Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

BV periodic solutions of an evolution problem associated with continuous moving convex sets

This paper is concerned with BV periodic solutions for multivalued perturbations of an evolution equation governed by the sweeping process (or Moreau's process). The perturbed equation has the form –DuN _C (t)(u(t))+F(t,u(t)), whereC is a closed convex valued continuousT-periodic multifunction from [0,T] to ^d ,N ^C (t)(u(t)) is the normal cone ofC(t) atu(t),F: [0,T]× ^{d d} is a compact convex valued multifunction and Du is the differential measure of the periodic BV solutionu. Several existence results for this differential inclusion are stated under various assumptions on the perturbationF.     

Dimension and entropy of a non-ergodic Markovian process and its relation to Rademacher Riesz Products

Given a Markov chain (not necessarily stationary or homogeneous) with finite state space and an initial distribution, we can construct a measure on the unit interval [0, 1]. In this work we examine the equality (up to a constant) of the Hausdorff dimension of and of a suitably defined entropy for the Markovian process. The results are applied to the so-called Rademacher-Riesz Products.     

Filtering on manifolds

We coasider a partially observable diffusion process (x _t,y_t)t60080x2715/xxlarge10878.gif" alt="ges" align="MIDDLE" BORDER="0">0 whose unobservable componentx _t lives on a submanifold M ofR ⁿ . We present some general conditions under which the conditional law ofx _t, given the observationsy _s ,s 60080x2715/xxlarge8712.gif" alt="isin" align="MIDDLE" BORDER="0"> [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

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 260.gif" alt="pgr" align="BASELINE" BORDER="0">j/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.     

Shift-coupling in continuous time

Summary The result linking shift-coupling to time-average total variation convergence and to the invariant -field is extended to continuous time and an analogous result established linking -couplings to smooth total variation convergence and to a smooth tail -field. Shift- and -coupling inequalities are presented.     

Shock fluctuations in the asymmetric simple exclusion process

Summary We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isv _{6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">,6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">} , a product measure with densities 6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0"> and 6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0"> to the left and right of the origin respectively, 6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0"><6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">, then there exists a (microscopic) shock for the system. A shock is a random positionX _t such that the system as seen from this position at timet has asymptotic product distributions with densities 6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0"> and 6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0"> to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim _{t6081/xxlarge8594.gif" alt="rarr" align="BASELINE" BORDER="0">6081/xxlarge8734.gif" alt="infin" align="MIDDLE" BORDER="0">} t ^–1(E(X _t )²–(EX _t )²) and findD=(p–q)(6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">–6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">)^–1(6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">(1–6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0">)+6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">(1–6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">)) as conjectured by Spohn (1991). We show that in the scale 6081/440_2005_Article_BF01199027_TeX2GIFIE1.gif" alt=" $$\sqrt t$$ " align="middle" border="0"> the position ofX _t is determined by the initial distribution of particles in a region of length proportional tot. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities 6081/xxlarge961.gif" alt="rgr" align="MIDDLE" BORDER="0"> and 6081/xxlarge955.gif" alt="lambda" align="BASELINE" BORDER="0">. This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.     

Predictable projections for point process filtrations

Summary We consider a point process with the Polish phase space (X,X) and a system of -fields (x),xX, generated by on certain sets (x)X. We define predictability for random processes indexed byX and for random measures onX and prove the existence and uniqueness of predictable and dual predictable projections under a regularity condition on . ForX= ₂ ⁺ and under monotonicity assumptions on the sets x we will identify the predictable projections of some simple processes as regular versions of certain martingales.     

The drift of a one-dimensional self-repellent random walk with bounded increments

Summary Consider a one-dimensional walk (S _k ) _k having steps of bounded size, and weight the probability of the path with some factor 1–(0,1) for every single self-intersection up to timen. We prove thatS _n /S S converges towards some deterministic number called the effective drift of the self-repellent walk. Furthermore, this drift is shown to tend to the basic drift as tends to 0 and, as tends to 1, to the self-avoiding walk's drift which is introduced in [10]. The main tool of the present paper is a representation of the sequence of the local times as a functional of a certain Markov process.Partially supported by Swiss National Sciences Foundation Grant 20-36305.92