strange non chaotic attractors in quasiperiodically forced circle maps the occurrence of strange non chaotic attractors sna in quasiperiodically forced systems has attracted considerable interest over the last two decades in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos their existence was first described by millionscikov and later by vinograd and also herman for quasiperiodic rm sl 2 mathbb r cocycles and by grebogi et al and later keller for so called pinched skew products however except for these two particular classes there are still hardly any rigorous results on the topic despite a large number of numerical studies confirming the widespread existence of sna in quasiperiodically forced systems here we prove the existence of sna in quasiperiodically forced circle maps under rather general conditions which can be stated in terms of mathcal c 1 estimates as a consequence we obtain the existence of sna for parameter sets of positive measure in suitable parameter families these sna carry the unique physical measure of the system which determines the behaviour of lebesgue almost all initial conditions finally we show that the dynamics are minimal in the considered situations the results apply in particular to a forced version of the arnold circle map for this example we also describe how the first arnold tongue collapses and looses its regularity due to the presence of strange non chaotic attractors and a related unbounded mean motion property

sur le theoreme d atiyah singer on donne une version simplifiee de la demonstration probabiliste du theoreme de l indice pour l operateur de dirac donnee par j m bismut au lieu d utiliser la construction du mouvement brownien au moyen du fibre des reperes on utilise la construction de schwartz en plongeant la variete ambiante dans un espace vectoriel de dimension plus grande on evite aussi l utilisation de la decomposition de l espace de wiener en deux utilisee par j m bismut le calcul des variations stochastiques etant notre outil principal de ce fait on perd la relation avec la cohomologie de l espace des lacets

resolution de l equation de schrodinger par integration stochastique du champ magnetique we solve the heat equation associated with a schrodinger operator with magnetic field onrd using a stochastic integral which involves only the magnetic field and does not use any potential of the field then we study a particular case on the torustd to show that our formula is no longer true and that an hypothesis like simple connexity is necessary

transition functions paths and path integrals the main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which kolmogorov achieved via pde theory the theory of partial differential equations and his criterion for path continuity section 2 describes how ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process x by solving an sde stochastic differential equation driven by brownian motion of course sdes have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems though 2 begins heuristically the mathematics is then tightened up this paper is after all a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation once kolmogorov s precise language is available 2 then takes a quick sight seeing trip through some of the great developments by doob ito and their successors in an age in which so many do simulations of ito equations i have explained precisely in the briefest possible fashion what the exact theory is it is easy and usable you will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems and of what the malliavin calculus is about you will however be advised on guide books on these and other areas including physicists favourites large deviations measure valued diffusions etc so that you can later explore at your leisure confession the paper consists very largely of selected tracks remixed from the album rogers williams 1987 diffusions markov processes and martingales chichester wiley tributes to kolmogorov s work in probability and statistics have appeared in every book ever written on probability and in ann probability 17 1989 815 964 ann statist 18 1990 987 1031 bull lond math soc 22 1990 31 100 teor veroyatnost primenen 34 1989 no 1 usp mat nauk 43 1988 no 6 the official biography by shiryaev will be a wonderful volume for me this paper is a further expression of my thanks to kolmogorov and as he would have wished to levy doob and ito too

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

symbol length in the brauer group of a field we bound the symbol length of elements in the brauer group of a field k containing a cm field for example any field containing an algebraically closed field or a finite field and solve the local exponent index problem for a cm field f in particular for a cm field f we show that every f central simple algebra of exponent p is similar to the tensor product of at most len p f t pm 1 1 symbol algebras of degree p we then use this bound on the symbol length to show that the index of such algebras is bounded by p 1 which in turn gives a bound for any algebra of exponent n via the primary decomposition finally for a field k containing a cm field f we show that every f central simple algebra of exponent p and degree p is similar to the tensor product of at most len p p k len p l symbol algebras of degree p where l is a cm edl a ps t 1 field

malliavin calculus with time dependent coefficients and application to nonlinear filtering in this paper we prove using malliavin calculus that under a local hormander condition the solution of a stochastic differential equation with time depending coefficients admits ac density with respect to lebesgue measure an application of this result to nonlinear filtering is developed in this paper to prove the existence of ac density for the filter associated with a correlated system whose observation is one dimensional with unbounded coefficients

decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

the malliavin calculus for pure jump processes and applications to local time on obtient trois resultats une formule d integration par parties des conditions pour qu un processus de markov de saut pur ait une densite en lp et des conditions pour qu une martingale purement discontinue ait un temps local

the period index problem for the brauer group of an algebraic surface in this paper we show that the period equals the index for elements of brauer groups of function fields of surfaces a key idea of the proof is that any azumaya algebra over a surface can be transformed into an azumaya algebra that is unobstructed

liouville theorems for non local operators abstract the paper characterizes some classes of pseudo differential operators for which there are or there are not non constant bounded harmonic functions non local perturbations of ornstein uhlenbeck operators and operators with dissipative coefficients are considered the methods used are probabilistic and based on the concept of absorption function and on a new extension of the bismut elworthy li formula the probabilistic interpretation of the liouville theorem by means of absorption functions for general markov processes is given as well

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

families index for manifolds with boundary superconnections and cones ii the chern character abstract this is part ii of a work in which we establish a formula for the chern character of a family of dirac operators of atiyah patodi and singer here we establish the existence of associated superconnection heat kernels and we prove the final index formula

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

smoothness of harmonic maps for hypoelliptic diffusions harmonic maps are viewed as maps sending a fixed diffusion to manifold valued martingales under a convexity condition we prove that the continuity of real valued harmonic functions implies the continuity of harmonic maps then we prove with a probabilistic method that continuous harmonic maps are smooth under hormander s condition the proof relies on the study of martingales with values in the tangent bundle

robustesse du filtre et calcul des variations stochastique abstract we show that under general assumptions the filter 1 is infinitely differentiable in the sense of stochastic calculus of variations for all in c which are bounded as well as their derivatives

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

applications of patching to quadratic forms and central simple algebras this paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields in particular we use a patching approach to reprove and generalize a recent result of parimala and suresh in preprint arxiv 0708 3128 2007 on the u invariant of p adic function fields p 2 the strategy relies on a local global principle for homogeneous spaces for rational algebraic groups combined with local computations

linear stochastic differential equations with boundary conditions we study linear stochastic differential equations with affine boundary conditions the equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution the solution is not adapted to the driving brownian motion and we use the extended stochastic calculus of nualart and pardoux 16 to analyse them we give analytical necessary and sufficient conditions for existence and uniqueness of a solution we establish sufficient conditions for the existence of probability densities using both the malliavin calculus and the co aera formula and give sufficient conditions that the solution be either a markov process or a markov field

last exit decompositions and regularity at the boundary of transition probabilities the purpose of this paper is to give a probabilistic approach to studying the regularity at the boundary of the transition probabilities of certain hypoelliptic diffusions with boundary conditions the main tools are last exit decompositions of brownian motion the malliavin calculus the theory of excursions and the calculus of variations on brownian excursions

a note on regime switching kolmogorov s forward and backward equations using stochastic flows abstract using stochastic flows probabilistic solutions of markovian regime switching forward and backward kolmogorov s equations are discussed these equations may be related to some pricing equations in mathematical finance their solutions are derived by differentiating a family of conditional expectations

some covariance inequalities in wiener space in this work we give an account of some covariance inequalities in abstract wiener space an fkg inequality is obtained with positivity and monotonicity being defined in terms of a given cone in the underlying cameron martin space the last part is dedicated to convex and log concave functionals including a proof of the gaussian conjecture for a particular class of log concave wiener functionals

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

dynamics of the quasi periodic schrodinger cocycle at the lowest energy in the spectrum in this paper we consider the quasi periodic schrodinger cocycle over mathbb t d d 1 and in particular its projectivization in the regime of large coupling constants and diophantine frequencies we give an affirmative answer to a question posed by m herman 21 p 482 concerning the geometric structure of certain strange non chaotic attractors which appear in the projective dynamical system we also show that for some phase the lowest energy in the spectrum of the associated schrodinger operator is an eigenvalue with an exponentially decaying eigenfunction this generalizes 39 to the multi frequency case d 1

