No abstract

We are interested in stationary ``fluid'' random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels. In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.Comment: Published at http://dx.doi.org/10.1214/009117904000000207 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

This paper is concerned with a general class of self-interacting diffusions {X t } t≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX t = Brownian increments + drift term depending on X t and µ t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {µ t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow = { t } t≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {µ t } are proved to be almost surely attractor free sets for . These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {µ t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.

This paper consists in the study of a stochastic differential equation on a metric graph, called an interface SDE (ISDE). To each edge of the graph is associated an independent white noise, which drives (ISDE) on this edge. This produces an interface at each vertex of the graph. We first do our study on star graphs with N ≥ 2 rays. The case N = 2 corresponds to the perturbed Tanaka's equation recently studied by Prokaj [18] and Le Jan-Raimond [12] among others. It is proved that (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 ϕ solving (ISDE). For N = 2, it is the only solution flow. For N ≥ 3, ϕ is not a strong solution and by filtering ϕ with respect to the family of white noises, we obtain a (Wiener) stochastic flow of kernels solution of (ISDE). There are no other Wiener solutions. Our previous results [8] 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 (X0, Y0) = (1, 0) and if S is the first time X hits 0, then Y 2 S is a beta random variable of the second kind. We also calculate E[Lσ 0 ], where L is the local time accumulated at the boundary, and σ0 is the first time (X, Y ) hits (0, 0).