multiple points of sample paths of markov processes 15 in 7 kahane treated symmetric stable levy processes iand obtained some zero probability and positive probability results the probability 1 result in the case k 2 was then proved by testard 16 in this paper we study this problem both positive probability and probability 1 for two large classes of markov processes namely diffusions driven by stochastic differential equations and levy processes with positive lower indices we first prove in theorem 2 1 a result for a general time homogeneous markov process with the transition density functions satisfying certain explicit conditions the existence of the k multiple points of x e is then ensured by the positivity of h0 e the hausdorff 01 measure of e where 0 0 01 1 and 0 is a

critical behaviour of the random field ising model we present an analytic real space renormalization group calculation for the random field ising model we apply the migdal kadanoff approximation for the renormalization of a cubic cell in dimensions d introducing a new field partitioning scheme which allows us to treat the random field fluctuations in a coherent manner our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality we expand analytically in further we show that and the magnitude of the inequality go to zero exponentially with we calculate the crossover exponent from pure to the random field system and find surprisingly good agreement with experimental values we find that satisfies the schwartz soffer inequality the susceptibility exponent of the pure system we expand in and find that the magnitude of the inequality varies exponentially in finally we find that dimensional reduction is satisfied to first order in with the reduced dimension

rationality of euler chow series and finite generation of cox rings abstract we study the rationality of the euler chow series e 1 x of codimension one cycles on a projective variety x and its relation with the effective cone and the cox ring of x among other things we prove that e 1 x is transcendental if the cone ne 1 x of pseudo effective divisors on x has infinitely many extremal rays generated by effective divisors on the other hand we give examples showing that the converse fails in addition we give an example where e 1 x is rational and cox x is infinitely generated finally we compute e 1 x for del pezzo surfaces x

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

self intersection gauge for random walks and for brownian motion a class of random fields associated with multiple points of a random walk in the plane is studied it is proved that these fields converge in distribution to analogous fields measuring self intersections of the planar brownian motion the concluding section contains a survey of literature on intersection local times and their renormalizations a brief look through the first pages of this section could provide the reader with additional motivation for the present work

renormalization of a self avoiding brownian motion in two dimensions abstract we obtain a measure defined on the space of continuous sample paths which describes a weakly self avoiding brownian motion in a 2 dimensional euclidean space our method is also applicable to a brownian motion with the dirichlet boundary conditions in a 2 dimensional domain for the new measures an inequality of simon lieb type is also obtained

the range of stable random walks limit theorems are proved for the range of d dimensional random walks in the domain of attraction of a stable process of index p3 the range rn is the number of distinct sites of zd visited by the random walk before time n our results depend on the value of the ratio 3 d the most interesting results are obtained for 2 3 1 which can only occur if d 1 we prove the convergence in distribution of rn e rn toward some constant times the lebesgue measure of the range of the limiting stable process some of our results require regularity assumptions on the characteristic function of x

a renormalization result for the intersection local time of lattice random walks ind 3 dimensions in this paper we derive some general conditions on stable walks inz d under which the central limit theorem holds for their normalized intersection local time in particular we prove that the process given by the normalized intersection local time of the simple random walk inz d withd 3 is weakly convergent to the standard brownian motion

large deviations for renormalized self intersection local times of stable processes we study large deviations for the renormalized self intersection local time of d dimensional stable processes of index 2d 3 d we find a difference between the upper and lower tail in addition we find that the behavior of the lower tail depends critically on whether d or d

a new discrete edwards model and a new polymer measure in two dimensions the new discrete edwards models in this paper are defined in terms of the so called restricted intersection local times of the lattice random walk in two dimensions we study the asymptotic behaviours of these new discrete edwards models in the superrenormalizable cases in particular by approximating these models we can construct new polymer measures in two dimensions which are different from the original polymer measures obtained by approximating the original discrete edwards models the new discrete edwards models can be thought of as zero component lattice 4 fields with different cutoffs in the free and interacting parts

sur la saucisse de wiener et les points multiples du mouvement brownien on considere b t t 0 un mouvement brownien a valeurs dans l espace euclidien a d dimensions r d d 2 issu de o on s interesse a la probabilite que b penetre avant un instant donne t dans une petite boule de l espace

