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


KeywordsScaling limits, Bessel process, stochastic reinforcement, self-similar Markov process


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.


[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