Mathematisches Forschungsinstitut Oberwolfach Probability and Analysis in the Context of Mathematical Physics and Biology

s Pavel Bleher Scaling Limits and Universality in the Matrix Model (joint work with Alexander Its) About thirty years ago Freeman Dyson found an exact solution for the scaling limit of correlations between eigenvalues in the Gaussian unitary ensemble of random matrices. He conjectured that this scaling limit should appear in a much broader class of non-Gaussian unitary ensembles of random matrices. This constitutes the famous conjecture of universality in the theory of random matrices. Dyson found also a remarkable formula which expresses the eigenvalue correlations in terms of orthogonal polynomials on the line with exponential weight. This enables to reduce the universality conjecture to semiclassical asymptotics of orthogonal polynomials. In the talk we will discuss a new approach to the semiclassical asymptotics of orthogonal polynomials on the line with respect to exponential weights. This approach is based on the methods of the theory of integrable systems and on an appropriate matrix Riemann-Hilbert problem. As an application of the semiclassical asymptotics, we prove the conjecture of universality for unitary ensembles of random matrices for models with quartic interaction. G erard Ben Arous Aging regime and dynamical phase transition for a spherical model of spin glasses We presented a survey about dynamics of Sherrington-Kirkpatrick spin-glasses, in Langevin form as recently worked out in a series of papers with A. Guionnet. We analyzed the longtime behaviour of the limiting self-consistent dynamics, which prove to be non-Markovian in a simpli ed model, i.e. the spherical one. In this context, in a joint work with A. Dembo and A. Guionnet we nd a dynamical phase transition, and one aging regime, i.e. for low temperature, the time-correlation of these limiting dynamics is such that lim s!1 t!1 ts=o(s) C(s; t) is a constant but C(s; t) t s s 3=4 if ts s: 2 Michiel van den Berg Heat Equation on the Arithmetic von Koch Snow ake Let 0 < s 1=3, and consider a unit square in R2 . Replacing repeatedly the middle proportion s of each side by the three other sides of a square results in the s-adic von Koch snow ake Ks. Let TKs be the rst exit time of Ks of a Brownian motion. We show that if (i) Ks is non-arithmetic (i.e., log(1 s 2 )= log(s) 62 Q) then ZKs Px[TKs t] dx = Cst1 ds=2 + o(t1 ds=2); t # 0 for some positive constant Cs; (ii) Ks is arithmetic with log(1 s 2 )= log(s) = pq , p 2 Z+, q 2 Z+, (p; q) = 1 then there exists a strictly positive, 2q log 1s -periodic function s such that ZKs Px[TKs t] dx = s( log t)t1 ds=2 + o(t1 ds=2); t # 0; ds is the interior Minkowski dimension of the boundary of Ks, and is given by the unique positive root of 3sd + 2 1 s 2 d = 1. Krzysztof Burdzy A new Ray-Knight-type theorem (joint work with Richard Bass) Let Bt be a Brownian motion, 1; 2 2 R, and let Xy t be the solution to dXy t dt = ( 1 if Xy t < Bt 2 if Xy t > Bt; Xy 0 = y: Let Lyt be the local time of Xy t Bt at 0. Fix some t > 0. Then dXy t =dy is a smooth function of Lyt . If 1; 2 > 0 and 1 2 > 0 then fLy1; y 0g and fL y 1 ; y 0g are di usions. The function y ! Xy t is C1; for < 1=2 but it is not C1;1=2. Terence Chan Dynamic scaling exponents for interacting KPZ equations Consider the following non-linear stochastic partial di erential equation (known as the Kardar-Parisi-Zhang equation) @u @t = u+ 2 jruj2 + W (t; x); u(0; x) = f(x); and more generally, the system of equations 3 @u0 @t = u0 + 2 jru0j2 + 2(jru1j2 + jru2j2) + 0W0(t; x); @ui @t = ui + ru0 rui + iWi(t; x); i = 1; 2: (Here, x 2 Rd, , , and i are constants and Wi are independent space-time white noises. The initial conditions fi(x) are given.) It has long been accepted in the physics literature that the solution u (and more generally ui) satisfy scaling properties of the form b 1E[u(b3=2t; bx)2] = E[u(t; x)2], or b 1=2u(b3=2t; bx) converges in some sense to a non-degenerate limit. However, except for the case of 1 spatial dimension (d = 1), the solution of such SPDEs are typically some kind of stochastic distribution (for example in the Hida sense) and cannot be interpreted as ordinary random variables. There is therefore a non-trivial problem in giving a meaning to the non-linear terms in the above equations and also to things like E[u(t; x)2] (except in the 1-dimensional case). The appropriate interpretation of the non-linear terms turns out to be Wick products (which is actually already implicit in much of the physics literature). This talk presents a family of weighted L2-like spaces of distributions to which solutions to the KPZ equations belong; moreover, the solutions admit a Wiener chaos expansion. This allows one to give a natural meaning to ideas like expectation and convergence in distribution { ideas which are crucial to any discussion of scaling properties. However, a mathematically rigorous proof of the aforementioned scaling properties is still an open problem. Phlippe Cl ement On the Attracting Orbit of a Nonlinear Transformation Arising