on simultaneous linearization of diffeomorphisms of the sphere let r1 r2 rm be rotations generating sod 1 d 2 and f1 f2 fm be their small smooth perturbations we show that f can be simultaneously linearized if and only if the associated random walk has zero lyapunov exponents as a consequence we obtain stable ergodicity of actions of random rotations in even dimensions 1 main results let f1 f2 fm be diffeomorphisms of s d 2 let i i n be a sequence of independent random variables uniformly distributed on 1 m consider the markov process on s 1 xn f nxn 1 if is an invariant measure for this process let 1 2 d be the lyapunov exponents of denote

polymer transport in random flow the dynamics of polymers in a random smooth flow is investigated in the framework of the hookean dumbbell model the analytical expression of the time dependent probability density function of polymer elongation is derived explicitly for a gaussian rapidly changing flow when polymers are in the coiled state the pdf reaches a stationary state characterized by power law tails both for small and large arguments compared to the equilibrium length the characteristic relaxation time is computed as a function of the weissenberg number in the stretched state the pdf is unstationary and exhibits multiscaling umerical simulations for the two dimensional navier stokes flow confirm the relevance of theoretical results obtained for the correlated model

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

superprocesses of stochastic flows we construct a continuous superprocess x on m r d which is the unique weak feller extension of the empirical process of consistent k point motions generated by a family of differential operators the process x differs from known dawson watanabe type fleming viot type and ornstein uhlenbeck type superprocesses this new type of superprocess provides a connection between stochastic flows and measure valued processes and determines a stochastic coalescence which is similar to those of smoluchowski moreover the support of x describes how an initial measure on r d is transported under the flow as an example the process realizes a viewpoint of darling about the isotropic stochastic flows under certain conditions

integration of brownian vector fields summary using the wiener chaos decomposition we show that strong solutions of non lipschitzian s d e s are given by random markovian kernels the example of sobolev flows is studied in some detail exhibiting interesting phase transitions

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

ergodicity of a measure valued markov chain induced by random transformations let z1 z2 be a sequence of i i d random transformations possibly discontinuous of a compact metric space m and let e denote the space of normalized mass distributions on m given in e let n denote the random measure z n z1 1 when well defined we construct the transition probability p of the e valued markov chain n and give a necessary and sufficient condition for p to have a unique invariant measure concentrated on the degenerate mass distributions convergence to statistical equilibrium of the associated discrete time stochastic flow is investigated

modulus of continuity of the canonic brownian motion on the group of diffeomorphisms of the circle abstract an explicit modulus of holder continuity is given for the flow associated to the canonic brownian motion on the diffeomorphism group of the circle

isotropic stochastic flows on considere les flots stochastiques isotropes dans un espace euclidien de d dimensions d 2 on relie la tendance des distances entre deux points et des vecteurs tangents a se borner ou s etendre a la dimension et a la proportion du flot solenoidal ou potentiel

kraichnan flow in a square an example of integrable chaos the kraichnan flow provides an example of a random dynamical system accessible to an exact analysis we study the evolution of the infinitesimal separation between two lagrangian trajectories of the flow its long time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is gaussian this does not have to be the case in the presence of an anisotropy we analyze in detail the flow in a two dimensional periodic square where the anisotropy generally persists at small scales the calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic schrodinger operator of the lame type the underlying integrability permits to explicitly exhibit the non gaussianity of the multiplicative large deviations and to analyze the time scales at which the large deviation regime sets in in particular we indicate how the divergence of some of those time scales when the two lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow the analysis of the two dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability

t e harris s contributions to recurrent markov processes and stochastic flows this is a brief survey of t e harris s work on recurrent markov processes and on stochastic flows and of some more recent work in these fields

asymptotic behaviour of stochastic flows of diffeomorphisms two case studies the two stochastic flows studied are i the canonical stochastic flow on the orthonormal frame bundle of hyperbolic space which gives stochastic parallel translation along brownian paths in hyperbolic space and ii a stochastic flow on the spheres n 1 arising from its embedding as the unit sphere in n both flows are controlled by the same stochastic differential equation in a finite dimensional lie group in each case the lyapunov exponents are computed and a complete description is given of the local and global stability of the flow

