. W. Rolles. How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126 (2003) 243-260. | MR 1990056 | Zbl 1029.60089

S. W. W. Rolles. On the recurrence of edge-reinforced random walk on ℤ×G. Probab. Theory Related Fields 135 (2006) 216-264. | MR 2218872 | Zbl pre05032374

T. Sellke. Recurrence of reinforced random walk on a ladder. Electron. J. Probab. 11 (2006) 301-310. | MR 2217818 | Zbl 1113.60048

T. Sellke. Reinforced random walks on the d-dimensional integer lattice. Technical Report 94-26, Purdue University, 1994. | Zbl 1154.82011

M. Takeshima. Behavior of 1-dimensional reinforced random walk. Osaka J. Math. 37 (2000) 355-372. | MR 1772837 | Zbl 0962.60017

P. Tarrès. Vertex-reinforced random walk on ℤ eventually gets stuck on five points. Ann. Probab. 32 (2004) 2650-2701. | MR 2078554 | Zbl 1068.60072

B. Tóth. The ‘true' self-avoiding walk with bond repulsion on ℤ: limit theorems. Ann. Probab. 23 (1995) 1523-1556. | MR 1379158 | Zbl 0852.60083

B. Tóth. Self-interacting random motions - a survey. Random Walks (Budapest, 1999), Bolyai Society Mathematical Studies 9 (1999) 349-384. | MR 1752900 | Zbl 0953.60027

B. Tóth and W. Werner. The true self-repelling motion. Probab. Theory Related Fields 111 (1998) 375-452. | MR 1640799 | Zbl 0912.60056

M. Vervoort. Games, walks and grammars: Problems I've worked on. PhD thesis, Universiteit van Amsterdam, 2000. | Zbl pre01797601

S. Volkov. Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29 (2001) 66-91. | MR 1825142 | Zbl 1031.60089

S. Volkov. Phase transition in vertex-reinforced random walks on ℤ with non-linear reinforcement. J. Theoret. Probab. 19 (2006) 691-700. | MR 2280515 | Zbl 1107.60068