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

minimal sets of fibre preserving maps in graph bundles topological structure of minimal sets is studied for a dynamical system e f given by a fibre preserving in general non invertible continuous selfmap f of a graph bundle e these systems include as a very particular case quasiperiodically forced circle homeomorphisms let m be a minimal set of f with full projection onto the base space b of the bundle we show that m is nowhere dense or has nonempty interior depending on whether the set of so called end points of m is dense in m or is empty if m is nowhere dense we prove that either a typical fibre of m is a cantor set or there is a positive integer n such that a typical fibre of m has cardinality n if m has nonempty interior we prove that there is a positive integer m such that a typical fibre of m in fact even each fibre of m over a dense open set fancyscript o subseteq b is a disjoint union of m circles moreover we show that each of the fibres of m over b setminus fancyscript o is a union of circles properly containing a disjoint union of m circles surprisingly some of the circles in such non typical fibres of m may intersect we also give sufficient conditions for m to be a sub bundle of e

path integral formulae for heat kernels and their derivatives the heat kernel and its derivatives of a vector laplacian on the sections of a bundle over a compact riemannian manifold are expressed as products of the scalar heat kernel of the manifold and path integrals over the brownian bridge the small time asymptotics of these integrals are computed

direct solutions of kolmogorov s equations by stochastic flows abstract solutions of kolmogorov s forward and backward equations are obtained by considering a family of conditional expectations and the use of stochastic flows to justify differentiation in the time variable

integrales oscillantes stochastiques estimation asymptotique de fonctionnelles caracteristiques abstract a general method for estimating e exp in for n is given using a stationary phase method in wiener space and stochastic variational calculus

existence and regularity of a solution of a kac equation without cutoff using the stochastic calculus of variations we consider a generalized kac equation without cutoff with which we associate a non standard nonlinear stochastic differential equation we adapt the techniques in bichteler and jacod 2 to prove that the law of a solution of the stochastic differential equation has a density which is a solution of the kac equation the initial law is very general we only assume it has second order moments and is not the dirac mass at 0 we thus generalize the analytical results of existence of a solution of this equation if we furthermore assume existence of all moments for the initial law we obtain as a corollary using the proof in desvillettes 6 that the density is smooth we prove a slightly better regularity result under more stringent assumptions using the stochastic calculus of variations adapting the methods in 1

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

laplacien hypoelliptique torsion analytique et theoreme de cheeger muller l objet de cette these est de demontrer une formule reliant les metriques de ray singer hypoelliptique et de milnor sur le determinant de la cohomologie d une variete riemannienne compacte par une deformation a la witten du laplacien hypoelliptique en theorie de de rham

positivity theorem in semi group theory we translate in semi group theory the proof of the theorem of ben arous and ourself got by the malliavin calculus giving a sufficient condition involved with the bismutian distance such that an heat kernel is strictly positive

divergence theorems in path space abstract we obtain divergence theorems on the solution space of an elliptic stochastic differential equation defined on a smooth compact finite dimensional manifold m the resulting divergences are expressed in terms of the ricci curvature of m with respect to a natural metric on m induced by the stochastic differential equation the proofs of the main theorems are based on the lifting method of malliavin together with a fundamental idea of driver

optimal control problems for stochastic reaction diffusion systems with non lipschitz coefficients by using the dynamic programming approach we study a control problem for a class of stochastic reaction diffusion systems with coefficients having polynomial growth in the cost functional a non lipschitz term appears and this allows us to treat the quadratic case which is of interest in the applications the corresponding hamilton jacobi bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates the main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

on the behaviour of brauer p dimensions under finitely generated field extensions abstract the present paper shows that if q p or q 0 where p is the set of prime numbers then there exist characteristic q fields e q k k n of brauer dimension brd e q k k and infinite absolute brauer p dimensions abrd p e q k for all p p not dividing q 2 q this ensures that brd p f q k p q 2 q for every finitely generated transcendental extension f q k e q k we also prove that each sequence a p b p p p satisfying the conditions a 2 b 2 and 0 b p a p equals the sequence abrd p e brd p e p p for a field e of characteristic zero

the atiyah singer index theorem for families of dirac operators two heat equation proofs the purpose of this paper is to give two heat equation proofs of the index theorem of atiyah singer for a family of dirac operators

