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

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 + ].

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 + ].

DOI : https://doi.org/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
@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},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1323-1346},
     doi = {10.1214/13-AIHP541},
     zbl = {06377556},
     mrnumber = {3269996},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 50 (2014) no. 4, pp. 1323-1346. doi : 10.1214/13-AIHP541. http://www.numdam.org/item/AIHPB_2014__50_4_1323_0/

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