On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Mouvement moyen et système dynamique gaussien

Summary In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn. From a given nonatomic probability measure on [0,], we construct a transformationT of the complex brownian path (B _u)_0u1 which preserves Wiener measure.T is defined as the limit of a sequenceT _n, whereT _n acts as the motion of 2ⁿ planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-,] obtained from . The transformationT can be inserted in a flow (T ^t) _t, and the orbitstZ _t=B ₁T ^t still have almost surely a mean motion, which is the mean of .     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

A study on Markovian maximality,change of probability and regularity

LetE be a rigid separable Banach space andm a bounded Borel measure onE. Let Ext denote the family of all gradient type Dirichlet forms onL ²(E, m) such that the domain of their extended generators (cf. Definition 1.1) contain the smooth functions. We prove three results. First, we prove the existence of the maximum element in Ext whenever Ext is not empty. Secondly, let be the maximum element in Ext (when Ext Ø) and let be a positive function in D(). We define a new measure =²·m and we consider the family Ext associated with the measure . We prove that if is associated with a diffusion process, Ext is not empty and its maximum element is also associated with a diffusion process. Finally, whenm is a centered Gaussian measure onE, we can prove that Ext contains exactly one element.     

Smoothness of Nonlinear Median-Interpolation Subdivision

We give a refined analysis of the Hölder regularity for the limit functions arising from a nonlinear pyramid algorithm for robust removal of non-Gaussian noise proposed by Donoho and Yu [6,7,17]. The synthesis part of this algorithm can be interpreted as a nonlinear triadic subdivision scheme where new points are inserted based on local quadratic polynomial median interpolation and imputation. We introduce the analogon of the Donoho–Yu scheme for dyadic refinement, and show that its limit functions are in C for >log₄(128/31)=1.0229.... In the triadic case, we improve the lower bound of >log₂(135/121)=0.0997... previously obtained in [6] to >log₃(135/53)=0.8510.... These lower bounds are relatively close to the anticipated upper bounds of log₂(16/7)=1.1982... in the dyadic, respectivly 1 in the triadic cases, and have been obtained by deriving recursive inequalities for the norm of second rather than first order differences of the sequences arising in the subdivision process.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

Some zero-one laws for additive functionals of Markov processes

Summary Let (X _t,P ^x ) be anm-symmetric Markov process with a strictly positive transition density. Consider the additive functionalA _t : = ₀ ^t f (X _s ) wheref:E[0, ] is a universally measurable function on the state spaceE. Among others, we prove thatP ^x (A _t <)=1, for somexE and somet>0, already impliesP ^x (A _t <)=1, for quasi everyxE and allt>0. The latter is also equivalent toP ^x (A _t <)>0, for quasi everyxE and allt>0, and to the analytic condition , for a sequence of finely open Borel setsF _n such thatEF _n is polar. In the special cases of Brownian motion and Bessel process, these results were obtained earlier by H.J. Engelbert, W. Schmidt, X.-X. Xue and the authors.     

On the semimartingale representation of reflecting Brownian motion in a cusp

Summary LetC be the symmetric cusp {(x, y)²:–x yx ,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too.     

Asymptotics of the generating function for the volume of the Wiener sausage

Summary We consider the generating function of the voltime of the Wiener sausageC (t), which is the -neighbourhood of the Wiener path in the time interval [0,t]. For <0, the limiting behavior fort, up to logarithmic equivalence, had been determined in a celebrated work of Donsker and Varadhan. For >0 it had been investigated by van den Berg and Tóth, but in contrast to the case <0, there is no simple expression for the exponential rate known. We determine the asymptotic behaviour of this rate for small and large .     

Estimates of transition densities for Brownian motion on nested fractals

Summary We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d _J, related to the shortest path metric and chemical exponent on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.     

Durch graduierte Lie-Algebren definierte Paar-Algebren

Pair algebras which have a non degenerate (left- and right-) invariant bilinear form and for which the inner derivation algebra is completely reducible are characterised by pairs (C,), where C is a n×n matrix satisfying certain conditions and is a sequence of n integers equal to 0 or 1. They occur as pair algebras of type (S(C,)_–1,S(C,)₁), xuy=[[x,u],y], where (S(C,)_r)_r is the gradation induced by . in the Kac-Moody algebraS(C). If C is an affin Cartan matrix (as in the case of Lie triple systems), there exists a finite dimensional simple Lie algebrag and a Aut (g), ord =m< such that the pair algebra is isomorphic to the pair algebra (g _–1,g ₁), xuy=[[x,u],y] (product ing), whereg _i. is the eigenspace of of eigenvalue ⁱ, a primitive m-th root of unity.     

