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     

On Toeplitz type quadratic functionals of stationary Gaussian processes

Summary A central limit theorem for Toeplitz type quadratic functionals of a stationary Gaussian processX(t),t, is proved, generalizing the result of Avram [1] for discrete time processes. The result is applied to the problem of nonparametric estimation of linear functionals of an unknown spectral density function. We give some upper bounds for the minimax mean square risk of the nonparametric estimators, similar to those by Ibragimov and Has'minskii [12] for a probability density function.     

A transmission problem in the calculus of variations

We study hölder regularity of minimizers of the functional , wherep(x) takes only two values and jumps across a Lipschitz surface. No restriction on the two values is imposed.This article was processed by the author using the Springer-Verlag T_EX PJourlg macro package 1991.     

Sur le comportement asymptotique des potentiels d'une chaîne de Markov sur ℝ d

Résumé Dans cet article j'étudie le comportement à l'infini des potentiels des chaînes de Markov sur ^d (d3) proches du mouvement brownien, tout spécialement le cas des marches aléatoires, ainsi que des critères de transience et de récurrence inspirés de la méthode utilisée.

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We study the asymptotic behaviour of potentials of Markov chains on ^d (d3), closed to Brownian motion, and particularly the case of random walks. Following a similar approach, we give transience and recurrence criteria.

     

Finely holomorphic functions and quasi-analytic classes

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M _n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function finely holomorphic onU satisfyingf(x)= (x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University.     

On a functional equation of Luce

Summary In a recent communication to J. Aczél, R. Duncan Luce asked about the functional equationU(x)U(G(x)F(y)) = U(G(x))U(xy) forx, y > 0, (1) which has arisen in his research on certainty equivalents of gambles. He was particularly interested in cases in which the unknowns (U, F andG) are strictly increasing functions from (0, + ) into (0, + ). In this paper we solve (1) in the case whereU, F andG are continuously differentiable with everywhere positive first derivatives. Our solution is perhaps novel in that in certain cases (1) reduces to a functional equation in a single variable and in other cases to a functional equation in several variables; see [1] for the terminology.