clustering of linearly interacting diffusions and universality of their long time limit distribution let k d d 1 be a compact convex set and a countable abelian group we study a stochastic process x in k equipped with the product topology where each coordinate solves a sde of the form dx i t j a j i x j t x i t dt x i t db i t here a is the kernel of a continuous time random walk on and is a continuous root of a diffusion matrix w on k if x t converges in distribution to a limit x and the symmetrized random walk with kernel a s i a i a i is recurrent then each component x i is concentrated on x k x 0 and the coordinates agree i e the system clusters both these statements fail if a s is transient under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of k we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components this distribution turns out to be universal in all recurrent kernels a s on abelian groups

interacting fisher wright diffusions in a catalytic medium we study the longtime behaviour of interacting systems in a randomly fluctuating space time medium and focus on models from population genetics there are two prototypes of spatial models in population genetics spatial branching processes and interacting fisher wright diffusions quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment catalytic medium

lyapunov exponent for the parabolic anderson model in rd abstract we consider the asymptotic almost sure behavior of the solution of the equation u t x u 0 x 2 0 t u s x d s 0 t u s x w x s where w x x r d is a field of brownian motions in fact we establish existence of the lyapunov exponent lim t 1 t log u t x we also show that c 1 1 3 c 2 1 5 as 0 under the assumption that the correlation function of the background field w x x r d is c for 1 2

renormalization of the voter model in equilibrium we consider the d dimensional voter model for d 3 our interest is the large scale limit of the equilibrium state of the voter model where we prove the d 3 results of 1 for d 4 which turn out to be of a different nature than for d 3 for this purpose we use the historical process we establish some surprising facts about the green s function of random walks in dimension d 4 which lead to the different features in d 3 versus d 4 secondly we prove an analogous result for the voter model on the hierarchical group

intermittency in a catalytic random medium in this paper we study intermittency for the parabolic anderson equation u t u u where u d 0 is the diffusion constant is the discrete laplacian and d 0 is a space time random medium we focus on the case where is times the random medium that is obtained by running independent simple random walks with diffusion constant starting from a poisson random field with intensity throughout the paper we assume that 0 the solution of the equation describes the evolution of a reactant u under the influence of a catalyst we consider the annealed lyapunov exponents that is the exponential growth rates of the successive moments of u and show that they display an interesting dependence on the dimension d and on the parameters with qualitatively different intermittency behavior in d 1 2 in d 3 and in d 4 special attention is given to the asymptotics of these lyapunov exponents for 0 and

a phase transition for the coupled branching process part i the ergodic theory in the range of finite second moments we consider a particular markov process tu on s s n the random variable tu x is interpreted as the number of particles atx at timet the initial distribution of this process is a translation invariant measure withf x d the evolution is as follows at rateb x a particle is born atx but moves instantaneously toy chosen with probabilityq x y all particles at a site die at ratepd withp 0 1 d and individual particles die independently from each other at rate 1 p d every particle moves independently of everything else according to a continuous time random walk

twisted loop groups and their affine flag varieties we develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field this can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case a consequence of our results is that our construction also includes the flag varieties for kac moody lie algebras of affine type we also give a coherence conjecture on the dimensions of the spaces of global sections of the natural ample line bundles on the partial flag varieties attached to a fixed group over k t and some applications to local models of shimura varieties

frobenius splitting of equivariant closures of regular conjugacy classes let g denote a connected semisimple and simply connected algebraic group over an algebraically closed field k of positive characteristic and let g denote a regular element of g let x denote any equivariant embedding of g we prove that the closure of the conjugacy class of g within x is normal and cohen macaulay moreover when x is smooth we prove that this closure is a local complete intersection as a consequence the closure of the unipotent variety within x shares the same geometric properties

on the flatness of local models for the symplectic group abstract we investigate the bad reduction of certain shimura varieties associated to the symplectic group more precisely we look at a model of the shimura variety at a prime p with parahoric level structure at p we show that this model is flat as conjectured by rapoport and zink ann of math stud 141 1996 and that its special fibre is reduced a crucial ingredient is faltings theorem on the normality of schubert varieties in the affine flag variety

