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 _, , a product measure with densities and to the left and right of the origin respectively, <, 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 and to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim _t t ^–1(E(X _t )²–(EX _t )²) and findD=(p–q)(–)^–1((1–)+(1–)) as conjectured by Spohn (1991). We show that in the scale 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 and . This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, yS ¹ randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle, we consider for each >0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|. With the further assumption that the density of is bounded away from zero, we show that when tends to zero the above point process scaled by –1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

An action functional for Lévy Brownian motion

Stoll's construction [7] of Lévy Brownian motion l on ^d as a white noise integral is used to obtain an action functional I(x) defined for the surfaces x of l. This provides a Cameron-Martin formula for translation of Lévy measure , and also a large deviation principle for scaled Lévy measures . Proofs follow the lines of [2], where nonstandard techniques were used to give natural proofs of the corresponding results for Wiener measure.The research for this paper was supported partly by a grant from the SERC.     

Converse results on equiconvergence of interpolating polynomials

, n- n- 1 , . , , ( « ») f .     

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.     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

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.     

Sample function behavior of increasing processes of class L

Summary We consider increasing processes {X(t)t0} of classL, that is, increasing self-similar processes with inswpendent increments. Leth(t) be an increasing positive function on (0,) withh(0+)=0 andh()=. By virtue of the zero-one laws, there existsc (resp.C) [0,] such that lim inf (resp. lim sup)X(t)/h(t)=c (resp.C) a.s. both ast tends to 0 and ast tends to . We decide a necessary and sufficient condition for the existence ofh(t) withc orC=1 and explicitly constructh(t) in caseh(t) exists withc orC=1. Moreover, we give a criterion to classify functionsh(t) withc (orC)=0 andh(t) withc (orC)= in caseh(t) does not exist withc (orC)=1.     

Some dimension results for super-Brownian motion

Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y _t)_t0 be a super-Brownian motion on ^d (d2) andH be a Borel subset of ^d . We determine the Hausdorff Dimension of {t0; SuppY _tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY _t as a function of the dimension ofB.     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

Spectral orders and convolution

. (R) _{– –} fg(y)h(x–y) dx dy _{– –} f ^{^} (x)g ^{^} (y)h ^{^} (x–y)dx dy (f,g0) —:f×gf ^{^} ×g ^{^}(f,g 0) f^{^} g^{^} f g -, X — . , - f ₁f ₂ , f ₁ ^{^} ×gf ₂×g 0g. .     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

Summability of multiple Fourier series. Growth of Lebesgue constants

. , . . , H ₂ . , .     

On polynomial variance functions

Summary Let be a natural exponential family on and (V, ) be its variance function. Here, is the mean domain of andV, defined on , is the variance of . A problem of increasing interest in the literature is the following: Given an open interval and a functionV defined on , is the pair (V, ) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, ) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.     

Склярные дифференциалвные инварианты третвего порядка риамнового простанства трех измерений

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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.     

A global and stochastic analysis approach to bosonic strings and associated quantum fields

We construct a probability measure giving a mathematical realization of Polyakov's heuristic measure for bosonic strings in space-time dimensions 3d13, having as world sheet compact Riemann surfaces of arbitrary genus. The measure involves the path space measures for scalar fields with exponential interaction on and a measure on Teichmüller space.Deceased 24 January 1988     

Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.