Three examples of brownian flows on
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1323-1346.

Nous montrons que le seul flot solution de l’équation différentielle stochastique (EDS) sur

dX t =1 {X t >0} W + (dt)+1 {X t <0} dW - (dt),
W + et W - sont deux bruits blancs indépendants, est un flot coalescent que nous noterons ϕ ± . Le flot ϕ ± est une solution Wiener de l’équation. De plus, K + =𝖤[δ ϕ ± |W + ] est l’unique solution (c’est aussi une solution Wiener) de l’EDS
K s,t + f(x)=f(x)+ s t K s,u (1 + f ' )(x)W + (du)+1 2 s t K s,u f``(x)du
pour tout s<t, x et f une fonction deux fois continûment mesurable. Un troisième flot ϕ + peut être construit à partir des mouvements à n points de K + . Ce flot est coalescent et ses mouvements à n points sont donnés par les mouvements à n points de K + jusqu’au premier temps de coalescence, avec comme condition que lorsque deux points se rencontrent, ils restent confondus. On remarquera finalement que K + =𝖤[δ ϕ + |W + ].

We show that the only flow solving the stochastic differential equation (SDE) on

dX t =1 {X t >0} W + (dt)+1 {X t <0} dW - (dt),
where W + and W - are two independent white noises, is a coalescing flow we will denote by ϕ ± . The flow ϕ ± is a Wiener solution of the SDE. Moreover, K + =𝖤[δ ϕ ± |W + ] is the unique solution (it is also a Wiener solution) of the SDE
K s,t + f(x)=f(x)+ s t K s,u (1 + f ' )(x)W + (du)+1 2 s t K s,u f``(x)du
for s<t, x and f a twice continuously differentiable function. A third flow ϕ + can be constructed out of the n-point motions of K + . This flow is coalescing and its n-point motion is given by the n-point motions of K + up to the first coalescing time, with the condition that when two points meet, they stay together. We note finally that K + =𝖤[δ ϕ + |W + ].

DOI : 10.1214/13-AIHP541
Classification : 60H25, 60J60
Mots clés : stochastic flows, coalescing flow, Arratia flow or brownian web, brownian motion with oblique reflection on a wedge
@article{AIHPB_2014__50_4_1323_0,
     author = {Le Jan, Yves and Raimond, Olivier},
     title = {Three examples of brownian flows on $\mathbb {R}$},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1323--1346},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     doi = {10.1214/13-AIHP541},
     mrnumber = {3269996},
     zbl = {06377556},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP541/}
}
TY  - JOUR
AU  - Le Jan, Yves
AU  - Raimond, Olivier
TI  - Three examples of brownian flows on $\mathbb {R}$
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 1323
EP  - 1346
VL  - 50
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP541/
DO  - 10.1214/13-AIHP541
LA  - en
ID  - AIHPB_2014__50_4_1323_0
ER  - 
%0 Journal Article
%A Le Jan, Yves
%A Raimond, Olivier
%T Three examples of brownian flows on $\mathbb {R}$
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 1323-1346
%V 50
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP541/
%R 10.1214/13-AIHP541
%G en
%F AIHPB_2014__50_4_1323_0
Le Jan, Yves; Raimond, Olivier. Three examples of brownian flows on $\mathbb {R}$. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1323-1346. doi : 10.1214/13-AIHP541. http://archive.numdam.org/articles/10.1214/13-AIHP541/

[1] R. Arratia. Brownian motion on the line. Ph.D. dissertation, Univ. Wisconsin, Madison, 1979.

[2] M. Benabdallah, S. Bouhadou and Y. Ouknine. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations with jumps. Preprint, 2011. Available at arXiv:1108.4016.

[3] K. Burdzy, W. Kang and K. Ramanan. The Skorokhod problem in a time-dependent interval. Stochastic Process. Appl. 119 (2009) 428-452. | MR | Zbl

[4] K. Burdzy and H. Kaspi. Lenses in skew Brownian flow. Ann. Probab. 32 (2004) 3085-3115. | MR | Zbl

[5] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web. Proc. Natl. Acad. Sci. USA 99 (2002) 15888-15893 (electronic). | MR | Zbl

[6] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. The Brownian web: Characterization and convergence. Ann. Probab. 32 (2004) 2857-2883. | MR | Zbl

[7] H. Hajri. Discrete approximations to solution flows of Tanaka's SDE related to Walsh Brownian motion. In Séminaire de Probabilités XLIV 167-190. Lecture Notes in Math. 2046. Springer, Heidelberg, 2012. | MR | Zbl

[8] H. Hajri. Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 (2011) 1563-1599 (electronic). | MR | Zbl

[9] W. Kang and K. Ramanan. A Dirichlet process characterization of a class of reflected diffusions. Ann. Probab. 38 (2010) 1062-1105. | MR | Zbl

[10] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springerg, New York, 1991. | MR | Zbl

[11] Y. Le Jan and O. Raimond. Integration of Brownian vector fields. Ann. Probab. 30 (2002) 826-873. | MR | Zbl

[12] Y. Le Jan and O. Raimond. Flows, coalescence and noise. Ann. Probab. 32 (2004) 1247-1315. | MR | Zbl

[13] Y. Le Jan and O. Raimond. Stochastic flows on the circle. In Probability and Partial Differential Equations in Modern Applied Mathematics 151-162. IMA Vol. Math. Appl. 140. Springer, New York, 2005. | MR | Zbl

[14] Y. Le Jan and O. Raimond. Flows associated to Tanaka's SDE. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 21-34. | MR | Zbl

[15] B. Micaux. Flots stochastiques d'opérateurs dirigés par des bruits gaussiens et poissonniens. Ph.D. dissertation, Univ. Paris-Sud, 2007.

[16] R. Mikulevicius and B. L. Rozovskii. Stochastic Navier-Stokes equations for turbulent flows. SIAM J. Math. Anal. 35 (2004) 1250-1310. | MR | Zbl

[17] R. Mikulevicius and B. L. Rozovskii. Global L 2 -solutions of stochastic Navier-Stokes equations. Ann. Probab. 33 (2005) 137-176. | MR | Zbl

[18] V. Prokaj. The solution of the perturbed Tanaka equation is pathwise unique. Preprint, 2011. Available at arXiv:1104.0740. | MR | Zbl

[19] K. Ramanan. Reflected diffusions defined via the extended Skorokhod map. Electron. J. Probab. 11 (2006) 934-992 (electronic). | MR | Zbl

[20] B. Tsirelson. Nonclassical stochastic flows and continuous products. Probab. Surv. 1 (2004) 173-298 (electronic). | MR | Zbl

[21] S. R. S. Varadhan and R. J. Williams. Brownian motion in a wedge with oblique reflection. Comm. Pure Appl. Math. 38 (1985) 405-443. | MR | Zbl

[22] R. J. Williams. Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete 69 (1985) 161-176. | MR | Zbl

Cité par Sources :