from Renormalization of Hierarchically Interacting Di usions (joint work with Jean-Bernard Baillon, Andreas Greven and Frank den Hollander) In this lecture the asymptotic behavior of the iterates of a non-linear transformation F acting on a class of functions g : [0;1)! [0;1) is considered. This problem arises in a probabilistic context, namely the study of systems of hierarchically interacting di usions discussed in the previous lecture by Professor Dawson. This study is a part of a larger area, where the goal is to understand universal behavior on large space-time scales of stochastic systems with interacting components. The transformation F under consideration plays the role of a renormalization transformation for an in nite system of di usions, taking values in [0;1) and interacting with each other in a hierarchical fashion. The iterates F ng (n 0) describe the behavior of this system along an in nite hierarchy of space-time scales indexed by n. The n-th iterate F ng is the local di usion rate of a typical block average on space-time scale n. If for some class of functions g, we have lim n!1F ng = g (in an appropriate sense) where g is a xed point of F and is independent of g in the class, then the spacetime scaling limit of the corresponding in nite system of interacting di usions on [0;1) has universal behavior independent of model parameters. 4 The Transformation Let H denote the class of functions g : [0;1)! [0;1) locally Lipschitz continuous satisfying g(0) = 0, g(x) > 0 for x > 0 and limx!1 x 2g(x) = 0. For g 2 H, let ( g ) 2[0;1) be the family of probability measures on [0;1) given by g 0 = 0 (point measure at 0) and g (dx) = 1 Zg 1 g(x) exp Z x y g(y) dy dx; > 0; where Zg is the normalizing ( nite) constant. The transformation F is de ned as (Fg)( ) = Z 1 0 g(x) g (dx); 2 [0;1): Since g is the equilibrium (probability) measure of a single di usion with drift towards and with local di usion rate given by g, (Fg)( ) is the average di usion rate in equilibrium as a function of the drift parameter . It appears that for g 2 H, (Fg)( ) is well-de ned for 2 [0;1) and that Fg 2 H. Fixed points of F From the moment relations: Z 1 0 g (dx) = 1; Z 1 0 x g (dx) = ; Z 1 0 x2 g (dx) = 2 + (Fg)( ); for all g 2 H and 2 [0;1), it follows in particular that the linear functions ga(x) = ax, a > 0, x 0, are xed points of F in the class H. We have: Theorem A There are no xed points in H other than (ga)a2(0;1). Universal behavior Theorem B If for g 2 H, limx!1 x 1g(x) = a 2 (0;1), then limn!1 F ng = ga uniformly on bounded intervals. Idea of the proof of Theorem B After observing that the map F : H ! H is order-preserving, i.e. g1; g2 2 H and g1 g2 implies Fg1 Fg2, we use a \monotone iterations" argument. Let g+ (resp. g ) denote the concave upper (resp. convex lower) enveloppe of g, then g+ Fg+, Fg g follow from the rst two moment relations and Jensen's inequality and from g+ g g follows F ng+ F ng F ng , n 1. Next it is shown that the monotone decreasing (resp. increasing) sequence F ng+ (resp. F ng ) converges to a xed point, which by Theorem A is a linear function ga+ (resp. ga ). From limx!1 x 1g+(x) = limx!1 x 1g (x) = a, we infer a+ = a = a. 5 Idea of the proof of Theorem A We use the \quasilinear" structure of the transformation F . Let B be the class of measurable functions h on (0;1) be such that there exist a; b 0 with jh(x)j a+ bx2, x 0. Then for each g 2 H, (Kgh)( ) := Z 1 0 h(x) g (dx); 0 is well-de ned and Kgh 2 B. Note that H B and Kgg = Fg. The linear operator Kg : B ! B can be shown to map convex functions into convex functions. Setting K(n) g := KFn 1g KF 0g, with F 0 = Id, n = 1, we have by iterating the moment relations: K(n) g e0 = e0; K(n) g e1 = e1 and K(n) g e2 = e2 + n(F ng); n 1; where ej(x) = xj, j = 0; 1; 2. If g 2 H is a xed point of F , we have g = lim n!1F ng = lim n!1 1 nK(n) g e2; which is convex since e2 is convex. By using a comparison argument with a translate of a linear function (supporting the convex function g ) and a kind of \strong" order preserving property of F one concludes that g has to be linear. Reference J. Funct. Anal. 146, No. 1, 236{298 (1997) where other properties of the transformation F are discussed. Ted Cox A spatial model for the abundance of species (joint work with Maury Bramson and Rick Durrett) In recent years theoretical biologists have begun to use interacting particle systems to model biological and ecological systems. These models are typically quite complicated, making mathematical analysis di cult. This talk summarizes recent work of Bramson, Cox and Durrett on a simple interacting particle system viewed as a model of speciation, for which rigorous results can be obtained. This work was motivated in part by recent data of Hubbell on species abundances of woody plants in a 50 hectare plot on Barro Colorado Island, Panama. In this study, counts of di erent plant species were made, and the data arranged in \octaves." That is, if N(i) is the number of di erent species which had exactly i representatives observed, and N(I) = Pi2I N(i), the data was recorded in the form N(Ij), with Ij = [2j; 2j+1), j = 0; 1; ::: The mathematical model used is the two-dimensional multitype voter model with mutation, t. This model is easily de ned. Let Z2 be the two-dimensional integer 6 lattice, and let t(x) denote the type at site x at time t, t(x) 2 (0; 1). At each site x, at rate 1, independently of all other sites, a site y is chosen at random from the nearest neighbors of x, and the type at site x is replaced by the type at site y. In addition, at rate (the mutation rate), the type at x is replaced by a new type not previously observed in the system. For strictly positive , there is a unique equilibrium 1 such that for any initial state 0, t ! 