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
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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://www.numdam.org/articles/10.1214/13-AIHP541/

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