asymptotic support theorem for planar isotropic brownian flows it has been shown by various authors that the diameter of a given nontrivial bounded connected set x grows linearly in time under the action of an isotropic brownian flow ibf which has a nonnegative top lyapunov exponent in case of a planar ibf with a positive top lyapunov exponent the precise deterministic linear growth rate k of the diameter is known to exist in this paper we will extend this result to an asymptotic support theorem for the time scaled trajectories of a planar ibf which has a positive top lyapunov exponent starting in a nontrivial compact connected set x r2 that is we will show convergence in probability of the set of time scaled trajectories in the hausdorff distance to the set of lipschitz continuous functions on 0 1 starting in 0 with lipschitz constant k

intermittency of the tracer gradient the problem of stirring a passive scalar tracer by a random velocity field is considered for gaussian velocity fields with infinitely small time and space scales it is shown that the tracer gradient is concentrated in a discrete set of points the points that form a random point process independent of the initial tracer distribution a complete description of this point process is given if the initial tracer field is a random function with homogeneous increments a full statistics of the jumps is also given

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

brownian motions on the homeomorphisms of the plane let z denote a point of r 2 we study random flows z st z in r 2 0 leq s leq t infty z in r 2 z tu z st z z su z for s leq t leq u such flows are called brownian if z is continuous in s t z and has appropriate spatial and temporal homogeneity properties and if z st z uv cdots are independent homeomorphisms of r 2 onto r 2 when s leq t leq u leq v leq cdots for a brownian flow the coordinates of any k points are a 2k dimensional continuous markov process k 1 2 cdots if these processes are diffusions whose diffusion matrices have bounded continuous derivatives of order leq 2 i e are c 2 bounded then the diffusion matrices are necessarily obtained in a certain way from the covariance tensor of the field of infinitesimal displacements a converse is given in the incompressible isotropic case given a c 2 bounded covariance tensor of an isotropic solenoidal r 2 valued field in r 2 there exists a corresponding incompressible isotropic brownian flow

polymer stretching in the inertial range of turbulence we study the deformation of flexible polymers whose contour length lies in the inertial range of a homogeneous and isotropic turbulent flow by using the elastic dumbbell model and a stochastic velocity field with nonsmooth spatial correlations we obtain the probability density function of the extension as a function of the weissenberg number and of the scaling exponent of the velocity structure functions in a spatially rough flow as in the inertial range of turbulence the statistics of polymer stretching differs from that observed in laminar flows or in smooth chaotic flows in particular the probability distribution of polymer extensions decays as a stretched exponential and the most probable extension grows as a power law of the weissenberg number furthermore the ability of the flow to stretch polymers weakens as the flow becomes rougher in space

on isotropic brownian motions this paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields characteristic exponents stability asymptotic behaviour statistical equilibrium are the topics on which the main results are obtained

aggregation of inertial particles in random flows we consider the trajectories of particles suspended in a randomly moving fluid if the lyapunov exponent of these trajectories is negative the paths of these particles coalesce so that particles aggregate here we give a detailed account of a method b mehlig and m wilkinson phys rev lett 92 250602 2004 for calculating this exponent it is expressed as the expectation value of a random variable evolving under a stochastic differential equation we analyze this equation in detail in the limit where the correlation time of the velocity field of the fluid is very short such that the stochastic differential equation is a langevin equation we derive an asymptotic perturbation expansion of the lyapunov exponent for particles suspended in three dimensional flows in terms of a dimensionless measure of the inertia of the particles epsilon and a measure of the relative intensities of potential and solenoidal components of the velocity field gamma we determine the phase diagram in the epsilon gamma plane

asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds extending the earlier results for analytic curve segments in this article we describe the asymptotic behaviour of evolution of a finite segment of a c n smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n manifold in particular we show that if the curve satisfies certain natural geometric conditions the pushforward of the parameter measure on the curve under the ge odesic flow converges to the normalized canonical riemannian measure on the tangent bundle in the limit we also study the limits of geodesic evolution of shrinking segments we use ratner s classification of ergodic invariant measures for unipo tent flows on homogeneous spaces of so n 1 and an observation re lating local growth properties of smooth curves and dynamics of linear sl 2 r actions