arXiv:math/0211387v1 [math.PR] 25 Nov 2002 Sticky flows on the circle. Yves Le Jan and Olivier Raimond February 1, 2008 Introduction The purpose of this note is to give an example of stochastic flows of kernels as defined in [3], which naturally interpolates between the Arratia coalescing flow associated with systems of coalescing independent Brownian particles on the circle and the deterministic diffusion flow (actually, the results are given in the slightly more general framework of symmetric Levy processes for which points are not polar). The construction is performed using Dirichlet form theory and the extension of De Finetti’s theorem given in [3]. The sticky flows of kernels are associated with systems of sticky independent Levy particles on the circle, for some fixed parameter of stickyness. Some elementary asymptotic properties of the flow are also given. 1 Compatible family of Dirichlet forms. Let (E n ) n≥1 be a family of Dirichlet forms 1 , respectively defined on L 2 (M n ,m n ), where M is a metric space and (m n ) n≥1 is a family of probability measures on M n . We will denote by D n the domain of the Dirichlet form E n . For all n ≥ 1, S n denotes the group of permutations of {1,...,n}. Let (P (n) t ) n≥1 be the family of Markovian semigroups associated with this family of Dirichlet forms. Definition 1.1 We will say that the family of Dirichlet forms (E n ) n≥1 is compatible if the family of Markovian semigroups (P (n) t ) n≥1 is compatible, that is if the following assertions are satisfied 1 we refer the reader not familiar with Dirichlet forms and symmetric Markov processes to [2] 1