Abstract
We introduce a continuous space limit of the vertex reinforced jump process (VRJP) in dimension one, which we call linearly reinforced motion (LRM) on \(\mathbb {R}\). It is constructed out of a convergent Bass–Burdzy flow. The proof goes through the representation of the VRJP as a mixture of Markov jump processes. As a by-product this gives a representation in terms of a mixture of diffusions of the LRM and of the Bass–Burdzy flow itself. We also show that our continuous space limit can be obtained out of the edge reinforced random walk (ERRW), since the ERRW and the VRJP are known to be closely related. Compared to the discrete space processes, the LRM has an additional symmetry in the initial local times (initial occupation profile): changing them amounts to a deterministic change of the space and time scales.
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References
Angel, O., Crawford, N.G., Kozma, G.: Localization for linearly edge reinforced random walks. Duke Math. J. 163(5), 889–921 (2014)
Attanasio, S.: Stochastic flow of diffeomorphisms for one-dimensional SDE with discontinuous drift. Electron. Commun. Probab. 15, 213–226 (2010)
Bass, R.F., Burdzy, K.: Stochastic bifurcation model. Ann. Probab. 27(1), 50–108 (1999)
Breiman, L.: Probability, Volume 7 of Classics in Applied Mathematics. SIAM, Philadelphia (1992)
Brox, T.: A one-dimensional diffusion process in a Wiener medium. Ann. Probab. 14(4), 1206–1218 (1986)
Basdevant, A.L., Singh, A.: Continuous-time vertex reinforced jump processes on Galton–Watson trees. Ann. Appl. Probab. 22(4), 1728–1743 (2012)
Coppersmith, D., Diaconis, P.: Random walks with reinforcement. Unpublished manuscript (1986)
Carmona, P., Petit, F., Yor, M.: Beta variables as times spent in \([0,+\infty [\) by certain perturbed Brownian motions. J. Lond. Math. Soc. Second Ser. 58(1), 239–256 (1998)
Davis, B.: Weak limits of perturbed random walks and the equation \({Y}_t={B}_t+\alpha \sup \lbrace {Y}_s: s\le t\rbrace + \beta \inf \lbrace {Y}_s: s\le t\rbrace \). Ann. Probab. 24(4), 2007–2023 (1996)
Davis, B., Dean, N.: Recurrence and transience preservation for vertex reinforced jump processes in one dimension. Ill. J. Math. 54(3), 869–893 (2010)
Diaconis, P.: Recent progress on de Finetti’s notions of exchangeability. In: Bernardo, J.M., Bayarri, M.J., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics (Valencia 1987), Volume 3 of Oxford Science Publications, pp. 111–125. Oxford University Press, New York (1988)
Diaconis, P., Rolles, S.W.W.: Bayesian analysis for reversible Markov chains. Ann. Stat. 34(3), 1270–1292 (2006)
Disertori, M., Spencer, T., Zirnbauer, M.R.: Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. Commun. Math. Phys. 300(2), 435–486 (2010)
Davis, B., Volkov, S.: Continuous time vertex-reinforced jump processes. Probab. Theory Relat. Fields 123(2), 281–300 (2002)
Davis, B., Volkov, S.: Vertex-reinforced jump processes on trees and finite graphs. Probab. Theory Relat. Fields 128(1), 42–62 (2004)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. Wiley, Blackwell (1986)
Föllmer, H.: Dirichlet processes. In: Williams, D. (ed), Stochastic Integrals: Proceedings of the LMS Durham Symposium, Volume 851 of Lecture Notes in Mathematics, pp. 476–478. Springer, New York (1981)
Hu, Y., Warren, J.: Ray–Knight theorems related to a stochastic flow. Stoch. Processes Appl. 86, 287–305 (2000)
Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths, Volume 125 of Grundlehren der Mathematischen Wissenschaften. Springer, New York (1974)
Keane, M.S., Rolles, S.W.W.: Edge-reinforced random walk on finite graphs. In: Clément, P., den Hollander, F., van Neerven, J., de Pagter, B. (eds.) Infinite Dimensional Stochastic Analysis (Amsterdam, 1999), pp. 217–234. Royal Netherlands Academy of Arts and Sciences, Netherlands (2000)
Merkl, F., Rolles, S.W.W.: A random environment for linearly edge-reinforced random walks on infinite graphs. Probab. Theory Relat. Fields 138, 157–176 (2007)
Pacheco, C. G.: From the Sinai’s walk to the Brox diffusion using bilinear forms. (2016). ArXiv preprint arXiv:1605.02826
Rolles, S.W.W.: On the recurrence of edge-reinforced random walk on \(\mathbb{Z} \times G\). Probab. Theory Relat. Fields 135(2), 216–264 (2006)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften, 3rd edn. Springer, New York (1999)
Schumacher, S.: Diffusions with random coefficients. In: Durrett, R. (ed.) Particle Systems, Random Media, and Large Deviations, Volume 41 of Contemporary Mathematics, pp. 351–356. American Mathematical Society, Providence (1985)
Seignourel, P.: Discrete schemes for processes in random media. Probab. Theory Relat. Fields 118(3), 293–322 (2000)
Sinai, Y.G.: The limiting behavior of a one-dimensional random walk in a random medium. Theory Probab. Appl. 27(2), 256–268 (1982)
Sabot, C., Tarrès, P.: Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. J. Eur. Math. Soc. 17(9), 2353–2378 (2015)
Sabot, C., Tarrès, P., Zeng, X.: The vertex reinforced jump process and random Schrödinger operator on finite graphs. Ann. Probab. 45(6), 3967–3986 (2017)
Tanaka, H.: Diffusion processes in random environments. In: Chatterji, S.D. (ed) Proceedings of the International Congress of Mathematicians, pp. 1047–1054. Birkhäuser, Basel (1995)
Van Zanten, H.: On uniform laws of large numbers for ergodic diffusions and consistency of estimators. Stat. Inference Stoch. Process. 6, 199–213 (2003)
Warren, J.: A stochastic flow arising in the study of local times. Probab. Theory Relat. Fields 133, 559–572 (2005)
Warren, J., Yor, M.: The Brownian Burglar: conditioning Brownian motion by its local time process. In: Azééma, J., Yor, M., Émery, M., Ledoux, M. (eds) Séminaire de Probabilités XXXII, volume 1686 of Lecture Notes in Mathematics, pp. 328–342. Springer, New York (1998)
Acknowledgements
This work was supported by the French National Research Agency (ANR) Grant within the project MALIN (ANR-16-CE93-0003). This work was partly supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). TL acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. PT acknowledges the support of the National Science Foundation of China (NSFC), Grant No. 11771293.
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Lupu, T., Sabot, C. & Tarrès, P. Fine mesh limit of the VRJP in dimension one and Bass–Burdzy flow. Probab. Theory Relat. Fields 177, 55–90 (2020). https://doi.org/10.1007/s00440-019-00944-y
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DOI: https://doi.org/10.1007/s00440-019-00944-y
Keywords
- Self-interacting diffusion
- Reinforcement
- Diffusion in random environment
- Local time
Mathematics Subject Classification
- Primary 60J60
- 60K35
- 60K37
- Secondary 60J55