1 in law as t!1. We are interested in the species abundance distribution of 1 for small . Inside B(L), the square of side L centered at the origin, we count the number of di erent types in 1 which are represented by exactly i sites, and denote this by NL(i). Let NL(I) = Pi2I NL(i), and assume that L = L( ) satis es L2 (log(1= ))4 for a positive constant . Thus, as ! 0, L ! 1. We show that as ! 0, for intervals Ij which are large but not too large, N(Ij) c( L2)j for an appropriate constant c, and this approximation is valid uniformly over the range considered. We also obtain another estimate for the very large Ij, and observe exponential fall o in this range. The limiting formulas obtained do not closely t Hubbell's data; this is to be expected, since one has gone \too far" in the limit ! 0. However, abundance data obtained from simulations of the two-dimensional model on large grids for small but positive seem to t very well. It also appears that the the two-dimensional model provides a much better t to Hubbell's data than does the corresponding mean eld model. Donald Dawson Multiple Space-Time Scale Analysis of Hierarchically Interacting Measurevalued Processes This work is motivated by the study of stochastic models of macroevolution. These models involve a collection of subpopulations fX (t) 2M1[E] : 2 S; t 0g where S is the set of population sites and E is the set of types of individuals. The dynamics includes local demographic uctations given either by measure-valued branching or Fleming-Viot sampling. We assume that S = N , the hierarchical group indexed by the positive integer N and that the migration is given by a random walk in which individual jumps to a randomly chosen point in the ball of radius k with rate ck=Nk. The recurrence-transience dichotomy for these random walks is related to the divergence or convergence of the series P(ck) 1. We then consider the in nite interacting system including local demographic uctuations and migration via the random walk. When the random walk is transient and the initial con guration is stationary and ergodic the system converges to a mean-preserving equilibrium. Otherwise, the only invariant measures are concentrated on con gurations consisting of a single type. The nal ingredient in our analysis is the asymptotic study of the system when the parameter N goes to 1. This leads to a M1[E]-valued reverse time Markov chain which describes the multiscale behavior of this system. It turns out that the measure-valued branching and Fleming-Viot systems arise as attractors for two \universality" classes. For example, in the special case in which E = f0; 1g and the resulting Fisher-Wright dynamics is replaced by a more general class of 7 di usions, the Fisher-Wright nevertheless emerges at large space-time scales in the recurrent case. This convergence has been studied by Baillon, Cl ement, Greven and den Hollander. Finally, we give a short discussion of the e ects of selection and mutation-selection on these systems. Jean-Dominique Deuschel Anharmonic Droplet Construction and Large Deviations (joint work with Giambattista Giacomin and Dmitry Io e) We consider a continuous S-O-S model with Hamiltonian HDN ( ) = X V ( (x) (y)) here the sum is over nearest neighbors and we set x 0 at the boundary of the set DN = ND \Zd. In case V ( ) = 2 this corresponds to a Gaussian, or harmonic model. We are looking at anharmonic models, assuming strict convexity of the function V . Let XN( )(x) = 1 N ([Nx]); x 2 D be the rescaled pro le. Our aim is to described the asymptotic of the law ofXN under a hard wall condition fXN( )(x) 0g and a volume condition fRDXN( ) dx vg for some v > 0. We prove that as N !1, the pro le converges to a deterministic shape v, solution of the variational problem inffID( ) = ZD (r (x)) dx : 2 H1 0 (D) ZD (x) dx vg: Here is the strictly convex surface tension. Our main step in the proof is the derivation of a large deviation principle for XN( ) with rate function ID. J urgen G artner Exact asymptotic formulas for moments and Lifshitz tails of the Anderson Hamiltonian (joint work with S. A. Molchanov) We consider the tails N( ), ! 