long time behavior for a class of feller processes in this paper as a main result we derive a chung fuchs type condition for the recurrence of feller processes associated with pseudo dierentia l operators in the l evy process case this condition reduces to the classical and well known chung fuchs condition further we also discuss the recurrence and transience of feller processes with respect to the dimension of the state space and pruitt indices and the recurrence and transience of feller dynkin diusions and stable like processes finally in the one dimensional symmetric case we study perturbations of feller processes which do not aect their recurrence and transience properties and we derive sucient conditions for their recurrence and transience in terms of the corresponding l evy measure in addition some comparison conditions for recurrence and transience also in terms of the l evy measures are obtained ams 2010 mathematics subject classication 60j75 60j25 60g17

on the linearization via a restricted class of dynamic feedback we deal with the problem of linearization of nonlinear systems using a restricted class of dynamic compensator which is composed of chains of integrators followed by static feedback in the vector field formulation we present an upper bound on the number of integrators to linearize nonlinear systems and show that the bound is sharp when n 2 spl ges m spl ges 2 this results in the checkable necessary and sufficient conditions

spectral signal space projection algorithm for frequency domain meg and eeg denoising whitening and source imaging abstract meg and eeg data contain additive correlated noise generated by environmental and physiological sources to suppress this type of spatially coloured noise source estimation is often performed with spatial whitening based on a measured or estimated noise covariance matrix however artifacts that span relatively small noise subspaces such as cardiac ocular and muscle artifacts are often explicitly removed by a variety of denoising methods e g signal space projection before source imaging here we introduce a new approach the spectral signal space projection s 3 p algorithm in which time frequency tf specific spatial projectors are designed and applied to the noisy tf transformed data and whitened source estimation is performed in the tf domain the approach can be used to derive spectral variants of all linear time domain whitened source estimation algorithms the denoised sensor and source time series are obtained by the corresponding inverse tf transform the method is evaluated and compared with existing subspace projection and signal separation techniques using experimental data altogether s 3 p provides an expanded framework for meg eeg data denoising and whitened source imaging in both the time and frequency scale domains

controllability and observability for affine nonlinear hamiltonian systems it is shown that an affine nonlinear hamiltonian system is controllable if and only if it is observable in the sense that strong accessibility implies local weak observability and vice versa furthermore it is shown that a nonminimal hamiltonian system can be reduced to a locally weakly observable and strongly accessible system in such a way that the reduced system is again hamiltonian

density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

controllability distributions for nonlinear control systems the purpose of this paper is to relate the notion of controlled invariance to a notion of controllability for nonlinear control systems by means of the linear ideas of a subsystem and a quotient system we derive a very structured description of nonlinear systems

zeros at infinity for affine nonlinear control systems a definition of zeros at infinity for affine nonlinear control systems is proposed the definition is local which means that we exclude certain singularities we argue the reasonableness of our definition by showing its relevance to the problem of nonlinear decoupling in particular we give a necessary and sufficient condition for the solvability of the general regular decoupling problem for affine systems in terms of the zeros at infinity

the multifractal nature of levy processes we show that the sample paths of most levy processes are multifractal functions and we determine their spectrum of singularities

asymptotically stable invariant manifold for coupled nonlinear parabolic hyperbolic partial differential equations abstract this article considers a coupled system of nonlinear parabolic and hyperbolic partial differential equations which arises in the study of wave phenomena which are heat generating or temperature related under appropriate conditions for example high thermal diffusivity it is proved that there exists an invariant manifold for the full system of equations the asymptotic stability of the invariant manifold is also considered moreover it is shown that an equilibrium which is asymptotically stable for flows on the invariant manifold will be asymptotically stable for the full system

