Three examples of brownian flows on ℝ

Yves Le Jan; Olivier Raimond

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 4, page 1323-1346
  • ISSN: 0246-0203

Abstract

top
We show that the only flow solving the stochastic differential equation (SDE) on ℝ d X t = 1 { X t g t ; 0 } W + ( d t ) + 1 { X t l t ; 0 } d W - ( d t ) , 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 ℝ + f ' ) ( x ) W + ( d u ) + 1 2 ∫ s t K s , u f ` ` ( x ) d u for s l t ; t , x ∈ ℝ 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 + ] .

How to cite

top

Le Jan, Yves, and Raimond, Olivier. "Three examples of brownian flows on $\mathbb {R}$." Annales de l'I.H.P. Probabilités et statistiques 50.4 (2014): 1323-1346. <http://eudml.org/doc/272064>.

@article{LeJan2014,
abstract = {We show that the only flow solving the stochastic differential equation (SDE) on $\mathbb \{R\}$\[\mathrm \{d\}X\_\{t\}=1\_\{\lbrace X\_\{t\}&gt;0\rbrace \}W^\{+\}(\mathrm \{d\}t)+1\_\{\lbrace X\_\{t\}&lt;0\rbrace \}\,\mathrm \{d\}W^\{-\}(\mathrm \{d\}t),\] 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, $K^\{+\}=\mathsf \{E\}[\delta _\{\varphi ^\{\pm \}\}|W^\{+\}]$ is the unique solution (it is also a Wiener solution) of the SDE \[K^\{+\}\_\{s,t\}f(x)=f(x)+\int \_\{s\}^\{t\}K\_\{s,u\}(1\_\mathbb \{R\}^\{+\}f^\{\prime \})(x)W^\{+\}(\mathrm \{d\}u)+\frac\{1\}\{2\}\int \_\{s\}^\{t\}K\_\{s,u\}f``(x)\,\mathrm \{d\}u\] for $s&lt;t$, $x\in \mathbb \{R\}$ 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 $K^\{+\}=\mathsf \{E\}[\delta _\{\varphi ^\{+\}\}|W^\{+\}]$.},
author = {Le Jan, Yves, Raimond, Olivier},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic flows; coalescing flow; Arratia flow or brownian web; brownian motion with oblique reflection on a wedge; stochastic differential equation; Brownian web; Brownian motion},
language = {eng},
number = {4},
pages = {1323-1346},
publisher = {Gauthier-Villars},
title = {Three examples of brownian flows on $\mathbb \{R\}$},
url = {http://eudml.org/doc/272064},
volume = {50},
year = {2014},
}

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
PB - Gauthier-Villars
VL - 50
IS - 4
SP - 1323
EP - 1346
AB - We show that the only flow solving the stochastic differential equation (SDE) on $\mathbb {R}$\[\mathrm {d}X_{t}=1_{\lbrace X_{t}&gt;0\rbrace }W^{+}(\mathrm {d}t)+1_{\lbrace X_{t}&lt;0\rbrace }\,\mathrm {d}W^{-}(\mathrm {d}t),\] 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, $K^{+}=\mathsf {E}[\delta _{\varphi ^{\pm }}|W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE \[K^{+}_{s,t}f(x)=f(x)+\int _{s}^{t}K_{s,u}(1_\mathbb {R}^{+}f^{\prime })(x)W^{+}(\mathrm {d}u)+\frac{1}{2}\int _{s}^{t}K_{s,u}f``(x)\,\mathrm {d}u\] for $s&lt;t$, $x\in \mathbb {R}$ 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 $K^{+}=\mathsf {E}[\delta _{\varphi ^{+}}|W^{+}]$.
LA - eng
KW - stochastic flows; coalescing flow; Arratia flow or brownian web; brownian motion with oblique reflection on a wedge; stochastic differential equation; Brownian web; Brownian motion
UR - http://eudml.org/doc/272064
ER -

References

top
  1. [1] R. Arratia. Brownian motion on the line. Ph.D. dissertation, Univ. Wisconsin, Madison, 1979. 
  2. [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. [3] K. Burdzy, W. Kang and K. Ramanan. The Skorokhod problem in a time-dependent interval. Stochastic Process. Appl.119 (2009) 428–452. Zbl1186.60035MR2493998
  4. [4] K. Burdzy and H. Kaspi. Lenses in skew Brownian flow. Ann. Probab.32 (2004) 3085–3115. Zbl1071.60073MR2094439
  5. [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). Zbl1069.60068MR1944976
  6. [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. Zbl1105.60075MR2094432
  7. [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. Zbl1261.60054MR2953347
  8. [8] H. Hajri. Stochastic flows related to Walsh Brownian motion. Electron. J. Probab. 16 (2011) 1563–1599 (electronic). Zbl1245.60067MR2835247
  9. [9] W. Kang and K. Ramanan. A Dirichlet process characterization of a class of reflected diffusions. Ann. Probab.38 (2010) 1062–1105. Zbl1202.60059MR2674994
  10. [10] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus, 2nd edition. Graduate Texts in Mathematics 113. Springerg, New York, 1991. Zbl0734.60060MR1121940
  11. [11] Y. Le Jan and O. Raimond. Integration of Brownian vector fields. Ann. Probab.30 (2002) 826–873. Zbl1037.60061MR1905858
  12. [12] Y. Le Jan and O. Raimond. Flows, coalescence and noise. Ann. Probab.32 (2004) 1247–1315. Zbl1065.60066MR2060298
  13. [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. Zbl1094.60041MR2202038
  14. [14] Y. Le Jan and O. Raimond. Flows associated to Tanaka’s SDE. ALEA Lat. Am. J. Probab. Math. Stat.1 (2006) 21–34. Zbl1105.60038MR2235172
  15. [15] B. Micaux. Flots stochastiques d’opérateurs dirigés par des bruits gaussiens et poissonniens. Ph.D. dissertation, Univ. Paris-Sud, 2007. 
  16. [16] R. Mikulevicius and B. L. Rozovskii. Stochastic Navier–Stokes equations for turbulent flows. SIAM J. Math. Anal.35 (2004) 1250–1310. Zbl1062.60061MR2050201
  17. [17] R. Mikulevicius and B. L. Rozovskii. Global L 2 -solutions of stochastic Navier–Stokes equations. Ann. Probab.33 (2005) 137–176. Zbl1098.60062MR2118862
  18. [18] V. Prokaj. The solution of the perturbed Tanaka equation is pathwise unique. Preprint, 2011. Available at arXiv:1104.0740. Zbl1284.60134MR3098074
  19. [19] K. Ramanan. Reflected diffusions defined via the extended Skorokhod map. Electron. J. Probab. 11 (2006) 934–992 (electronic). Zbl1111.60043MR2261058
  20. [20] B. Tsirelson. Nonclassical stochastic flows and continuous products. Probab. Surv. 1 (2004) 173–298 (electronic). Zbl1189.60082MR2068474
  21. [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. Zbl0579.60082MR792398
  22. [22] R. J. Williams. Reflected Brownian motion in a wedge: Semimartingale property. Z. Wahrsch. Verw. Gebiete69 (1985) 161–176. Zbl0535.60042MR779455

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.