Une application de Corps des Normes

Let k be a perfect field of characteristic p > 0, K₀ = Frac(W(k)), a uniformizer in K₀ and _n K ₀ (n N) such that ₀ = and _n+1 ^p = _n. We write K = _nN K₀ (_n), H = Gal (K₀/ K and G = Gal(K₀/ K₀). The main result of this paper is that the functor restriction of the Galois action from the category of crystalline representations of G with Hodge–Tate weights in an interval of length p - 2 to the category of p-adic representations of H is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H: one thanks to the norm field of K, the other thanks to some categories of filtered modules with divided powers previously introduced by the author.     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.     

Spitzer's condition and ladder variables in random walks

Summary Spitzer's condition holds for a random walk if the probabilities _n =P{ _n > 0} converge in Cèsaro mean to , where 0<<1. We answer a question which was posed both by Spitzer [12] and by Emery [5] by showing that whenever this happens, it is actually true that n converges to . This also enables us to give an improved version of a result in Doney and Greenwood [4], and show that the random walk is in a domain of attraction, without centering, if and only if the first ladder epoch and height are in a bivariate domain of attraction.     

Beitrag zur Reduktion des Entscheidungsproblems auf Klassen von Hornformeln mit kurzen Alternationen

Ohne Zusammenfassung Zusatz bei der Korrektur: Ein vollständiger und korrekter Beweis für die Entscheidbarkeit der eingangs angeführten Aanderaaschen Klasse ((0, ), (, , ...)) erscheint demnächst im JSL (S.O. Aanderaa/H.R.Lewis: Prefix classes of Krom formulas). Ebendort wird auch die Reduktionstypeneigenschaft für ((0, ), (0, 0, )) und ((0, )), (0, 0, )) nachgewiesen, während ((0, ), (, )) sich als entscheidbar herausgestellt hat (s. E. Börger: Eine entscheidbare Klasse von Kromformeln. ZMLG 19 (1973), 117–120.) Der Kromsche Reduktionstyp konnte mittlerweile einerseits zu ((0, ), (0, 4)) verschärft werden (s. D. Rödding, E. Börger: The undecidability of (0, 4)-formulae with binary disjunctions, vorgetragen auf dem Logic Coll. Bristol 1973, ein abstract erscheint im JSL), andererseits kündigt H.R.Lewis die Reduktionstypeneigenschaft für ((0, ), (0, 1)) an (s. H.R.Lewis: Krom formulas with one dyadic predicate letter. Notices AMS 20, 5 (1973) A-500, abstr. no. 73T-E78.)Dieser Aufsatz geht aus der Dissertation [2] hervor, die dem Fachbereich Mathematik der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Münster im Sommersemester 1971 vorgelegt worden ist. Die Ergebnisse stammen aus dem Wintersemester 1970/71. Eine Ankündigung der hauptsächlichen Resultate ist in den Notices of the American Mathematical Society 19, 2 (1972) A-333 unter der abstract no. * 72T-E24 erschienen.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

On measures that agree on balls

The fundamental result: if and v are two finite Borel measures, defined in the spaceL ^p[0, 1] (1p<) or in C(K) (K is a metric compactum without isolated points), then from the equalities (B)=v(B) for all balls B of radius 1 there follows that =v. In addition, in the spaces C(K) and _p (1p<) from the inequalities (B) v(B) there follows that v.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 122–128, 1989.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Kennzeichnungen von Minimalschraubflächen Mittels Vertikalisophoten und Horizontalschattengrenzen

In Euclidean space E³, let be a (regular C-) minimal surface without planar points having locally (without loss of generality) the spherical representation n(u,v)=(cos v/cosh u, sin v/cosh u, tanh u), (u,v)G². The corresponding (isothermal) parametrization : x(u,v), (u,v)G can be expressed using agenerating Function (u,v) which satisfies _{uu + vv} – 2_utanh u + =0; the v-curves (coordinate curves u=u₀) in , along each of which the angle between the normal n(u,v) of and the x₃-axis is constant, are thevertical- isophotes of , the u-curves (v=v₀) being their orthogonal trajectories (theorems 1, 2). Considering u-curves and/or v-curves of having additional geometric properties (curves of constant/steepest slope, curves of constant Gaussian curvature, asymptotic curves, lines of curvature or geodesies of ) we prove many newgeometric characterizations of theright helicoid, thecatenoid andScherk's second surface (theorems 3–7). All of these surfaces areminimal hélicoidal surfaces.