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Three examples of brownian flows on

Yves Le JanOlivier Raimond — 2014

Annales de l'I.H.P. Probabilités et statistiques

We show that the only flow solving the stochastic differential equation (SDE) on 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 + f ' ) ( x ) W + ( d u ) + 1 2 s t K s , u f ` ` ( x ) d u for s l t ; 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...

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