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.

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 $K^{+}=𝖤\left[{\delta }_{{\varphi }^{+}}\right|W^{+}]$.

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 $K^{+}=𝖤\left[{\delta }_{{\varphi }^{+}}\right|W^{+}]$.

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