1, of the spectral distribution function for the Hamiltonian H = + in l2(Zd). Here (x), x 2 Zd, is a eld of i.i.d. random variables with tails F ( ) = P ( (0) > ) which decay slower than double exponential tails. This means that the overwhelming part of the high exceedances of the potential consists of single site `peaks'. Under these assumptions we show that the tail behavior of N( ) is determined by the tails of the principle eigenvalue of H in a ball of xed large radius (with Dirichlet and re ection boundary conditions, respectively). In particular, this allows to derive exact asymptotic formulas for N( ) as !1 in the case of fractional exponential tails F ( ) = expf g, 0 < <1. A similar approach allows to nd corresponding formulas for the moments of the solution to the Cauchy problem @u=@t = Hu, u(0; x) 1. 8 Kenneth Hochberg The Longtime Behaviour of Multilevel Branching Systems (joint work with Andreas Greven) We consider (Yt)t 0, a critical two-level superprocess on Rd , that is the high-density, short-lifetime, small-mass di usion limit of the following particle system. Initially, particles are situated in Rd and are grouped into superparticles. Since the superparticles can be viewed as being elements of M(Rd), the state of the two-level system is an element of M(M(Rd)), with M denoting the set of positive Radon measures. Individual particles move according to a Brownian motion, and they split or die according to a critical branching mechanism with nite variance in which the o spring always belong to the same parent superparticle. In addition, every superparticle is replaced by two copies of itself or becomes extinct, with equal probability. Our focus is on the behaviour of the process (Yt)t 0 as t!1 and, in particular, on the features that are di erent from those of a single-level system|i.e., superBrownian motion. Speci cally, we discuss classes of initial distributions for which the behaviour of the law of Yt as t!1 can be determined. The longtime behaviour depends on the relative strength of the two competing forces, migration and branching|one attening the mass distribution and the other causing local accumulation of mass. In low dimensions, the branching dominates, while in higher dimensions, the migration tends to dominate. In dimensions d 4, the limit of L(Y (t)) will be 0 or 1 where 0 denotes the zero measure and 1 denotes the measure that satis es 1(A) = 1 for every Borel set with positive Lebesgue measure. Because of the in uence of the level-1 branching, each of the dimensions 1,2,3,4 has a distinct way of approaching 0 or 1, respectively. In dimensions d > 4, the limit is either a nondegenerate equilibrium state with nite intensity or has the degenerate form 0 or 1; the determination as to which limit occurs depends on two parameters|-the initial intensity of the particles and the spatial distribution of the superparticles. In particular, in d > 4, for a given particle intensity there are di erent equilibrium states, depending on the subdivision into superparticles. Nonetheless, it is also possible that in d > 4 the system becomes extinct, in the event that the particles are initially grouped into superparticles that are too \large" in a certain well-de ned sense. In contrast, for super-Brownian motion, the intensity is the only relevant parameter determining the longtime behaviour of an initial law in high dimensions. Achim Klenke Branching Random Walk in a Catalytic Medium (joint work with Andreas Greven and Anton Wakolbinger) Consider (critical binary) branching random walk ( t)t 0 on Zd in a space-time varying catalytic medium ( t)t 0. The medium determines the local branching rate of ( t)t 0. We consider the special case where ( t)t 0 is itself ordinary branching random walk. Our aim is to investigate the longtime behavior of ( t)t 0 if both ( t)t 0 and ( t)t 0 are started in Poison eld eld with intensity 1. 9 (1) If ( t)t 0 is persistent (i.e. if its symmetrized motion is recurrent), ( t)t 0 shows the classical dichotomy of extinction and persistence. (2) If both ( t)t 0 and ( t)t 0 perform symmetric simple random walk in Z, ( t)t 0 dies out fast enough such that ( t)t 0 converges to a Poisson mean 1 eld. However, if ( t)t 0 has a drift it hits all the catalyst clusters and dies out (in distribution). (3) If ( t)t 0 and ( t)t 0 perform both according to symmetric simple random walk on Z2, a new phenomenon occurs. Since ( t)t 0 dies out locally only in distribution and not almost surely, ( t)t 0 converges to a limit that re ects the history of the catalytic medium: a mixed Poisson eld. The law of the mixture can be understood in terms of the density of catalytic super-Brownian motion, investigated in a recent paper with Klaus Fleischmann. Wolfgang K onig Moment Asymptotics for the Continuous Anderson Model (joint work with J urgen Gartner) We consider the parabolic problem @u(t; x) = u(t; x) + u(t; x) (x); t > 0; x 2 Rd ; u(0; ) = 0; where = f (x); x 2 Rdg is a random stationary Holder continuous potential. We write h i for expectation w.r.t. and assume the niteness of H(t) = loghet (0)i for all t > 0. Furthermore, we assume that the potential has high peaks on small islands with certain probabilities; more precisely, for some scale function (t) ! 