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 .     

On the occupation times of cones by Brownian motion

Summary We study some features concerning the occupation timeA _t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A₁u0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf _t A _t/(tf(t))=0 or for a decreasing functionf.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

The behavior of solutions of stochastic differential inequalities

Summary LetX andZ be ^d -valued solutions of the stochastic differential inequalities dX _t a(t,X _t )dt+(t,X _t )dW _t andb(t, Z _t )dt+(t, Z _t )dW _t dZ _t , respectively, with a fixed ^m -valued Wiener processW. In this paper we give conditions ona, b and under which the relationX ₀Z ₀ of the initial values leads to the same relation between the solutions with probability one. Further we discuss whether in general our conditions can be weakened or not. Then we deal with notions like maximal/minimal solution of a stochastic differential inequality. Using the comparison result we derive a sufficient condition for the existence of such solutions as well as some Gronwall-type estimates.     

Sample path large deviations for super-Brownian motion

Summary We derive two large deviation principles of Freidlin-Wentzell type for rescaled super-Brownian motion. For one of the appearing rate functions an integral representation is given and interpreted as Kakutani-Hellinger energy. As a tool we develop estimates for the Laplace functionals of (historical) super-Brownian motion and certain maximal inequalities. Also it is shown that the Hölder norm of index <1/2 of the processtf, X _t possesses some finite exponential moments provided the functionf is smooth.This work was supported in part by the Graduiertenkolleg Algebraische, analytische und geometrische Methoden und ihre Wechselwirkung in der modernen Mathematik, Bonn     

Brownian processes in infinite dimension

LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW _t with covariance . The main goal is to solve the SDE of Langevin type dX _t= dW _t–AX _t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)= u–Av(t).     

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.     

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 .     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Non-equilibrium entropy on stationary Markov processes

We study the evolution of probability measures under the action of stationary Markov processes by means of a non-equilibrium entropy defined in terms of a convex function . We prove that the convergence of the non-equilibrium entropy to zero for all measures of finite entropy is independent of for a wide class of convex functions, including ₀(t)=t log t. We also prove that this is equivalent to the convergence of all the densities of a finite norm to a uniform density, on the Orlicz spaces related to , which include the L ^p -spaces for p>1. By means of the quadratic function ₂(t)=t ²–1, we relate the non-equlibrium entropies defined by the past -algebras of a K-dynamical system with the non-equilibrium entropy of its associated irreversible Markov processes converging to equilibrium.Partially supported by DIB Universidad de Chile, E19468412.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.

     

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

The limit of the spectral radius of block Toeplitz matrices with nonnegative entries

A bi-infinite sequence ...,t _–2,t _–1,t ₀,t ₁,t ₂,... of nonnegativep×p matrices defines a sequence of block Toeplitz matricesT _n =(t _ik ),n=1,2,...,, wheret _ik =t _k–i ,i,k=1,...,n. Under certain irreducibility assumptions, we show that the limit of the spectral radius ofT _n , asn tends to infinity, is given by inf{()[0,]}, where () is the spectral radius of _jz t _j ^j .Supported by SFB 343 Diskrete Strukturen in der Mathematik, Universität Bielefeld     

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.     

Sample path behaviour in connection with generalized arcsine laws

Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA ⁺=(A ⁺ (t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law.     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant