Tanaka's equation on the circle and stochastic flows

We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels $K$ solution of Tanaka's equation can be classified by pairs of probability measures $(m^+,m^-)$ on $[0,1]$, with mean 1/2. What happens at the first vertex is governed by $m^+$, and at the second by $m^-$. For each vertex $P$, we construct a sequence of stopping times along which the image of the whole circle by $K$ is reduced to $P$. We also prove that the supports of these flows contains a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.Comment: To appear in ALEA Lat. Am. J. Probab. Math. Sta

Excited Brownian motions as limits of excited random walks

We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say $\phi$, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by T\'oth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when $\phi$ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.Comment: v.3: main result improved: hyothesis of recurrence removed. To appear in P.T.R.

Stochastic flows and an interface SDE on metric graphs

This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE $(\hbox{ISDE})$. To each edge of the graph is associated an independent white noise, which drives $(\hbox{ISDE})$ on this edge. This produces an interface at each vertex of the graph. We first do our study on star graphs with $N\ge 2$ rays. The case $N=2$ corresponds to the perturbed Tanaka's equation recently studied by Prokaj \cite{MR18} and Le Jan-Raimond \cite{MR000} among others. It is proved that $(\hbox{ISDE})$ has a unique in law solution, which is a Walsh's Brownian motion. This solution is strong if and only if $N=2$. Solution flows are also considered. There is a (unique in law) coalescing stochastic flow of mappings \p solving $(\hbox{ISDE})$. For $N=2$, it is the only solution flow. For $N\ge 3$, \p is not a strong solution and by filtering \p with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of $(\hbox{ISDE})$. There are no other Wiener solutions. Our previous results \cite{MR501011} in hand, these results are extended to more general metric graphs. The proofs involve the study of $(X,Y)$ a Brownian motion in a two dimensional quadrant obliquely reflected at the boundary, with time dependent angle of reflection. We prove in particular that, when $(X\_0,Y\_0)=(1,0)$ and if $S$ is the first time $X$ hits $0$, then $Y\_S^2$ is a beta random variable of the second kind. We also calculate \EE[L\_{\sigma\_0}], where $L$ is the local time accumulated at the boundary, and $\sigma\_0$ is the first time $(X,Y)$ hits $(0,0)$.Comment: Submitte