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

lower functions for processes with stationary independent increments let x t be ar 1 valued process with stationary independent increments and a t mathop sup limits s leqq t x s in this paper we find a sufficient condition for there to exist nonnegative and nondecreasing functionh t such that lim infa t h t c a s ast 0 andt for some positive finite constantc whenh t takes a particular form also two analytic conditions are considered as application

slow manifold for a nonlocal stochastic evolutionary system with fast and slow components abstract this work is devoted to investigating invariant manifolds for a fast slow stochastic evolutionary system with nonlocal diffusion we establish the slow reduction via a random slow manifold which captures slow dynamics of the original stochastic fast slow system a simple example is shown to illustrate this slow reduction method

a remark on nonlinear accessibility conditions and infinite prolongations the lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order it yields to a new lie algebraic criterion and to the consideration of first integrals which apply to nonlinear systems in quite general form the latter characterization in particular is valid without distinguishing between the system variables and is independent of time scalings weak accessibility is also considered

necessary and sufficient conditions for state equivalence to a nonlinear discrete time observer canonical form in this technical note we obtain necessary and sufficient conditions for a multi input multi output discrete time nonlinear system to be state equivalent to a nonlinear observer form and for an uncontrolled multi output system to be state equivalent to a linear observer form we adopt a geometric approach and the proofs are constructive with respect to the required coordinate change

growth and holder conditions for the sample paths of feller processes let a d a be the infinitesimal generator of a feller semigroup such that cc n d a and a cc n is a pseudo differential operator with symbol p x satisfying p c 1 2 and imp x c0rep x we show that the associated feller process xt t 0 on n is a semimartingale even a homogeneous diffusion with jumps in the sense of 21 and characterize the limiting behaviour of its trajectories as t 0 and to this end we introduce various indices e g x 0 lim x y 2 p y 0 or x 0 liminf x y 2 e 1 p y e 0 and obtain a s x that lim t 0t 1 s t xs x 0 or according to x or x similar statements hold for the limit inferior and superior and also for t our results extend the constant coefficient i e levy case considered by w pruitt 27

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

observability and controllability for smooth nonlinear systems the definition of a smooth nonlinear system as proposed recently by willems is elaborated as a natural generalization of the more common definitions of a smooth nonlinear input output system mini

upper semicontinuity of attractors and synchronization abstract in this paper we prove that diffusively coupled abstract semilinear parabolic systems synchronize we apply the abstract results obtained to a class of ordinary differential equations and to reaction diffusion problems the technique consists of proving that the attractors for the coupled differential equations are upper semicontinuous with respect to the attractor of a limiting problem explicitly exhibited in the diagonal

estimates of transition densities and their derivatives for jump levy processes abstract we give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding levy measure and the levy khinchin exponent we obtain also estimates of derivatives of densities

global bilinearization of non linear systems and the existence of volterra series a global volterra series is developed for an analytic system defined on an analytic manifold m local bilinearizations are pieced together using the theory of fibre bundles giving rise to twisted bilinear systems denned globally on m

controlled invariance for nonlinear systems necessary and sufficient conditions are derived for a b invariance called here controlled invariance for nonlinear systems dot x f x u the obtained results generalize and elucidate already known results about systems dot x a x sum min i 1 max m u i b i x a new and direct differential geometric interpretation of the concept of controlled invariance and the derived conditions is given

path regularity for feller semigroups via gaussian kernel estimates and generalizations to arbitrary semigroups on c0 given 1 1 we present a dyadic growth condition open image in new window on the finite dimensional distributions of operator semigroups on c0 e which for 0 and feller semigroups assures that the corresponding feller process has paths in local holder spaces and in weighted besov spaces of order we show that for operator semigroups satisfying gaussian kernel estimates of order m 1 condition open image in new window holds for all open image in new window and even for all open image in new window in the case of feller semigroups such gaussian kernel estimates are typical for feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on d of order m d

assessing and improving the spatial accuracy in meg source localization by depth weighted minimum norm estimates cerebral currents responsible for the extra cranially recorded magnetoencephalography meg data can be estimated by applying a suitable source model a popular choice is the distributed minimum norm estimate mne which minimizes the l2 norm of the estimated current under the l2 norm constraint the current estimate is related to the measurements by a linear inverse operator however the mne has a bias towards superficial sources which can be reduced by applying depth weighting we studied the effect of depth weighting in mne using a shift metric we assessed the localization performance of the depth weighted mne as well as depth weighted noise normalized mne solutions under different cortical orientation constraints source space densities and signal to noise ratios snrs in multiple subjects we found that mne with depth weighting parameter between 0 6 and 0 8 showed improved localization accuracy reducing the mean displacement error from 12 mm to 7 mm the noise normalized mne was insensitive to depth weighting a similar investigation of eeg data indicated that depth weighting parameter between 2 0 and 5 0 resulted in an improved localization accuracy the application of depth weighting to auditory and somatosensory experimental data illustrated the beneficial effect of depth weighting on the accuracy of spatiotemporal mapping of neuronal sources

