On the local times of noise reinforced Bessel processes
Annales Henri Lebesgue, Volume 5 (2022), pp. 1277-1294.

We investigate the effects of noise reinforcement on a Bessel process of dimension d(0,2), and more specifically on the asymptotic behavior of its additive functionals. This leads us to introduce a local time process and its inverse. We identify the latter as an increasing self-similar (time-homogeneous) Markov process, and from this, several explicit results can be deduced.

On étudie les effects du renforcement du bruit sur un processus de Bessel de dimension d(0,2), et plus précisément sur le comportement asymptotique de ses fonctionnelles additives. Cela nous conduit à introduire le processus du temps local et son inverse. Nous identifions ce dernier comme un processus de Markov (homogène en temps) auto-similaire, ce qui conduit à plusieurs résultats explicites.

Published online:
DOI: 10.5802/ahl.151
Classification: 60J55, 60J60
Keywords: Scaling limits, Bessel process, stochastic reinforcement, self-similar Markov process
Bertoin, Jean 1

1 Institute of Mathematics, University of Zurich, (Switzerland)
     author = {Bertoin, Jean},
     title = {On the local times of noise reinforced {Bessel} processes},
     journal = {Annales Henri Lebesgue},
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     year = {2022},
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Bertoin, Jean. On the local times of noise reinforced Bessel processes. Annales Henri Lebesgue, Volume 5 (2022), pp. 1277-1294. doi : 10.5802/ahl.151. http://www.numdam.org/articles/10.5802/ahl.151/

[BB16] Baur, Erich; Bertoin, Jean Elephant random walks and their connection to Pólya-type urns, Phys. Rev. E, Volume 94 (2016) no. 5, 052134 | DOI

[BC02] Bertoin, Jean; Caballero, María E. Entrance from 0+ for increasing semi-stable Markov processes, Bernoulli, Volume 8 (2002) no. 2, pp. 195-205 | MR | Zbl

[Ber90] Bertoin, Jean Excursions of a BES 0 (d) and its drift term (0<d<1), Probab. Theory Relat. Fields, Volume 84 (1990) no. 2, pp. 231-250 | DOI | MR | Zbl

[Ber20] Bertoin, Jean Noise reinforcement for Lévy processes, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 3, pp. 2236-2252 | DOI | Zbl

[Ber21] Bertoin, Jean Universality of noise reinforced Brownian motions, In and out of equilibrium. 3. Celebrating Vladas Sidoravicius (Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M., eds.) (Progress in Probability), Volume 77, Birkhäuser/Springer, 2021, pp. 147-161 | DOI | MR | Zbl

[Ber22] Bertoin, Jean Counting the zeros of an elephant random walk, Trans. Am. Math. Soc., Volume 375 (2022) no. 8, pp. 5539-5560 | MR | Zbl

[BG68] Blumenthal, Robert M.; Getoor, Ronald K. Markov processes and potential theory, Pure and Applied Mathematics, 29, Academic Press Inc., 1968 | MR | Zbl

[BHZ02] Bai, Zhi-Dong; Hu, Feifang; Zhang, Li-Xin Gaussian approximation theorems for urn models and their applications, Ann. Appl. Probab., Volume 12 (2002) no. 4, pp. 1149-1173 | DOI | MR | Zbl

[BO22] Bertenghi, Marco; Ortiz, Alejandro Rosales Joint Invariance Principles for Random Walks with Positively and Negatively Reinforced Steps, J. Stat. Phys., Volume 189 (2022) no. 3, 35 | DOI | MR | Zbl

[BY02] Bertoin, Jean; Yor, Marc The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes, Potential Anal., Volume 17 (2002) no. 4, pp. 389-400 | DOI | MR | Zbl

[CC06] Caballero, María E.; Chaumont, Loïc Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab., Volume 43 (2006) no. 4, pp. 967-983 | DOI | Zbl

[CGS17] Coletti, Cristian F.; Gava, Renato; Schütz, Gunter M. A strong invariance principle for the elephant random walk, J. Stat. Mech. Theory Exp. (2017) no. 12, p. 123207, 8 | DOI | MR | Zbl

[CPY94] Carmona, Philippe; Petit, F.; Yor, Marc Sur les fonctionnelles exponentielles de certains processus de Lévy, Stochastics Stochastics Rep., Volume 47 (1994) no. 1-2, pp. 71-101 | DOI | Zbl

[CY03] Chaumont, Loïc; Yor, Marc Exercises in probability. A guided tour from measure theory to random processes, via conditioning, Cambridge Series in Statistical and Probabilistic Mathematics, 13, Cambridge University Press, 2003 | DOI | MR | Zbl

[DK57] Darling, Donald A.; Kac, Mark On occupation times for Markoff processes, Trans. Am. Math. Soc., Volume 84 (1957), pp. 444-458 | DOI | MR | Zbl

