Almost sure behaviour ofF-valued random fields

Summary LetX _n, n ^d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields _n, ^d of positive numbers, the almost sure behaviour of _knX_k/_n, n ^d is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.Supported in part by the Youth Science Foundation of China, No. 19001018Supported by the National Natural Science Foundation of China     

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.     

Combinatorial decompositions and homogeneous geometrical processes

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR ². An expression for the probabilities ,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a typical line of direction (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a typical edge having direction in terms of the process {P _i , _i } is found, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction and the mark _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.     

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.     

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.     

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 .     

A new approach to the single point catalytic super-Brownian motion

Summary A new approach is provided to the super-Brownian motionX with a single point-catalyst _c as branching rate. We start from a superprocessU with constant branching rate and spatial motion given by the 1/2-stable subordinator. We prove that the occupation density measure ^c ofX at the catalystc is distributed as the total occupation time measure ofU. Furthermore, we show thatX _t is determined from ^c by an explicit representation formula. Heuristically, a mass ^c (ds) of particles leaves the catalyst at times and then evolves according to Itô's Brownian excursion measure. As a consequence of our representation formula, the density fieldx ofX satisfies the heat equation outside ofc, with a noisy boundary condition atc given by the singularly continuous random measure ^c . In particular,x isC outside the catalyst. We also provide a new derivation of the singularity of the measure ^c .     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

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     

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     

Measures induced on Wiener space by monotone shifts

Summary In this work we study the absolute continuity of the image of the Wiener measure under the transformations of the formT()=+u(), the shiftu is a random variable with values in the Cameron-Martin spaceH and is monotone in the sense that (T(+h-T(),h) _H 0 a.s. for allh inH.     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Summary LetL(x, T),xR ^d ,TR ₊ ^N , be the local time of theN-parameter Wiener processW taking values inR ^d . Even in the distribution valued casedd2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|0 and/orT and of related occupation integrals asT. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.X _T (f), and of the second order, i.e. normalized convergence laws forL(x, T)–E(L(x, T)) resp.X _T (f)–E(X _T (f)).This research was made during a stay at the LMU in München supported by DAAD     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

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     

Large indecomposables over representation-infinite orders and algebras

Let R be a complete discrete valuation domain with quotient field K, and let be an R-order in a semisimple K-algebra. Butler, Campbell, and Kovács have shown that R-free -modules decompose into -lattices when is representation-finite. Using the theory of ladder functors, we prove the converse by constructing indecomposable R-free -modules of infinite rank if is not representation-finite.Received: 23 March 2004     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

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     

Riemann-measurable selections

LetX be a Polish space equipped with a -finite regular Borel measure . IfE is a metric space andF a set-valued function:X 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.     

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.