Three examples of brownian flows on $ℝ$

Le Jan, Yves  ; Raimond, Olivier 

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1323-1346.

Abstract (VO)

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

$$\mathrm{d}{X}_t=1_{\{X_t>0\}}{W}^{+}\left(\mathrm{d}t\right)+1_{\{X_t<0\}}\mathrm{d}{W}^{-}\left(\mathrm{d}t\right),$$

where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by ${\varphi }^{\pm }$. The flow ${\varphi }^{\pm }$ is a Wiener solution of the SDE. Moreover, $K^{+}=𝖤\left[{\delta }_{{\varphi }^{\pm }}\right|W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE

$$K_{s,t}^{+}f\left(x\right)=f\left(x\right)+{\int }_s^t{K}_{s,u}\left(1_{ℝ}^{+}{f}^{\text{'}}\right)\left(x\right)W^{+}\left(\mathrm{d}u\right)+\frac{1}{2}{\int }_s^t{K}_{s,u}f``\left(x\right)\mathrm{d}u$$

for $s<t$, $x\in ℝ$ and $f$ a twice continuously differentiable function. A third flow ${\varphi }^{+}$ 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 $K^{+}=𝖤\left[{\delta }_{{\varphi }^{+}}\right|W^{+}]$.

Résumé

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

$$\mathrm{d}{X}_t=1_{\{X_t>0\}}{W}^{+}\left(\mathrm{d}t\right)+1_{\{X_t<0\}}\mathrm{d}{W}^{-}\left(\mathrm{d}t\right),$$

où $W^{+}$ et $W^{-}$ sont deux bruits blancs indépendants, est un flot coalescent que nous noterons ${\varphi }^{\pm }$. Le flot ${\varphi }^{\pm }$ est une solution Wiener de l’équation. De plus, $K^{+}=𝖤\left[{\delta }_{{\varphi }^{\pm }}\right|W^{+}]$ est l’unique solution (c’est aussi une solution Wiener) de l’EDS

$$K_{s,t}^{+}f\left(x\right)=f\left(x\right)+{\int }_s^t{K}_{s,u}\left(1_{ℝ}^{+}{f}^{\text{'}}\right)\left(x\right)W^{+}\left(\mathrm{d}u\right)+\frac{1}{2}{\int }_s^t{K}_{s,u}f``\left(x\right)\mathrm{d}u$$

pour tout $s<t$, $x\in ℝ$ et $f$ une fonction deux fois continûment mesurable. Un troisième flot ${\varphi }^{+}$ 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 $K^{+}=𝖤\left[{\delta }_{{\varphi }^{+}}\right|W^{+}]$.

Détail

DOI : 10.1214/13-AIHP541

Classification : 60H25, 60J60
Keywords: stochastic flows, coalescing flow, Arratia flow or brownian web, brownian motion with oblique reflection on a wedge

Bibliographie

[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. | Zbl | MR

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

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[17] R. Mikulevicius and B. L. Rozovskii. Global $L_2$-solutions of stochastic Navier-Stokes equations. Ann. Probab. 33 (2005) 137-176. | Zbl | MR

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

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

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

[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. | Zbl | MR

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