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

harmonic analysis of additive levy processes let x1 xn denote n independent d dimensional levy processes and consider the n parameter random field mathfrak x t x 1 t 1 cdots x n t n first we demonstrate that for all nonrandom borel sets f subseteq bf r d the minkowski sum mathfrak x bf r n oplus f of the range mathfrak x bf r n of mathfrak x with f can have positive d dimensional lebesgue measure if and only if a certain capacity of f is positive this improves our earlier joint effort with yuquan zhong by removing a certain condition of symmetry in khoshnevisan et al ann probab 31 2 1097 1141 2003 moreover we show that under mild regularity conditions our necessary and sufficient condition can be recast in terms of one potential densities this rests on developing results in classical non probabilistic harmonic analysis that might be of independent interest as was shown in khoshnevisan et al ann probab 31 2 1097 1141 2003 the potential theory of the type studied here has a large number of consequences in the theory of levy processes presently we highlight a few new consequences

predictability and stopping on lattices of sets as a first step in the development of a general theory of set indexed martingales we define predictability on a general space with respect to a filtration indexed by a lattice of sets we prove a characterization of the predictable algebra in terms of adapted and left continuous processes without any form of topology for the index set we then define a stopping set and show that it is a natural generalization of the stopping time in particular the predictable algebra can be characterized by various stochastic intervals generated by stopping sets

random walks and intersection local time with each random walk on z 2 we associate a functional related to the number of steps which the walk spends in sites occupied at least k times we show that if our random walk is in the domain of attraction of a stable process of order greater than 2 2k 2 2k 1 then our functional converges to the intersection local time of the limiting process

real space renormalization group analysis of the random field ising model we present an analytic migdal kadanoff renormalization group analysis of the random field ising model the renormalization flows to a zero temperature critical point from which we calculate independently three critical exponents in arbitrary dimension in three dimensions the magnetization exponent and the schwartz soffer inequality is almost satisfied as an equality expanding analytically in we find that and the distance from the upper bound of the equality go to zero exponentially with

local times of fractional brownian sheets let b0h b0h t t i mathbb r n be a real valued fractional brownian sheet consider the n d gaussian random field bh defined by bh t bh1 t bhd t t i mathbb r n where bh1 bhd are independent copies of b0h in this paper the existence and joint continuity of the local times of bh are established

measure of the multiple self intersection set of a markov process let x t t 0 be a markov process in rm with homogeneous transition density p t x y for a closed bounded set b rm x is said to have a self intersection of order r 2 in b if there are distinct points t1 tr such that x t1 b and x tj x t1 for j 2 r the focus of this work is the hausdorff measure suitably defined of the set of such r tuples the main result is that under general conditions on p t x y as well as the specific condition there is a measure function m t defined in terms of the integral above such that the corresponding hausdorff measure of self intersection set is positive with positive probability the results are applied to levy and diffusion processes and are shown to extend recent results in this area

an almost sure invariance principle for the range of planar random walks for a symmetric random walk in z 2 with 2 moments we represent r n the cardinality of the range in terms of an expansion involving the renormalized intersection local times of a brownian motion we show that for each k 1 logn k 1 n r n 1 j 1 2 logn c x j j n 0 a s where w t is a brownian motion w n t w nt n j n is the renormalized intersection local time at time 1 for w n and c x is a constant depending on the distribution of the random walk

some exact inequalities for the random field ϕ4 system exact lower bound on the correlation function and susceptibility are obtained in terms of the quantity r r0 12u 2 d ddl l l where l l are the exact averages further lower bounds are obtained on r relating it to an exactly soluble model

an extension of a result of burdzy and lawler it is shown that for all mean zero finite variance random walks the critical non intersection exponents are equal to those for brownian motion the method uses the local time of intersection

lack of self averaging in random systems liability or asset abstract the study of quenched random systems is facilitated by the idea that the ensemble averages describe the thermal averages for any specific realization of the couplings provided that the system is large enough careful examination suggests that this idea might have a flaw when the correlation length becomes of the order of the size of the system we find that certain bounded quantities are not self averaging when the correlation length becomes of the order of the size of the system this suggests that the lack of self averaging expressed in terms of properly chosen signal to noise ratios may serve to identify phase boundaries this is demonstrated by using such signal to noise ratios to identify the boundary of the ferromagnetic phase of the random field ising system and compare the findings with more traditional measures