0 and all test measures 2 Pc(Rd), the limit J( ) = lim t!1 2(t) t log het R ( x) (dx)ie H(t) is assumed to exist. Then our main result says that, for all p 2 [1;1), the asymptotic expansion hu(t; 0)pi = exp H(t) t 2(t)( + o(1)) holds where the convergence parameter is given by = inffJ( ) + S( ) : 2 Pc(Rd)g, and S is the Donsker-Varadhan rate function for the Brownian occupationtimes measures. We explain the result in terms of the largest eigenvalue of the random operator + . 10 Gregory F. LawlerIntersection Exponent and Multifractal Spectrum for Brownian PathsThe intersection exponent for Brownian motion is a measure of how likely Brow-nian motion paths in two and three dimensions do not intersect. We consider theintersection exponent ( ) = d(k; ) as a function of and show that has a con-tinuous, negative second derivative. One major application of this result is to themultifractal spectrum of harmonic measure on a Brownian path; we show that themultifractal spectrum is nontrivial and give a formula for the spectrum in terms ofthe intersection exponent.Thomas M. LiggettStochastic Growth Models on TreesThe contact process on a graph G is a continuous time Markov process with statespace the collection of all subsets of G. Points are removed from the set at rate one,and points are added to the set at a rate that is a constant multiple of the numberof neighbors in the set. When G = Zd, there is a critical value that separates theregimes of survival and extinction. It was proved by Bezuidenhout and Grimmettthat in the subcritical regime, extinction occurs at a time with exponential tails,while in the supercritical regime, survival occurs in a strong sense, expressed forexample by the complete convergence theorem. In particular, there are alwayseither exactly one or exactly two extremal invariant measures.In 1992, Pemantle proved that the picture is richer on (most) homogeneous trees.There are now two critical values, and three types of behavior: extinction, weaksurvival, and strong survival. This talk is a survey of the dozen+ papers that haveappeared since 1992 on this process. Greatest interest centers on the intermediatephase of weak survival. Here there are in nitely many extremal invariant measures.Two families of invariant measures are constructed.Jean-Francois Le GallBranching processes, supoerprocesses and L evy processesWe discuss some recent developments concerning the genealogy of continuous-statebranching processes, and their applications to superprocesses and interacting parti-cle systems. The genealogy of a discrete-time Galton-Watson branching process isdescribed by a discrete tree, the genealogical tree of the population. One of the goalsof this talk is to study the analogous description for the genealogy of continuous-statebranching processes, which are the possible scaling limits of discrete-time Galton-Watson branching processes. To this end, we determine the so-called contour processassociated with the genealogical structure of a continuous-state branching process.Informally, the contour process of a tree gives the motion of a particle that exploresthe tree by moving up and down along its branches. In the special case of Feller'sdi usion, the simplest continuous-state branching process, it has been known for along time that the associated contour process is re ecting Brownian motion on the11 positive real line. A consequence of this fact is the Brownian snake constructionof superprocesses with a quadratic branching mechanism. This construction can beapplied to various properties of solutions of the partial di erential equation u = u2in a domain, including the existence of solutions with boundary blow-up and theclassi cation of general nonnegative solutions.As another application of the description of the genealogy of Feller's di usion,we investigate certain limit theorems for systems of coalescing random walks andfor the voter model. In particular, for the voter model in Zd starting initially withall types di erent, we show that the random measure describing the positions attime t of all individuals with the same type as the origin converges (after a suitablerescaling) asymptotically to a simple functional of super-Brownian motion (jointwork with M. Bramson and T. Cox).For a general continuous-state branching process, the contour process can bedetermined as a simple functional of a spectrally positive Levy process (joint workwith Yves Le Jan). This leads to a new connection between branching processesand Levy processes, which can be used to derive new results in both theories. Inparticular, we get an extension for general spectrally positive Levy processes of theclassical Ray-Knight theorem on Brownian local times. We also derive a path-valuedprocess construction of superprocesses with a general branching mechanism, whichextends the Brownian snake approach of the quadratic case.John T. LewisLarge Deviation Theory and Statistical MechanicsThe insights which Large Deviation Theory has provided in Statistical Mechan-ics have far-reaching consequences; I describe some of these including the use ofequipartition measures in information theory and ergodic theory.ReferencesJ.T. Lewis, Ch.-E. P ster, W.G. Sullivan: Entropy, Concentration of Proba-bility and Conditional Limit Theorems, Markov Processes and Related Fields1, 319-386 (1995)J.T. Lewis, Ch.