a stroock varadhan support theorem in non linear filtering theory let t 0 t 1 be the unnormalized filter arising in the filtering theory of correlated diffusions in this article is considered as a stochastic process taking values inc n a description of the support of its law in the frechet spacec 0 1 c n r is given this result is the analogue for stochastic partial differential equations of the celebrated stroock varadhan theorem for diffusion processes the support of the law of the filter is shown to be the closure of the set of trajectories obtained from the zakai equation by replacing the stratonovitch differentialdy by anh1 control herey denotes the observation process

nonlinear reflecting diffusion process and the propagation of chaos and fluctuations associated abstract a nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven then a gaussian limit for the fluctuation field which is defined in l 0 2 b of a wiener type space b is obtained the covariance of the gaussian limit is computed in terms of a hilbert schmidt operator on l 0 2 b

denjoy constructions for fibered homeomorphisms of the torus we construct different types of quasiperiodically forced circle homeomorphisms with transitive but non minimal dynamics concerning the recent poincare like classification by jager and stark for this class of maps we demonstrate that transitive but non minimal behaviour can occur in each of the different cases this closes one of the last gaps in the topological classification actually we are able to get some transitive quasiperiodically forced circle homeomorphisms with rather complicated minimal sets for example we show that in some of the examples we construct the unique minimal set is a cantor set and its intersection with each vertical fibre is uncountable and nowhere dense but may contain isolated points we also prove that minimal sets of the latter kind cannot occur when the dynamics are given by the projective action of a quasiperiodic sl 2 r cocycle more precisely we show that for a quasiperiodic sl 2 cocycle any minimal proper subset of the torus either is a union of finitely many continuous curves or contains at most two points on generic fibres

the composition of wiener functionals with non absolutely continuous shifts in this paper we study some classes of wiener functionals whose elements can be composed with a non linear non absolutely continous transformation of the form of perturbation of identity in the direction of cameron martin space we show that under certain conditions the image of the wiener measure under the above transformation induces a generalized wiener functional on certain sobolev spaces generalizing the radon nikodym relation to non absolutely continuous transformations a series representation for the generalized radon nikodym derivative is presented and conditional expectations of some generalized random variables are considered

local times for a class of purely discontinuous martingales suppose xtis a purely discontinuous martingale a sufficient condition for xtto have a local time is given in terms of the local characteristics of x an example is constructed to show that this condition is nearly optimal

the analysis of elliptic families ii dirac operators eta invariants and the holonomy theorem in this paper we specialize the results obtained in bf1 to the case of a family of dirac operators we first calculate the curvature of the unitary connection on the determinant bundle which we introduced in bf1

explicit stochastic integral representations for historical functionals it is known from previous work of the authors that any square integrable functional of a superprocess may be represented as a constant plus a stochastic integral against the associated orthogonal martingale measure here we give for a large class of such functionals an explicit description of the integrand that is analogous to clark s formula for the representation of certain brownian functionals as a consequence we develop a partial analogue of the wiener chaos expansion in the superprocess setting rather than work with superprocesses per se our results are stated and proved in the richer and more natural context of the associated historical process

a version of hormander s theorem for the fractional brownian motion it is shown that the law of an sde driven by fractional brownian motion with hurst parameter greater than 1 2 has a smooth density with respect to lebesgue measure provided that the driving vector fields satisfy hormander s condition the main new ingredient of the proof is an extension of norris lemma to this situation

homogenization of periodic linear degenerate pdes it is well known under the name of periodic homogenization that under a centering condition of the drift a periodic diffusion process on r d converges under diffusive rescaling to a d dimensional brownian motion existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion in this paper we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set as a consequence it is no longer the case that the effective diffusivity matrix is necessarily non degenerate it turns out that provided that some very weak regularity conditions are met the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion in particular this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank we also discuss the application of our results to the homogenization of a class of elliptic and parabolic pdes c 2008 elsevier inc all rights reserved

analytic semigroups and degenerate elliptic operators with unbounded coefficients a probabilistic approach abstract by using techniques derived from the theory of stochastic differential equations we prove that a class of second order degenerate elliptic operators having unbounded coefficients generates analytic semigroups in c b r d the space of uniformly continuous and bounded functions from r d into r

reducibility or nonuniform hyperbolicity for quasiperiodic schrodinger cocycles we show that for almost every frequency r q for every c o potential v r z r and for almost every energy e the corresponding quasiperiodic schr nodinger cocycle is either reducible or nonuniformly hyperbolic this result gives very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic schr nodinger operator and allows us to complete the proof of the aubry andr le conjecture on the measure of the spectrum of the almost mathieu operator