gambler s ruin and the first exit position of random walk from large spheres let t r be the first time a sum s n of nondegenerate i i d random vectors in r d leaves the sphere of radius r in some given norm we characterize in terms of the distribution of the individual summands the following probabilistic behavior s tr s tr has no subsequential weak limit supported on a closed half space in one dimension this result solves a very general form of the gambler s ruin problem we also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence r k finally we compute the limiting joint distribution of s tr r s tr s tr

controlled invariance by static output feedback for nonlinear systems the paper deals with the notion of static output feedback for nonlinear systems necessary and sufficient conditions are derived for c a b invariance here called measured controlled invariance for nonlinear control systems

singularity sets of levy processes we completely describe the size and large intersection properties of the holder singularity sets of levy processes we also study the set of times at which a given function cannot be a modulus of continuity of a levy process the holder singularity sets of the sample paths of certain random wavelet series are investigated as well

equivalence of nonlinear systems to input output prime forms the problem of transforming nonlinear control systems into input output prime forms is dealt with using state space static state feedback and also output space transformations necessary and sufficient geometric conditions for the solvability of this problem are obtained the results obtained generalize well known results both on feedback linearization as well as input output decoupling of nonlinear systems it turns out that from a computational point of view the output space transformation is the crucial step that is performed by constructing rectifying coordinates for a nested sequence of distributions on the output manifold

controllability canonical phase variable form for non linear time variable systems a controllability canonical form for non linear time variable systems with a control variable appearing linearly is introduced the global controllability of this nonlinear phase variable form is given independently of the non linearities a transformation to the controllability canonical form requires the regularity of a matrix which can be considered the controllability matrix of a non linear time variable system a problem which still remains unsolved is the actual computation of the non linear controllability canonical form for a given system

packing and covering indices for a general levy process there has been substantial interest in the indices 0 2 defined by blumenthal and getoor determined by a general levy process in r d pruitt defined an index which determines the covering dimension and taylor showed that an index first considered by hendricks determines the packing dimension for the trajectory in the present paper we prove that b 2 min d and give examples to show that the whole range is attainable however we cannot completely determine the set of values of which can be attained as indices of some levy process

symmetries conservation laws and time reversibility for hamiltonian systems with external forces a system theoretic framework is given for the description of hamiltonian systems with external forces and partial observations of the state it is shown how symmetries and conservation laws can be defined within this framework a generalization of noether s theorem is obtained finally a precise definition of time reversibility is given and its consequences are explored

linearization and input output decoupling for general nonlinear systems necessary and sufficient conditions are obtained for the linearization and input decoupling by state feedback of general nonlinear systems it is shown how these conditions can be derived from the already known conditions for affine nonlinear systems thereby also elucidating the existing theory for affine systems

the growth of additive processes let x t be any additive process in r d there are finite indices i i i 1 2 and a function u all of which are defined in terms of the characteristics of x t such that limt 0 inf t 1 x t 0 if 1 if 2 a s if 1 where x t sup 0 s t x s when x t is a levy process with x 0 0 1 2 1 2 and u t t this is a special case obtained by pruitt when x t is not a levy process its characteristics are complicated functions of t however there are interesting conditions under which u becomes sharp to achieve 1 2 1 2

barriers exit time and survival probability for unimodal levy processes we give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal levy processes in smooth domains

the persistence of synchronization under environmental noise it is shown that the synchronization of dissipative systems persists when they are disturbed by additive noise no matter how large the intensity of the noise provided asymptotically stable stationary stochastic solutions are used instead of asymptotically stable equilibria