-E. P ster, R. Russell and W.G. Sullivan: ReconstructionSequences and Equipartition Measures: An Examination of the AsymptoticEquipartition Property, IEEE Transactions of Information Theory (Nov. 1997)J.T. Lewis, Ch.-E. P ster, W.G. Sullivan: Large Deviations and GenericPoints (DIAS preprint, 1998)Terry LyonsBiological Models for Solving Stochastic PDEsThe Zakai/Kushner Stratonovich SPDE of nonlinear ltering is typical of a class ofparabolic pdes where there is a practical interest in obtaining numerical solutions12 in moderately high dimensions and where the solution is a probability measure.In this case e ective algorithms can be built from Branching particle systems. Thegreatest e ciency is achieved if the variance of these random algorithms can beminimised) this has been been achieved in some directions but the question ofhow to recombine particles e ciently has not been settled completely satisfactorilyat the present time.The methods developed to date are e ective, and of value. They are due to anumber of people Relevant references are Grisan, Lyons PTRF 1997, Grisan, Gaines,Lyons SIAM 1998, and papers of del Moral, Guionnet, Grisan, Smith, Gordon,Cli ord,...Stefano OllaEquilibriumFluctuations for the Ginzburg-Landau r' Interference Model(joint work with G. Giacomin and H. Spohn)Consider an e ective interface in d + 1 dimensions f'(x) 2 R; x 2 Zdg with inter-ference energyH(') = dX=1Xx V ('(x + ex) '(x))(massless eld model). We consider the corresponding Langevin dynamicsd't(x) = @H@'(x)dt+ dWt(x):The Gibbs measure e H=Z is reversible with respect to this dynamics.We prove that the uctuation eld"(f; t) ="1+d=2Xx f("x)'" 2t(x)converges in law, as " # 0, to the in nite dimensional Ornstein-Uhlenbeck processd (f; t) = (Af; t)dt+ dW (x; t)with drift operator A =Pi;j @iqij j; and we give a variational characterizationand upper and lower bounds for the di usion matrix qij.George PapanicolaouA survey of some recent work on waves in random media with applica-tions to seismologyI presented three problems: The O'Doherty-Anstey theory for the behavior of thefront of pulses travelling in randomly layerd media, the universality theory for wavelocalization in the time domain (also for randomly layered media), and the univer-sality theory for the P to S wave energy ratio in the deep coda of elastic waves ingeneral random media.13 Charles-Edouard PfisterLarge deviations and surface phase transition(joint work with Y. Velenik)We study the 2D Ising model in a square box L of linear size L, when the tem-perature is below the critical one. There is a real boundary magnetic eld h actingon one side of the box. We determine the exact asymptotic behaviour of the largedeviations of the magnetizationPt2 L (t) when L tends to in nity. Scaling thelengths by 1=L, the model is de ned in a xed box B of the Euclidean plane. Themain result is the following one.Let > c; h a real number; m < m < m (spontaneous magnetization) and0 < c < 1=4. Then there exist a, 0 < a < 1 and L0 such that for all L L0,ProbL; ;hh Xt2 L (t) mj Lj j LjLc i = exp L infK BV olK=m m2m F(@K) +O(La) ;where F(@K) is a functional on curves in B, given byF(@K) =Z@K (ns) ds+ ( ; h)j@K \W j :Here (ns) is the surface tension of an interface perpendicular to ns; ( ; h) is aboundary free energy coming from the action of the magnetic eld h on the side Wof the box B; j@K \W j is the Lebesgue measure of @K \W .There are two regimes. There exists hw( ) such that if h > hw( ), then ( ; h) =0. If h < hw( ), then ( ; h) < 0, so that the solution of the isoperimetric problemdepends explicitly on h. The solution is a convex body whose boundary has anon-empty intersection with the side W of B.References:1. P ster C.-E., Velenik Y.: Mathematical theory of the wetting phenomenon inthe 2D Ising model Helv. Phys. Acta 69 949-973 (1996).2. P ster C.-E., Velenik Y.: Large deviations and the continuum limit in the 2DIsing model Probab. Theory Relat. Fields 109 435-506 (1997).Michael RocknerAnalysis and Geometry on Con guration Spaces: the Gibbsian Case(joint work with Sergio Albeverio and Yuri Kondratiev)Using a natural \Riemannian{geometry{like" structure on the con guration spaceover Rd , we prove that for a large class of potentials the corresponding canonicalGibbs measures on can be completely characterized by an integration by partsformula. That is, if r is the gradient of the Riemannian structure on one can14 de ne a corresponding divergence div such that the canonical Gibbs measures areexactly those measures for which r and div are dual operators on L2( ; ).One consequence is that for such the corresponding Dirichlet forms E  arede ned. In addition, each of them is shown to be associated with a conservativedi usion process on with invariant measure . The corresponding generators areextensions of the operator := div r . The di usions can be characterized interms of a martingale problem and they can be considered as a Brownian motion onperturbed by a singular drift. Another main result of this paper is the following:if is a canonical Gibbs measure, then