logo Institut Camille Jordan

CONFLUENTES MATHEMATICI

logo UMPA
no. 1
Iterated Brownian motion ad libitum is not the pseudo-arc
Jérôme Casse1 ; Nicolas Curien2
1 CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016 Paris, France
2 Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France Institut Universitaire de France
Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 35-42.

The construction of a random continuum 𝒞 from independent two-sided Brownian motions as considered in [11] almost surely yields a non-degenerate indecomposable continuum. We show that 𝒞 is not-hereditarily indecomposable and, in particular, it is (unfortunately) not the pseudo-arc.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/cml.70
Classification : 54F15, 60J65
Mots clés : continuum, iterated Brownian motions, pseudo-arc
Jérôme Casse 1 ; Nicolas Curien 2

1 CEREMADE, CNRS, UMR 7534, Université Paris-Dauphine, PSL University, 75016 Paris, France
2 Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France Institut Universitaire de France
Licence : CC-BY-NC-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CML_2021__13_1_35_0,
     author = {J\'er\^ome Casse and Nicolas Curien},
     title = {Iterated {Brownian} motion ad libitum is not the pseudo-arc},
     journal = {Confluentes Mathematici},
     pages = {35--42},
     publisher = {Institut Camille Jordan},
     volume = {13},
     number = {1},
     year = {2021},
     doi = {10.5802/cml.70},
     language = {en},
     url = {https://cml.centre-mersenne.org/articles/10.5802/cml.70/}
}
TY  - JOUR
AU  - Jérôme Casse
AU  - Nicolas Curien
TI  - Iterated Brownian motion ad libitum is not the pseudo-arc
JO  - Confluentes Mathematici
PY  - 2021
SP  - 35
EP  - 42
VL  - 13
IS  - 1
PB  - Institut Camille Jordan
UR  - https://cml.centre-mersenne.org/articles/10.5802/cml.70/
DO  - 10.5802/cml.70
LA  - en
ID  - CML_2021__13_1_35_0
ER  - 
%0 Journal Article
%A Jérôme Casse
%A Nicolas Curien
%T Iterated Brownian motion ad libitum is not the pseudo-arc
%J Confluentes Mathematici
%D 2021
%P 35-42
%V 13
%N 1
%I Institut Camille Jordan
%U https://cml.centre-mersenne.org/articles/10.5802/cml.70/
%R 10.5802/cml.70
%G en
%F CML_2021__13_1_35_0
Jérôme Casse; Nicolas Curien. Iterated Brownian motion ad libitum is not the pseudo-arc. Confluentes Mathematici, Tome 13 (2021) no. 1, pp. 35-42. doi : 10.5802/cml.70. https://cml.centre-mersenne.org/articles/10.5802/cml.70/

[1] Jean Bertoin Iterated Brownian motion and stable (1 4) subordinator, Statist. Probab. Lett., Volume 27 (1996) no. 2, pp. 111-114 | DOI | MR | Zbl

[2] R. H. Bing A homogeneous indecomposable plane continuum, Duke Math. J., Volume 15 (1948), pp. 729-742 http://projecteuclid.org/euclid.dmj/1077475025 | MR | Zbl

[3] R. H. Bing Concerning hereditarily indecomposable continua, Pacific J. Math., Volume 1 (1951), pp. 43-51 http://projecteuclid.org/euclid.pjm/1102613150 | DOI | MR | Zbl

[4] Krzysztof Burdzy Some path properties of iterated Brownian motion, Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992) (Progr. Probab.), Volume 33 (1993), pp. 67-87 | DOI | MR | Zbl

[5] Krzysztof Burdzy; Davar Khoshnevisan The level sets of iterated Brownian motion, Séminaire de Probabilités, XXIX (Lecture Notes in Math.), Volume 1613, Springer, Berlin, 1995, pp. 231-236 | DOI | Numdam | MR | Zbl

[6] Jérôme Casse; Jean-François Marckert Processes iterated ad libitum, Stochastic Process. Appl., Volume 126 (2016) no. 11, pp. 3353-3376 | DOI | MR | Zbl

[7] Nicolas Curien; Takis Konstantopoulos Iterating Brownian motions, ad libitum, J. Theoret. Probab., Volume 27 (2014) no. 2, pp. 433-448 | DOI | MR | Zbl

[8] Nathalie Eisenbaum; Zhan Shi Uniform oscillations of the local time of iterated Brownian motion, Bernoulli, Volume 5 (1999) no. 1, pp. 49-65 | DOI | MR | Zbl

[9] Tadahisa Funaki Probabilistic construction of the solution of some higher order parabolic differential equation, Proc. Japan Acad. Ser. A Math. Sci., Volume 55 (1979) no. 5, pp. 176-179 http://projecteuclid.org/euclid.pja/1195517312 | MR | Zbl

[10] Davar Khoshnevisan; Thomas M. Lewis Iterated Brownian motion and its intrinsic skeletal structure, Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996) (Progr. Probab.), Volume 45 (1999), pp. 201-210 | DOI | MR | Zbl

[11] Viktor Kiss; Sławomir Solecki Random continuum and Brownian motion, arXiv preprint arXiv:2004.01367 (2020)

[12] Bronisław Knaster Un continu dont tout sous-continu est indécomposable, Fundamenta Mathematicae, Volume 1 (1922) no. 3, pp. 247-286 | DOI | Zbl

[13] Wayne Lewis; Piotr Minc Drawing the pseudo-arc, Houston J. Math, Volume 36 (2010), pp. 905-934 | MR | Zbl

[14] Edwin E. Moise An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc., Volume 63 (1948), pp. 581-594 | DOI | MR | Zbl

[15] Peter Mörters; Yuval Peres Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, 30, Cambridge University Press, Cambridge, 2010, xii+403 pages (With an appendix by Oded Schramm and Wendelin Werner) | DOI | MR | Zbl

[16] Sam B. Nadler Continuum theory : an introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992, xiv+328 pages | MR | Zbl

[17] Enzo Orsingher; Luisa Beghin Fractional diffusion equations and processes with randomly varying time, Ann. Probab., Volume 37 (2009) no. 1, pp. 206-249 | DOI | MR | Zbl

[18] Yimin Xiao Local times and related properties of multidimensional iterated Brownian motion, J. Theoret. Probab., Volume 11 (1998) no. 2, pp. 383-408 | DOI | MR | Zbl

Cité par Sources :