On démontre que, dans , les marches aléatoires en milieu aléatoire i.i.d. de Dirichlet – ou, de façon équivalente, les marches renforcées par arêtes orientées – ont presque sûrement une direction asymptotique égale à la direction de la dérive initiale, c’est-à-dire que converge vers quand , à moins que cette dérive soit nulle. Ceci est obtenu en généralisant le résultat de transience directionnelle de (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). De plus, on explicite la valeur ou la loi de certaines probabilités, ce qui démontre et généralise une conjecture de ce dernier article.
We prove that, on , random walks in i.i.d. Dirichlet environment – or equivalently oriented-edge reinforced random walks – have almost surely an asymptotic direction equal to the direction of the initial drift, i.e. converges to as , unless this drift is zero. This is obtained by generalizing the result of directional transience from (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of that paper.
Mots clés : random walk, random environment, Dirichlet distribution, reinforced random walk, asymptotic direction, time reversal
@article{AIHPB_2015__51_2_716_0, author = {Tournier, Laurent}, title = {Asymptotic direction of random walks in {Dirichlet} environment}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {716--726}, publisher = {Gauthier-Villars}, volume = {51}, number = {2}, year = {2015}, doi = {10.1214/13-AIHP582}, mrnumber = {3335022}, zbl = {1319.60095}, language = {en}, url = {http://www.numdam.org/articles/10.1214/13-AIHP582/} }
TY - JOUR AU - Tournier, Laurent TI - Asymptotic direction of random walks in Dirichlet environment JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 716 EP - 726 VL - 51 IS - 2 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/13-AIHP582/ DO - 10.1214/13-AIHP582 LA - en ID - AIHPB_2015__51_2_716_0 ER -
%0 Journal Article %A Tournier, Laurent %T Asymptotic direction of random walks in Dirichlet environment %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 716-726 %V 51 %N 2 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/13-AIHP582/ %R 10.1214/13-AIHP582 %G en %F AIHPB_2015__51_2_716_0
Tournier, Laurent. Asymptotic direction of random walks in Dirichlet environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 716-726. doi : 10.1214/13-AIHP582. http://www.numdam.org/articles/10.1214/13-AIHP582/
[1] Sub-ballistic random walk in Dirichlet environment. Electron. J. Probab. 18 (58) (2013) 1–25 (electronic). | MR | Zbl
.[2] Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4 (1991) 3–36. | DOI | MR | Zbl
and .[3] Asymptotic direction in random walks in random environment revisited. Braz. J. Probab. Stat. 24 (2) (2010) 212–225. | MR | Zbl
and .[4] Edge oriented reinforced random walks and RWRE. C. R. Math. Acad. Sci. Paris 335 (11) (2002) 941–946. | MR | Zbl
and .[5] Random walks in a Dirichlet environment. Electron. J. Probab. 11 (31) (2006) 802–817 (electronic). | MR | Zbl
and .[6] A zero–one law for planar random walks in random environment. Ann. Probab. 29 (4) (2001) 1716–1732. | MR | Zbl
and .[7] Ballistic random walks in random environments at low disorder. Ann. Probab. 32 (4) (2004) 2996–3023. | MR | Zbl
.[8] Random walks in random Dirichlet environment are transient in dimension . Probab. Theory Related Fields 151 (1–2) (2009) 297–317. | MR | Zbl
.[9] Random Dirichlet environment viewed from the particle in dimension . Ann. Probab. 41 (2) (2013) 722–743. | MR | Zbl
.[10] Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment. Ann. Inst. Henri Poincaré Probab. Stat. 47 (1) (2011) 1–8. | Numdam | MR | Zbl
and .[11] Asymptotic direction for random walks in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 43 (6) (2007) 751–761. | MR | Zbl
.[12] Integrability of exit times and ballisticity for random walks in Dirichlet environment. Electron. J. Probab. 14 (16) (2009) 431–451 (electronic). | MR | Zbl
.[13] The zero–one law for planar random walks in i.i.d. random environments revisited. Electron. Commun. Probab. 12 (2007) 326–335 (electronic). | DOI | MR | Zbl
.Cité par Sources :