it is extreme (or a \pure phase") if andonly if the corresponding weak Sobolev space W 1;2 = ( ; ) on is irreducible.As a consequence we prove that for extreme canonical Gibbs measures the abovementioned di usions are time{ergodic. In particular, this holds for tempered grandcanonical Gibbs measures (\Ruelle measures") provided the activity constant issmall enough. We also include a complete discussion of the free case (i.e.,0)where the underlying space Rd is even replaced by a Riemannian manifold X.Hermann RostStationary Non-equilibrium States in a Random EnvironmentOne perturbs the selfadjoint (=reversible) Markov chain with random jump ratesa(x; y) = a(y; x) on Zd by passing to ea(x; y) = exp(u (y x)), where u 2 Rd isa xed vector, interpreted as external eld if the chain describes the position of acharged particle. The problem is to nd an invariant measure u(x), x 2 Zd, which isspatially stationary, jointly with the environment a( ; ). In generality, the patternis not yet solved. Partial solutions for d = 1 or a periodic environment are knownto exist. Here we show that on a strip Z nite set a solution exists and that thecurrent induced by the eld is bounded from below, independently of the width ofthe strip. The method of showing that relies on a model study for nite Markovchains, in which reversibility is perturbed by adding an \external term" which isnot of the gradient type.Alexander SchiedA Rademacher Type Theorem on Con guration Space and some Appli-cations(joint work with Michael Rockner)We consider an L2-Wasserstein type distance on the con guration space X over aRiemannian manifold X. Typically the distance between two con gurations will bein nite a situation reminiscent of the Cameron-Martin norm on Wiener space. Weprove that -Lipschitz functions are contained in a certain Dirichlet space associatedwith a measure on X satisfying certain assumptions. They are in particular ful lledby a large class of tempered grandcanonical Gibbs measures with respect to a super-stable lower regular pair potential. Both examples rely on the recent integration byparts formula of Albeverio, Kondratiev and Rockner (1997). As an application we15 show that, if A is a set with full measure, then the set of all con gurations havingnon-zero -distance to A is exceptional. This immediately implies, for instance, aquasi-sure version of the spatial ergodic theorem. We also show that our Dirichletform is quasi-regular, which implies the existence of an associated process. Finallywe prove in the case dimX 2 that the distance is optimal in the sense that it isthe so-called intrinsic metric of our Dirichlet form.Karl-Theodor SturmDirichlet forms and harmonic mapsIn this talk, two problems and partial solutions related to generalized harmonicmaps between singular spaces were presented.The rst problem is how to construct a reversible di usion process Xt on a givenmetric space (M; d). The solution consists in constructing a regular local Dirichletform as a -limit of certain non-local Dirichlet forms de ned in terms of the metricd and the reversible measure m, see [1].The second problem is how to de ne and approximate the energy of a map fwith values in a metric space N . This leads to the question whether12tEm d2(f(X0); f(Xt))as a function of t is always decreasing in t (or whether at least it converges fort ! 0). A rmative answers can be given either if Xt is BM on M = Rm (witharbitrary f;N; d) or if the space (N; d) has nonnegative curvature (with arbitraryM;Xt; f), see [2].References.[1] K.T.Sturm: Di usion processes and heat kernels on metric spaces. Ann. Prob.26 (1998), 1-55[2] K.T.Sturm: Monotone approximation of energy functionals for mappings intometric spaces I. J. reine angew. math. 486 (1997), 129-151Domokos Sz aszBall-Avoiding TheoremsAccording to the Boltzmann-Sinai Ergodic hypothesis, the system of N hard ballson the -dimensional torus is ergodic on submanifolds of the phase space speci edby the trivial conserved quantities of mechanics.A cornerstone of establishing the hypothesis for concrete systems is the handlingof ball-avoiding sets; in particular, the proof of smallness of the subset of those phasepoints where the set of the balls can be partitioned into two non-trivial classes suchthat the two subgroups of balls do not interact in the future.16 Weak ball-avoiding theorems claim that the measure of these ball-avoiding tra-jectories is zero, and strong ones claim that the subset of these orbits has topologicalcodimension at least two. Strong ball-avoiding theorems are necessary in proofs ofergodicity of hard ball systems (and, in general, of semi-dispersing billiards), whereasweak ones are su cient when one proves that the system in question is hyperbolic,a property implying that the ergodic components are open.Examples of ball-avoiding theorems are presented, for instance, a weak theoremof N andor Sim anyi and myself, which is needed in our recent proof of the followingtheorem: the system of N hard balls of masses m1; : : : ; mN and of radii r given onthe -torus is hyperbolic | apart from a countable union of analytic submanifoldsof the