phase transitions for the long time behavior of interacting diffusions let xi t i zd t 0 be the system of interacting diffusions on 0 defined by the following collection of coupled stochastic differential equations dxi t j zd a i j xj t xi t dt bxi t dwi t i z t 0 here a is an irreducible random walk transition kernel on z z b 0 is a diffusion parameter and wi t i zd t 0 is a collection of independent standard brownian motions on r the initial condition is chosen such that xi 0 i zd is a shift invariant and shift ergodic random field on 0 with mean 0 the evolution preserves the mean we show that the long time behaviour of this system is the result of a delicate interplay between a and b in contrast to systems where the diffusion function is subquadratic in particular let i j 12 a i j a j i i j z denote the symmetrised transition kernel we show that a if is recurrent then for any b 0 the system locally dies out b if is transient then there exist b b2 0 such that b1 the system converges to an equilibrium with mean if 0 b b3 has a finite 2 nd moment if and only if 0 b2 the equilibrium is shown to be associated and mixing for all 0 b2 we further conjecture that the system locally dies out at b b for the case where a is symmetric and transient we further show that c there exists a sequence b2 b3 b4 0 such that c1 has a finite m th moment if and only if 0 bm c3 b2 m 1 bm 2 c4 limm m 1 bm c supm 2 m 1 bm mathematisches institut universitat erlangen nurnberg bismarckstrasse 1 1 2 d 91504 erlangen germany greven mi uni erlangen de mathematical institute leiden university p o box 9512 2300 ra leiden the netherlands denholla math leidenuniv nl eurandom p o box 513 5600 mb eindhoven the netherlands

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

central limit theorem for an infinite lattice system of interacting diffusion processes on donne une demonstration basee sur une representation par integrale stochastique de dimension infinie des fonctionnelles lisses des processus de diffusion

on the flatness of models of certain shimura varieties of pel type consider a pel shimura variety associated to a unitary group that splits over an unramified extension of q p rapoport and zink have defined a model of the shimura variety over the ring of integers of the completion of the reflex field at a place lying over p with parahoric level structures at p we show that this model is flat as conjectured by rapoport and zink and that its special fibre is reduced

renormalization of hierarchically interacting isotropic diffusions we study a renormalization transformation arising in an infinite system of interacting diffusions the components of the system are labeled by the n dimensional hierarchical lattice n 2 and take values in the closure of a compact convex set bar d subset mathbb r d d geqslant 1 each component starts at some d and is subject to two motions 1 an isotropic diffusion according to a local diffusion rate g bar d to 0 infty chosen from an appropriate class 2 a linear drift toward an average of the surrounding components weighted according to their hierarchical distance in the local mean field limit n block averages of diffusions within a hierarchical distance k on an appropriate time scale are expected to perform a diffusion with local diffusion rate f k g where f k g f c k circ circ f c 1 g is the kth iterate of renormalization transformations fc c 0 applied to g here the ck measure the strength of the interaction at hierarchical distance k we identify fc and study its orbit f k g k 0 we show that there exists a fixed shape g such that limk kf k g g for all g where the k are normalizing constants in terms of the infinite system this property means that there is complete universal behavior on large space time scales our results extend earlier work for d 1 and bar d 0 1 resp 0 the renormalization transformation fc is defined in terms of the ergodic measure of a d dimensional diffusion in d 1 this diffusion allows a yamada watanabe type coupling its ergodic measure is reversible and the renormalization transformation fc is given by an explicit formula all this breaks down in d 2 which complicates the analysis considerably and forces us to new methods part of our results depend on a certain martingale problem being well posed

global observability of a class of nonlinear discrete time systems abstract it is shown that discrete time systems whose dynamics are governed by piecewise monotone maps of the interval are globally observable under a general class of observations the results rest on the fact that the dynamics are ergodic

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

generalized potlatch and smoothing processes we consider simple generalizations of the potlatch and smoothing processes which were introduced in 8 and studied in 5 these generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition the original potlatch and smoothing processes do not exhibit phase transition our results show that for the generalized processes phase transition does not usually occur in one or two dimensions but usually does occur in higher dimensions upper and lower bounds for the relevant critical values are obtained as one application of our results we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d thus answering a question we raised in 3 this is done by comparing the contact process with an appropriate generalized smoothing process

on the moments and the interface of the symbiotic branching model in this paper we introduce a critical curve separating the asymptotic behavior of the moments of the symbiotic branching model introduced by etheridge and fleischmann stochastic process app 114 2004 127 160 into two regimes using arguments based on two different dualities and a classical result of spitzer trans amer math soc 87 1958 187 197 on the exit time of a planar brownian motion from a wedge we prove that the parameter governing the model provides regimes of bounded and exponentially growing moments separated by subexponential growth the moments turn out to be closely linked to the limiting distribution as time tends to infinity the limiting distribution can be derived by a self duality argument extending a result of dawson and perkins ann probab 26 1998 1088 1138 for the mutually catalytic branching model as an application we show how a bound on the 35th moment improves the result of etheridge and fleischmann stochastic process appl 114 2004 127 160 on the speed of the propagation of the the interface of the symbiotic branching model

on compactifications of the steinberg zero fiber abstract let g be a connected semisimple linear algebraic group over an algebraically closed field k of positive characteristic and let x denote an equivariant embedding of g we define a distinguished steinberg fiber n in g called the zero fiber and prove that the closure of n within x is normal and cohen macaulay furthermore when x is smooth we prove that the closure of n is a local complete intersection