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

master slave synchronization and invariant manifolds for coupled stochastic systems we deal with abstract systems of two coupled nonlinear stochastic infinite dimensional equations subjected to additive white noise type process this kind of systems may describe various interaction phenomena in a continuum random medium under suitable conditions we prove the existence of an exponentially attracting random invariant manifold for the coupled system and show that this system can be reduced to a single equation with modified nonlinearity this result means that under some conditions we observe nonlinear synchronization phenomena in the coupled system our applications include stochastic systems consisting of i parabolic and hyperbolic equations ii two hyperbolic equations and iii klein gordon and schrodinger equations we also show that the random manifold constructed converges to its deterministic counterpart when the intensity of noise tends to zero

local observability of nonlinear differential algebraic equations daes from the linearization along a trajectory the main results of this note establish sufficient rank conditions for local observability of nonlinear differential algebraic equations dae systems near a known trajectory associated with a given control verification of the observability rank condition is addressed and the relationship with previous work on observability of dae is indicated

heat content for convolution semigroups abstract let x x t t 0 be a levy process in r d and be an open subset of r d with finite lebesgue measure in this article we consider the quantity h t p x x t c d x related to x which is called the heat content we study its asymptotic behaviour as t goes to zero for isotropic levy processes under some mild assumptions on the characteristic exponent we also treat the class of levy processes with finite variation in full generality

time frequency mixed norm estimates sparse m eeg imaging with non stationary source activations magnetoencephalography meg and electroencephalography eeg allow functional brain imaging with high temporal resolution while solving the inverse problem independently at every time point can give an image of the active brain at every millisecond such a procedure does not capitalize on the temporal dynamics of the signal linear inverse methods minimum norm dspm sloreta beamformers typically assume that the signal is stationary regularization parameter and data covariance are independent of time and the time varying signal to noise ratio snr other recently proposed non linear inverse solvers promoting focal activations estimate the sources in both space and time while also assuming stationary sources during a time interval however such an hypothesis only holds for short time intervals to overcome this limitation we propose time frequency mixed norm estimates tf mxne which use time frequency analysis to regularize the ill posed inverse problem this method makes use of structured sparse priors defined in the time frequency domain offering more accurate estimates by capturing the non stationary and transient nature of brain signals state of the art convex optimization procedures based on proximal operators are employed allowing the derivation of a fast estimation algorithm the accuracy of the tf mxne is compared to recently proposed inverse solvers with help of simulations and by analyzing publicly available meg datasets

passage of levy processes across power law boundaries at small times we wish to characterise when a l e vy process x t crosses boundaries like t kappa kappa 0 in a one or two sided sense for small times t thus we enquire when limsup t downarrow 0 x t t kappa limsup t downarrow 0 x t t kappa and or liminf t downarrow 0 x t t kappa are almost surely a s finite or infinite necessary and sufficient conditions are given for these possibilities for all values of kappa 0 often for many values of kappa when the limsups are finite a s they are in fact zero as we show but the limsups may in some circumstances take finite nonzero values a s in general the process crosses one or two sided boundaries in quite different ways but surprisingly this is not so for the case kappa 1 2 an integral test is given to distinguish the possibilities in that case some results relating to other norming sequences for x and when x is centered at a nonstochastic function are also given

small time compactness and convergence behavior of deterministically and self normalised levy processes the first author s research was partially supported by arc grant dp0664603 the second author s research was partially supported by nsf grant dms 0503908

on the limiting behaviour of levy processes at zero we find an analytical condition characterising when the probability that a levy process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero we show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results for each 0 we find an analytical condition equivalent to x t r t r 1 alpha stackrel p longrightarrow infty and x t t 1 alpha stackrel p longrightarrow infty as r t to 0 where x is a levy process and t r the time it first leaves an interval of radius r

proportional integral controllers for minimum phase nonaffine in control systems we show that stabilizing tracking proportional integral pi controllers can be constructed for minimum phase nonaffine in control systems the constructed pi controller is an equivalent realization of an approximate dynamic inversion controller this equivalence holds only for the time response when applied to the unperturbed system even when restricted to unperturbed minimum phase linear time invariant systems their closed loop robustness properties differ this shows that in general properties that do not define the equivalence relation for systems controllers are not preserved under such equivalence transformations

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

darling erdős theorems for normalized sums of i i d variables close to a stable law let 1 be i i d real valued random variables and s n 1 n in the case when the distribution of is close to a stable law for some 0 1 1 2 we investigate the asymptotic behavior in distribution of the maximum of normalized sums max k 1 k 1 s k this completes the darling erdos limit theorem for the case 2

the lower limit of a normalized random walk soit sn une marche aleatoire et n une suite de constantes telles que n n est non decroissante on donne un test integral universel qui determine la limite inferieure de sn n jusqu a une echelle constante pour lim sup 2n n