quasi invariance and integration by parts for determinantal and permanental processes abstract determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent their atoms exhibit mutual attraction of repulsion thus these processes are very far from the uncorrelated situation encountered in poisson models we establish a quasi invariance result we show that if atom locations are perturbed along a vector field the resulting process is still a determinantal respectively permanental process the law of which is absolutely continuous with respect to the original distribution based on this formula following bismut approach of malliavin calculus we then give an integration by parts formula

bond pricing under a markovian regime switching jump augmented vasicek model via stochastic flows in this article we shall explore the state of art of stochastic flows to derive an exponential affine form of the bond price when the short rate process is governed by a markovian regime switching jump diffusion version of the vasicek model we provide the flexibility that the market parameters including the mean reversion level the volatility rate and the intensity of the jump component switch over time according to a continuous time finite state markov chain the states of the chain may be interpreted as different states of an economy or different stages of a business cycle we shall provide a representation for the exponential affine form of the bond price in terms of fundamental matrix solutions of linear matrix differential equations

diffusions conditionnelles ii generateur conditionnel application au filtrage abstract this paper is a continuation of diffusions conditionelles i if x t z t is a two component diffusion process it is shown that under appropriate conditions the process x t t t given z s s t is a nonhomogeneous strong markov process whose generator is explicitly found by using the theory of stochastic flows the filtering equation is reduced to an ordinary partial differential equation

absolute continuity of distributions of solutions of anticipating stochastic differential equations abstract we consider a stochastic differential equation with anticipating initial value and drift and give two types of sufficient conditions for the absolute continuity of the distribution of the solution with respect to the lebesgue measure using the malliavin calculus

markovian forward backward stochastic differential equations and stochastic flows abstract markovian forward backward stochastic differential equations mfbsdes are discussed by exploiting techniques of stochastic flows using martingale representation a differentiation rule stochastic flows of diffeomorphisms and the unique decomposition of special semimartingales we identify the solution of the backward system of the fbsde applications of the result to convex risk measures are discussed

period index and u invariant questions for function fields over complete discretely valued fields let k be a complete discretely valued field with residue field and f the function field of a curve over k let p be the characteristic of and l a prime not equal to p if the brauer l dimensions of all finite extensions of are bounded by d and the brauer l dimensions of all extensions of of transcendence degree at most 1 are bounded by d 1 then it is known that the brauer l dimension of f is at most d 2 lieblich in j reine angew math 659 1 41 2011 saltman in j ramanujan math soc 12 25 47 1997 harbater et al in invent math 178 231 263 2009 in this paper we give a bound for the brauer p dimension of f in terms of the p rank of as an application we show that if is a perfect field of characteristic 2 then any quadratic form over f in at least 9 variables is isotropic this leads to the fact that every element in h 3 f mathbb z 2 mathbb z is a symbol if is a finite field of characteristic 2 u f 8 is a result of heath brown leep heath brown in compos math 146 271 287 2010 leep in j reine angew math 2013 to appear

comparison of some field invariants in all this paper p is a prime number and f is a field of characteristic 6 p the invariants we are interested in are the p cohomological dimension cdp f 16 the diophantine dimension dd f inf i f is ci 5 for p 2 the quadratic diophantine dimension ddq f inf i f is c i where c q i is the condition introduced by pfister 13 for p 2 the u invariant u f 8 ch 11 the p invariant 7 for an element c pbr f h f p p c inf n c is a sum of n classes of algebras of degree p p f sup p c c pbr f the p invariant 7 for c as above p c logp ind c where ind c is the schur index of any central simple algebra representing c p f sup p c c pbr f to muddy water a little more we shall also consider the stable p and p invariants p f sup p e e f finite separable e f p 1 p f sup p e e f finite separable e f p 1

a generalized formula of ito and some other properties of stochastic flows a stochastic differential equation with smooth coefficients is considered which defines a continuous flow t of c 8 mappings of rd in rd if zt is a continuous semi martingale t zt s is proved to be a semi martingale for which an ito type formula is established it is shown that a s for any t t is an onto diffeomorphism if zt is a continuous semi martingale t 1 zt is proved to be a semi martingale whose ito decomposition is explicitly found