holder conditions for the local times and the hausdorff measure of the level sets of gaussian random fields let y t t n be a real valued strongly locally nondeterministic gaussian random field with stationary increments and y 0 0 consider the n d gaussian random field defined by where x 1 x d are independent copies of y the local and global holder conditions in the set variable for the local time of x t are established and the exact hausdorff measure of the level set x 1 x is evaluated

large and moderate deviations for intersection local times we study the large and moderate deviations for intersection local times generated by respectively independent brownian local times and independent local times of symmetric random walks our result in the brownian case generalizes the large deviation principle achieved in mansmann 1991 for the l 2 norm of brownian local times and coincides with the large deviation obtained by csorgo shi and yor 1991 for self intersection local times of brownian bridges our approach relies on a feynman kac type large deviation for brownian occupation time certain localization techniques from donsker varadhan 1975 and mansmann 1991 and some general methods developed along the line of probability in banach space our treatment in the case of random walks also involves rescaling spectral representation and invariance principle the law of the iterated logarithm for intersection local times is given as an application of our deviation results

local times on curves and uniform invariance principles sufficient conditions are given for a family of local times l t ofd dimensional brownian motion to be jointly continuous as a function oft and then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of brownian motion uniformly over a large family of measures applications included some new results for intersection local times for brownian motions on 2 and 2

wiener sausage and self intersection local times abstract let b bt t 0 be a standard brownian motion in r 2 for every e 0 and every compact subset k of r 2 the wiener sausage of radius e associated with k is defined as the union of the sets bs ek s 0 1 the present paper gives full asymptotic expansions for the area of the wiener sausage when the radius e goes to 0 the kth term of the expansion is of order log e k and involves a random variable which measures the number of k multiple self intersections of the process such random variables are called renormalized self intersection local times and have been recently introduced and studied by e b dynkin a self contained construction of these local times is given together with a number of new approximations

rationality of motivic chow series modulo a1 homotopy abstract consider the formal power series c p x t called the motivic chow series where c p x c p x is the chow variety of x parametrizing the p dimensional effective cycles on x with c p x its connected components and c p x its class in k c h m a 1 the k ring of chow motives modulo a 1 homotopy using the picard product formula and torus action we will show that the motivic chow series is rational in many cases

random fields associated with multiple points of the brownian motion abstract let xt be the brownian motion in r d the random set t1 tn z xtl xtn z in r d n is empty a s except in the following cases a n 1 d 1 2 b d 2 n 2 3 c d 3 n 2 in each of these cases a family of random measures m concentrated on is constructed takes values in a certain class of measures on r d measures m characterize the time space location of self intersections for brownian paths if n d 1 then m dt dz dz nz dt where n2 is the local time at z in the case n 2 the set can be identified with the set of brownian loops the measure m explodes on the diagonal t1 t2 and to study small loops a random distribution which regularizes m is constructed

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

self intersection local time critical exponent large deviations and laws of the iterated logarithm if t is renormalized self intersection local time for planar brownian motion we characterize when ee 1 is finite or infinite in terms of the best constant of a gagliardo nirenberg inequality we prove large deviation estimates for 1 and 1 we establish lim sup and lim inf laws of the iterated logarithm for t as t

some properties of the fractional ornstein uhlenbeck process we consider the fractional analogue of the ornstein uhlenbeck process i e the solution of the langevin equation driven by a fractional brownian motion in place of the usual brownian motion we establish some properties of these processes we show that the process is local nondeterminism for a two dimensional process we show that its renormalized self intersection local time exists in l2 if and only if

irrationality of motivic series of chow varieties in this paper we extend the euler chow series for chow varieties to chow motives in both series it is very natural to ask when the series is rational we give an example where the extended series is not rational

real space renormalization group approach to the random field ising model the authors discuss results from two types of real space renormalization group rsrg calculations applied to the random field ising model in three dimensions starting from a lattice of size l the rsrg is used to reduce the lattice to a size l 2 on which the trace is done exactly in this way thermodynamic properties such as the magnetization and susceptibility can be determined approximately they find that for a given size the susceptibility increases as the temperature t is reduced down to the transition temperature tc and becomes essentially independent of temperature below tc both in the vicinity of tc and at lower temperatures there are large sample to sample fluctuations in the susceptibility which grow with increasing system size they interpret these results in terms of the droplet theory of the transition