ergodic theorems for coupled random walks and other systems with locally interacting components in 5 the second author introduced a variety of new infinite systems with locally interacting components on the basis of computations for the finite analogues of these systems he made conjectures ragarding their limiting behavior as t this paper is devoted to the construction of these processes and to the proofs of these conjectures we restrict ourselves primarily to spatially homogeneous situations interesting problems remain unsolved in inhomogeneous cases two features distinguish these processes from most other infinite particle systems which have been studied one is that the state spaces of these systems are noncompact the other that even though the invariant measures are not generally of product form one can nevertheless compute explicitly the first and second moments of the number of particles per site in equilibrium the second moment computations are of inherent interest of course and they play an important role in the proofs of the ergodic theorems as well

an example of almost universally observable vector fields on p 1 r abstract using methods of dynamical systems we construct examples of smooth almost universally observable vector fields on the projective 1 space these vector fields are non linear non autonomous and their time dependence is almost periodic

affine demazure modules and t fixed point subschemes in the affine grassmannian let g be a simple algebraic group defined over c and t be a maximal torus of g for a dominant coweight of g the t fixed point subscheme formula of the schubert variety formula in the affine grassmannian grg is a finite scheme we prove that for all such if g is of type a or d and for many of them if g is of type e there is a natural isomorphism between the dual of the level one affine demazure module corresponding to and the ring of functions twisted by certain line bundle on grg of formula we use this fact to give a geometrical proof of the frenkel kac segal isomorphism between basic representations of affine algebras of a d e type and lattice vertex algebras

long time behavior and coexistence in a mutually catalytic branching model we study a system of two interacting populations which undergo random migration and mutually catalytic branching the branching rate of one population at a site is proportional to the mass of the other population at the site the system is modelled by an infinite system of stochastic differential equations allowing symmetric markov migration if the set of sites is discrete z d or by a stochastic partial differential equation with brownian migration if the set of sites is the real line a duality technique of leonid mytnik which gives uniqueness in law is used to examine the long time behavior of the solutions for example with uniform initial conditions the process converges to an equilibrium distribution as t and there is coexistence of types in the equilibrium iff the random migration is transient

observability of minimal distal systems in a recent series of papers drager and martin have shown global observability for a variety of particular classes of systems including both discrete and continuous time systems the underlying philosophy behind their methods is apparently that if a flow is ergodic and the system observation has a sufficiently special value the system is completely observable in this paper we prove observability for observed systems whose underlying flow is minimal and distal while our results hold also in discrete time we illustrate this result for invariant ergodic systems on nilmanifolds i e those invariant flows with an infinitesimal generator that is irrational modulo the commutator one well known example of such a flow is the irrational flow on a torus studied by drager and martin

on a grauert riemenschneider vanishing theorem for frobenius split varieties in characteristic p it is known that the grauert riemenschneider vanishing theorem is not valid in characteristic p 1 here we show that it may be restored in the presence of a suitable frobenius splitting the proof uses interchanging two projective limits one involving iterated frobenius maps cf 2 and 4 the other coming from grothendieck s theorem on formal functions that leads to the following general vanishing theorem which we then apply in the situation of the grauert riemenschneider theorem

the space of triangles vanishing theorems and combinatorics abstract we consider compactifications of pn 3 ij the space of triples of distinct points in projective space one such space is a singular variety of configurations of points and lines another is the smooth compactification of fulton and macpherson and a third is the triangle space of schubert and semple we compute the sections of line bundles on these spaces and show that they are equal as gl n representations to the generalized schur modules associated to bad generalized young diagrams with three rows borel weil theorem on the one hand this yields weyl type character and dimension formulas for the schur modules on the other a combinatorial picture of the space of sections cohomology vanishing theorems play a key role in our analysis

the parabolic anderson model in a dynamic random environment space time ergodicity for the quenched lyapunov exponent we continue our study of the parabolic anderson equation u x t t u x t x t u x t x2 z d t 0 where 2 0 1 is the diusion constant is the discrete laplacian and plays the role of a dynamic random environment that drives the equation the initial condition u x 0 u0 x x2 z d is taken to be non negative and bounded the solution of the parabolic anderson equation describes the evolution of a eld of particles performing independent simple random walks with binary branching particles jump at rate 2d split into two at rate 0 and die at rate 0 we assume that is stationary and ergodic under translations in space and time is not constant and satises e j 0 0 j e 0 0 for 2 0 1 in the present paper we show that lim 1 0 e 0 0 under an additional

etale splittings of certain azumaya algebras on toric and hypertoric varieties in positive characteristic abstract for a smooth toric variety x over a field of positive characteristic a t equivariant etale cover y t x 1 trivializing the sheaf of crystalline differential operators on x is constructed this trivialization is used to show that d is a trivial azumaya algebra along the fibers of the moment map t x 1 t 1 this result is then extended to certain azumaya algebras on hypertoric varieties whose global sections are analogous to central reductions of the hypertoric enveloping algebra a criterion for a derived beilinson bernstein localization theorem is then formulated