Abstract
The problems considered in the present paper have their roots in two different cultures. The `true’ (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit et al. (Phys Rev B 27:1635–1645, 1983). This is a nearest neighbor non-Markovian random walk in \({{\mathbb Z}^d}\) which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (Probab Theory Relat Fields 92:337–349, 1992) (independently of the physics community), is the continuous space–time counterpart: a diffusion in \({{\mathbb R}^d}\) pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a path-wise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally ‘valid’ for the TSAW and SRBP cases), in Amit et al. (1983). The proof of the CLT follows the non-reversible version of Kipnis–Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.
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References
Amit D., Parisi G., Peliti L.: Asymptotic behavior of the ‘true’ self-avoiding walk. Phys. Rev. B 27, 1635–1645 (1983)
Bobkov S.G., Ledoux M.: From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10, 1028–1052 (2000)
Cranston M., Le Jan Y.: Self-attracting diffusions: two case studies. Math. Ann. 303, 87–93 (1995)
Cranston M., Mountford T.S.: The strong law of large numbers for a Brownian polymer. Ann. Probab. 24, 1300–1323 (1996)
Dobrushin, R.L., Suhov, Yu.M., Fritz, J.: A.N. Kolmogorov—the founder of the theory of reversible Markov processes. Uspekhi Mat. Nauk 43(6), 167–188 (1988) [English translation: Russ. Math. Surv. 43(6), 157–182]
Durrett R.T., Rogers L.C.G.: Asymptotic behavior of Brownian polymers. Probab. Theory Relat. Fields 92, 337–349 (1992)
Funaki, T.: Stochastic interface models. In: Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol. 1869. Springer, Brelin (2005)
Gordin M.I., Lifshits B.A.: Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239, 766–767 (1978)
Horváth, I., Tóth, B., Vető, B.: Diffusive limit for self-repelling Brownian polymers in d ≥ 3. http://arxiv.org/abs/0912.5174 (2009)
Janson S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)
Kipnis C., Varadhan S.R.S.: Central limit theorem for additive functionals of reversible Markov processes with applications to simple exclusion. Commun. Math. Phys. 106, 1–19 (1986)
Komorowski, T., Landim, C., Olla, S.: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Springer, New York (2011, to appear)
Komorowski T., Olla S.: On the sector condition and homogenization of diffusions with a Gaussian drift. J. Funct. Anal. 197, 179–211 (2003)
Kozlov S.M.: The method of averaging and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40, 61–120 (1885)
Landim C., Yau H.-T.: Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Relat. Fields 108, 321–356 (1997)
Mountford T.S., Tarrès P.: An asymptotic result for Brownian polymers. Ann. Inst. H. Poincaré Probab. Stat. 44, 29–46 (2008)
Norris J.R., Rogers L.C.G., Williams D.: Self-avoiding walk: a Brownian motion model with local time drift. Probab. Theory Relat. Fields 74, 271–287 (1987)
Obukhov S.P., Peliti L.: Renormalisation of the “true” self-avoiding walk. J. Phys. A 16, L147–L151 (1983)
Olla, S.: Central limit theorems for tagged particles and for Diffusions in random environment. In: Comets, F., Pardoux, É. (eds.) Milieux aléatoires Panor. Synthèses 12. Soc. Math. France, Paris (2001)
Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Mathematics, vol. 1021, pp. 507–517. Springer, Berlin (1983)
Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Fritz, J., Szász, D., Lebowitz, J.L. (eds.) Random Fields (Esztergom, 1979). Colloq. Math. Soc. János Bolyai, vol. 27, pp. 835–873. North-Holland, Amsterdam (1981)
Peliti L., Pietronero L.: Random walks with memory. Riv. Nuovo Cimento 10, 1–33 (1987)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vols. 1, 2. Academic Press, New York (1972–1975)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales. Ito Calculus, vol. 2. Wiley, New York (1987)
Sethuraman S., Varadhan S.R.S., Yau H.-T.: Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Commun. Pure Appl. Math. 53, 972–1006 (2000)
Simon B.: The \({P(\phi)_2}\) Euclidean (Quantum) Field Theory. Princeton University Press, Princeton (1974)
Tarrès, P., Tóth, B., Valkó, B.: Diffusivity bounds for 1d Brownian polymers. Ann. Probab. (2011, to appear)
Tóth B.: Persistent random walk in random environment. Probab. Theory Relat. Fields 71, 615–625 (1986)
Tóth B.: The ‘true’ self-avoiding walk with bond repulsion on \({{\mathbb Z}}\): limit theorems. Ann. Probab. 23, 1523–1556 (1995)
Tóth, B., Valkó, B.: Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2. http://arxiv.org/abs/1012.5698 (2011, submitted)
Tóth B., Vető B.: Continuous time ‘true’ self-avoiding random walk on \({{\mathbb Z}}\). ALEA, Lat. Am. J. Probab. Math. Stat. 8, 59–75 (2011)
Tóth B., Werner W.: The true self-repelling motion. Probab. Theory Relat. Fields 111, 375–452 (1998)
Varadhan S.R.S.: Self-diffusion of a tagged particle in equilibrium of asymmetric mean zero random walks with simple exclusion. Ann. Inst. H. Poincaré Probab. Stat. 31, 273–285 (1996)
Yaglom A.M.: On the statistical treatment of Brownian motion. Dokl. Akad. Nauk SSSR 56, 691–694 (1947) (in Russian)
Yaglom A.M.: On the statistical reversibility of Brownian motion. Mat. Sbornik 24, 457–492 (1949) (in Russian)
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Horváth, I., Tóth, B. & Vető, B. Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d ≥ 3. Probab. Theory Relat. Fields 153, 691–726 (2012). https://doi.org/10.1007/s00440-011-0358-3
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DOI: https://doi.org/10.1007/s00440-011-0358-3