geometric parameters m1; : : : ; mN ; r.Balint T oth and Wendelin WernerSelf-Repelling Motions I. & II.We construct and study a continuous real-valued random process, which is of anew type: It is self-interacting (self-repelling) but only in a local sense: it onlyfeels the self-repellance due to its occupation-time measure density at `immediateneighbourhood' of the point it is just visiting. We focus on the most natural processwith these properties that we call `true self-repelling motion'. This is the continuouscounterpart to the integer-valued `true' self-avoiding walk, which had been studiedamong others by the rst author. One of the striking properties of true self-repellingmotion is that, although the couple (Xt; occupation-time measure of X at time t) isa continuous Markov process, X is not driven by a stochastic di erential equationand is not a semi-martingale. It turns out, for instance, that it has a nite variationof order 3/2, which contrasts with the nite quadratic variation of semi-martingales.One of the key-tools in the construction of X is a continuous system of coalescingBrownian motions similar to those that have been constructed by Arratia. We derivevarious properties of X (existence and properties of the occupation time densitiesLt(x), local variation, etc.) and an identity that shows that the dynamics of X canbe very loosely speaking described as follows: dXt is equal to the gradient (inspace) of Lt(x), in a generalized sense, even if x 7! Lt(x) is not di erentiable.S.R.S. VaradhanHydrodynamic scaling, recent developments and open problemsHydrodynamic scaling is the problem of tracking, over time, the spatial distributionof conserved quantities in a large system. Space and time are to be suitable rescaledbefore the limiting behavior is established.The simple exclusion models provide a convenient class of examples that illustratethe various phenomena that arise. The number of particles is the only conservedquantity and the quantity to be tracked is the local density over macroscopic scalesof space and time.17 In the asymmetric case the limiting equation is a hyperbolic nonlinear conserva-tion equation that develops shocks. the interesting question is how well the micro-scopic particle system tracks the shock, especially how the entropy loss associatedwith the shock is mirrored in the particle system.There are issues of large deviation that arise in both the symmetric as well as theasymmetric case. If we look at the case of Kawasaki dynamics relative to a Gibbsmeasure, the system turns out to be "non gradient" and consequently much morecomplex.Wendelin WernerSelf-Repelling Motions I. & II.(see B alint Toth and Wendelin Werner)Marc YorRanked Functionals of Brownian Excursions(joint work with Jim Pitman)The lecture aimed at presenting descriptions of some important random measurein terms of the laws of ranked lengths of excursions of Brownian motion or, moregenerally, Bessel processes.De ne F (dx)=P1i=1 V 0iXi(dx), where the Xi's are iid, with common distribu-tion on [0; 1], and V 0 = (V 01 ; V 02 ; : : : ) is a random sequence of masses which addup to 1. As far as the law of F is concerned, one may replace the (V 0i ) sequence byits decreasing reordering V1 V2 : : : Important random measures may be con-structed by the stick-breaking-procedure starting from beta variables. This givesthe two-parameter GEM distributions, whose reordering are the Poisson-Dirichletlaws PD( ; ), for 0 < 1, > .In terms of stochastic processes, PD(0; ) is the law of nVn( 0)0; n 1o where( 0;0) denotes the gamma subordinator, whereas PD( ; 0) is the law ofVn( ( )1 )( )1 ; n 1 where ( ( )s ; s 0) denotes the stable ( ) subordinator. Inboth cases, the above notation nVn(t)t ; n 1o indicates the normalized sequence ofranked lengths of component intervals of [0; 1] n Z, where Z is the closure of therange of the corresponding subordinator ( s; s 0).References about this work, done with J. Pitman, are found inJ. Pitman, M. Yor: Ranked Functionals of Brownian excursions. To appear inComptes Rendus Acad. Sci. Paris; end of December 1997 or January 1998.18 E-mail list of participantsNameFirst nameE-mail [email protected] [email protected] .chvan den Berg [email protected]@cucc.ruk.cuni.czBleherPavelbleher@[email protected]@math.washington.eduChanTerenceterence@[email protected]@math.ubc.caDawsonDonddawson@ elds.utoronto.caDeuschelJean-Dominiquedeuschel@math.tu-berlin.deFleischmann [email protected]artnerJ[email protected]@mi.informatik.uni-frankfurt.deGeorgiiHans-Ottogeorgii@rz.mathematik.uni-muenchen.deGrevenAndreasgreven@mi.uni-erlangen.deGundlachMatthiasgundlach@[email protected] Hollander [email protected][email protected]@[email protected][email protected]@[email protected]@[email protected]@[email protected]@paris.polytechnique.frPapanicolaou [email protected] sterCharles-Ed.charles.p [email protected] .chR[email protected]@math.uni-heidelberg.deSchiedAlexanderschied@mathematik.hu-berlin.deSturmKarl-Theodorsturm@[email protected] [email protected] [email protected]@cims.nyu.eduWernerWendelinwerner@stats.math.u-psud.frWinterAnitawinter@[email protected]