Three examples of brownian flows on $\mathbb {R}$

Le Jan, Yves ; Raimond, Olivier

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 1323-1346 / Harvested from Numdam

Résumé

Nous montrons que le seul flot solution de l’équation différentielle stochastique (EDS) sur $\mathbb {R}$ dX t =1 {X t >0} W + (dt)+1 {X t <0} dW - (dt), 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, $K^{+}=\mathsf {E}[\delta _{\varphi ^{\pm }}|W^{+}]$ 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 ℝ + f ' )(x)W + (du)+1 2∫ s t K s,u f``(x)du pour tout $s<t$, $x\in \mathbb {R}$ 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 $K^{+}=\mathsf {E}[\delta _{\varphi ^{+}}|W^{+}]$.

Abstract

We show that the only flow solving the stochastic differential equation (SDE) on $\mathbb {R}$ 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 $\varphi ^{\pm }$. 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)+∫ s t K s,u (1 ℝ + f ' )(x)W + (du)+1 2∫ s t K s,u f``(x)du for $s<t$, $x\in \mathbb {R}$ 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 $K^{+}=\mathsf {E}[\delta _{\varphi ^{+}}|W^{+}]$.

Détail

Publié le : 2014-01-01

MR 3269996 | Zbl 06377556

DOI : https://doi.org/10.1214/13-AIHP541
Classification:  60H25,  60J60

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