[DMRVY08] Donati-Martin, Catherine; Roynette, Bernard; Vallois, Pierre; Yor, Marc On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension d=2(1-α),0<α<1, Stud. Sci. Math. Hung., Volume 45 (2008) no. 2, pp. 207-221 | DOI | Zbl

[FP99] Fitzsimmons, Patrick J.; Pitman, Jim Kac’s moment formula and the Feynman–Kac formula for additive functionals of a Markov process, Stochastic Processes Appl., Volume 79 (1999) no. 1, pp. 117-134 | DOI | MR | Zbl

[Gou93] Gouet, Raúl Martingale functional central limit theorems for a generalized Pólya urn, Ann. Probab., Volume 21 (1993) no. 3, pp. 1624-1639 | MR | Zbl

[Haa21] Haas, Bénédicte Precise asymptotics for the density and the upper tail of exponential functionals of subordinators (2021) (https://arxiv.org/abs/2106.08691v1)

[HNX14] Hu, Yaozhong; Nualart, David; Xu, Fangjun Central limit theorem for an additive functional of the fractional Brownian motion, Ann. Probab., Volume 42 (2014) no. 1, pp. 168-203 | DOI | MR | Zbl

[HY13] Hirsch, Francis; Yor, Marc On the remarkable Lamperti representation of the inverse local time of a radial Ornstein–Uhlenbeck process, Bull. Belg. Math. Soc. Simon Stevin, Volume 20 (2013) no. 3, pp. 435-449 | MR | Zbl

[KK79] Kasahara, Yuji; Kotani, Shin’ichi On limit processes for a class of additive functionals of recurrent diffusion processes, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 49 (1979), pp. 133-153 | DOI | MR | Zbl

[KM96] Kasahara, Yuji; Matsumoto, Yuki On Kallianpur–Robbins law for fractional Brownian motion, J. Math. Kyoto Univ., Volume 36 (1996) no. 4, pp. 815-824 | DOI | MR | Zbl

[KP22] Kyprianou, Andreas E.; Pardo, Juan C. Stable Lévy Processes via Lamperti-Type Representations, Institute of Mathematical Statistics Monographs, 7, Cambridge University Press, 2022 | DOI | Zbl

[KR53] Kallianpur, Gopinath; Robbins, Herbert E. Ergodic property of the Brownian motion process, Proc. Natl. Acad. Sci. USA, Volume 39 (1953), pp. 525-533 | DOI | MR | Zbl

[Kôn96] Kôno, Norio Kallianpur-Robbins law for fractional Brownian motion, Probability theory and mathematical statistics. Proceedings of the seventh Japan-Russia symposium, Tokyo, Japan, July 26–30, 1995, World Scientific, 1996, pp. 229-236 | Zbl

[Lam72] Lamperti, John Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 22 (1972), pp. 205-225 | DOI | MR | Zbl

[Law19] Lawler, Gregory F. Notes on the Bessel processes (2019) (available at: https://math.uchicago.edu/~lawler/bessel18new.pdf)

[MS21] Minchev, Martin; Savov, Mladen Asymptotic of densities of exponential functionals of subordinators (2021) (https://arxiv.org/abs/2104.05381)

[MY08] Mansuy, Roger; Yor, Marc Aspects of Brownian motion, Universitext, Springer, 2008 | DOI | MR | Zbl

[Pem07] Pemantle, Robin A survey of random processes with reinforcement, Probab. Surveys, Volume 4 (2007), pp. 1-79 | DOI | MR | Zbl

[PRVS13] Pardo, Juan C.; Rivero, Victor; Van Schaik, Kees On the density of exponential functionals of Lévy processes, Bernoulli, Volume 19 (2013) no. 5A, pp. 1938-1964 | DOI | Zbl

[PS18] Patie, Pierre; Savov, Mladen Bernstein-gamma functions and exponential functionals of Lévy processes, Electron. J. Probab., Volume 23 (2018), 75 | DOI | Zbl

[PSV77] Papanicolaou, George C.; Stroock, Daniel W.; Varadhan, Srinivasa R. S. Martingale approach to some limit theorems, Papers from the Duke Turbulence Conference (Duke University, Durham, N.C., 1976), Duke University, Durham, N.C., 1977, p. ii+120 | Zbl

[RW00] Rogers, L. C. G.; Williams, David Diffusions, Markov processes, and martingales. Vol. 1 Foundations, Cambridge Mathematical Library, Cambridge University Press, 2000 reprint of the second (1994) edition | DOI | MR | Zbl

[RY99] Revuz, Daniel; Yor, Marc Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer, 1999 | DOI | MR | Zbl

[Sat99] Sato, Ken-iti Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, 1999 (translated from the 1990 Japanese original, Revised by the author) | MR | Zbl

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