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

weak limits of perturbed random walks and the equation y_t b_t alpha sup y_s colon s leq t beta inf y sb s colon s leq t let and be real numbers and f c o 0 we study the existence and uniqueness of solutions g of the equation g t f t sup o 8 t g s inf o 8 t g s carmona petit le gall and yor have shown existence or nonexistence and uniqueness for some we settle the remaining cases we study the nearest neighbor walk on the integers which behaves just like fair random walk unless one neighbor has been visited and the other has not when it jumps to the unvisited neighbor with probability p if p 2 3 we show these processes scaled converge to the solution of the equation above for brownian paths with 2p 1 p

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

some extensions of the arc sine law as partial consequences of the scaling property of brownian motion the scaling property of brownian motion is exploited systematically in order to extend paul levy s arc sine law to brownian motion perturbed by its local time at 0 other important ingredients of the proofs are some ray knight theorems for local times

robust criterion for the existence of nonhyperbolic ergodic measures we give explicit c 1 open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy actually our criterion provides the existence of a partially hyperbolic compact set with one dimensional center and positive topological entropy on which the center lyapunov exponent vanishes uniformly

enlacements du mouvement brownien autour des courbes de l espace limit theorems are proved for the winding numbers of a three dimensional brownian motion around certain curves in space in particular the joint asymptotic distribution of the winding numbers around two curves is obtained this joint distribution generalizes the asymptotic law of the winding numbers of a planar brownian motion around two points which has recently been given by pitman and yor the limiting distributions are closely related to the time spent by a linear brownian motion above and below a multiple of its maximum process proofs rely on stochastic calculus for continuous semi martingales

a volume preserving diffeomorphism with essential coexistence of zero and nonzero lyapunov exponents we show that there exists a c volume preserving topologically transitive diffeomorphism of a compact smooth riemannian manifold which is ergodic indeed is bernoulli on an open and dense subset mathcal g of not full volume and has zero lyapunov exponent on the complement of mathcal g

brownian motion and random walk perturbed at extrema let bt be brownian motion we show there is a unique adapted process xt which satisfies dxt dbt except when xt is at a maximum or a minimum when it receives a push the magnitudes and directions of the pushes being the parameters of the process for some ranges of the parameters this is already known we show that if a random walk close to bt is perturbed properly its paths are close to those of xt

doubly perturbed jump diffusion processes in this paper we study the existence and uniqueness of the solution to a class of doubly perturbed jump diffusion processes with non lipschitz coefficients some well known results are improved and generalized

oscillator synchronisation under arbitrary quasi periodic forcing we study the problem of existence of response solutions for a real analytic one dimensional system consisting of a rotator subject to a small quasi periodic forcing with bryuno frequency vector we prove that at least one response solution always exists without any assumption on the forcing besides smallness and analyticity this strengthens the results available in the literature where generic non degeneracy conditions are assumed the proof is based on a diagrammatic formalism and relies on renormalisation group techniques which exploit the formal analogy with problems of quantum field theory a crucial role is played by remarkable identities between classes of diagrams

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

doubly perturbed neutral stochastic functional equations in this paper we prove the existence and uniqueness of the solution to a class of doubly perturbed neutral stochastic functional equations dpnsfes in short under some non lipschitz conditions the solution is constructed by successive approximation furthermore we give the continuous dependence of the solution on the initial value by means of the corollary of bihari inequality

fluids elasticity geometry and the existence of wrinkled solutions we are concerned with underlying connections between fluids elasticity isometric embedding of riemannian manifolds and the existence of wrinkled solutions of the associated nonlinear partial differential equations in this paper we develop such connections for the case of two spatial dimensions and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the gauss codazzi equations through examples for the euler equations for fluids and the euler lagrange equations for elastic solids these results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

perturbed brownian motions we study perturbed brownian motions that can be loosely speaking described as follows they behave exactly as linear brownian motion except when they hit their past maximum or and maximum where they get an extra push we define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain natural class of processes in the case where both perturbations at the maximum and at the minimum are self repelling we show that in fact more is true such a process can almost surely be constructed from brownian paths by a one to one measurable transformation this generalizes some results of carmona petit yor and davis we also derive some fine properties of perturbed brownian motions hausdorff dimension of points of monotonicity for example

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

pathwise uniqueness for perturbed versions of brownian motion and reflected brownian motion any solution of the functional equation wta bta asup st wsa 1 l w t 0 where b is a brownian motion behaves like a reflected brownian motion except when it attains a new maximum we call it an a per turbed reflected brownian motion similarly any solution of xta bta asup st xsa binft xs t 0 behaves like a brownian motion except when it attains a new maxi mum or minimum we call it an a b doubly perturbed brownian motion we complete some recent investigations by showing that for all permissible values of the parameters a a and b respectively these equations have pathwise unique solutions and these are adapted to the filtration of b