the trace of spatial brownian motion is capacity equivalent to the unit square we show that with probability 1 the trace b 0 1 of brownian motion in space has positive capacity with respect to exactly the same kernels as the unit square more precisely the energy of occupation measure on b 0 1 in the kernel f x y is bounded above and below by constant multiples of the energy of lebesgue measure on the unit square the constants are random but do not depend on the kernel as an application we give almost sure asymptotics for the probability that an stable process approaches within of b 0 1 conditional on b 0 1

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

path integral representation for fractional brownian motion fractional brownian motion fbm is a generalization of the usual brownian motion a path integral representation that has recently been suggested for it is shown to be not for the fbm but for a different generalization of the brownian motion a new path integral representation is given and its measure has fractional derivatives of the path in it the measure shows that the process is gaussian but is in general non markovian even though brownian motion itself is markovian it is shown how the propagator for the motion of free fbm may be evaluated this is somewhat more complex than for the usual path integrals due to the occurrence of fractional derivatives we also find the propagator in the presence of a linear absorption potential and for fbm on a ring

new extremal characterizations of generalized inverses of linear operators abstract in 1956 r penrose studied best approximate solutions of the matrix equation ax b he proved that a b where a is the moore penrose inverse is the unique matrix of minimal frobenius norm among all matrices which minimize the frobenius norm of ax b in particular a is the unique best approximate solution of ax i the vector version of penrose s result that is the fact that the vector a b is the best approximate solution in the euclidean norm of the vector equation ax b has long been generalized to infinite dimensional hilbert spaces in this paper an infinite dimensional version of penrose s full result is given we show that a straightforward generalization is not possible and provide new extremal characterizations in terms of the hermitian order of a and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least squares solutions for a certain class of operators we can phrase our characterizations in terms of a whole class of norms including the hilbert schmidt and the trace norms thus providing new extremal characterizations even in the matrix case we treat both operators with closed range and with not necessarily closed range finally we characterize a as the unique inner inverse of minimal hilbert schmidt norm if a 2

unbounded symmetrizable idempotents abstract the relationship between closed unbounded idempotents and dense decompositions of a hilbert space is explored by extending the notion of compatibility between closed subspaces and positive bounded operators

sample function properties of multi parameter stable processes the almost sure behaviour of the sample functions of the vectorvalued n parameter wiener process and its stable analogues is investigated especially their small fluctuations in particular laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times these results give precise information about the minimum oscillation of the sample functions in addition the hausdorff measure problem for the graph and the range of these processes is solved

on the fock space representation of functionals of the occupation field and their renormalization on fournit une representation du champ d occupation et de ses puissances renormalisees dans un bon espace de fock

occupation time large deviations of two dimensional symmetric simple exclusion process we prove a large deviations principle for the occupation time of a site in the two dimensional symmetric simple exclusion process the decay probability rate is of order t logt and the rate function is given by 2 sin 1 2 1 sin 1 2 1 2 the proof relies on a large deviations principle for the polar empirical measure which contains an interesting log scale spatial average a contraction principle permits us to deduce the occupation time large deviations from the large deviations for the polar empirical measure

dimension results for gaussian vector fields and index alpha stable fields the hausdorff dimension and packing dimension of the image and graph of gaussian vector fields and index a stable fields are obtained under general conditions which allow for different local behavior of the components and for dependence among them these results correct some errors in print the condition for an index a stable field to have k multiple points are considered and in the case of nk k 1 e 1ap the hausdorff dimension of the set of k multiple times is given

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

weighted least squares solutions of the equation axb c 0 abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a b l h with closed range and c l h we study the following weighted approximation problem analyze the existence of 0 1 m i n x l h a x b c p w where x p w w 1 2 x p we also study the related operator approximation problem analyze the existence of 0 2 m i n x l h a x b c w a x b c where the order is the one induced in l h by the cone of positive operators in this paper we prove that the existence of the minimum of 0 2 is equivalent to the existence of a solution of the normal equation a w a x b c 0 we also give sufficient conditions for the existence of the minimum of 0 1 and we characterize the operators where the minimum is attained

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

superprocesses and their linear additive functionals let x x t p be a measure valued stochastic process linear functionals of x are the elements of the minimal closed subspace l of l 2 p which contains all x t b with x t b 2 dp we represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their laplace transforms we consider markov processes with nonstationary transition functions to reveal better the principal role of the backward equations

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

some norm inequalities concerning generalized inverses 2 abstract we extend to the von neumann schatten classes c p and norms psome inequalities concerning generalized inverses we extend penrose s result on minimizing ax b 2 to the p norm we also minimize x p where x varies such that 1 b a ax a ab and 2 a ax a in both cases a b fixed local considerations are adduced in each result here global minima and critical points coincide

linear problems in linear problems in pth order and stable processes this work extends to processes with finite moments of order p 1 p 2 and to symmetric alpha stable processes 1 alpha 2 some of the basic linear theory known for processes with finite second moments p 2 and for gaussian processes alpha 2 here the covariation plays a role analogous to the covariance specifically stochastic integrals of two types are introduced and studied for pth order processes and in particular for symmetric stable processes regression estimates and linear estimates on certain symmetric stable processes are evaluated including regression and linear filtering of signal in noise also for certain symmetric stable inputs the identification of a linear system from the input covariation and the input output cross covariation is considered and the way the distribution of the output depends on the linear system is studied

renormalized self intersection local time for fractional brownian motion let b h t be a d dimensional fractional brownian motion with hurst parameter h 0 1 assume d 2 we prove that the renormalized self intersection local time l t 0 t 0 b h t b h s ds dt e t 0 t 0 b h t b h s ds dt exists in l 2 if and only if h h 3 2d r e l e converges in distribution to a normal law n 0 t 2 as e tends to zero where l e is an approximation of l defined through 2 and r e loge 1 if h 3 2d and r e e d 3 2h if 3 2d h

the asymptotics of stable sausages in the plane in this paper we develop an asymptotic expansion for the e neighborhood of the symmetric stable process of order 3 1 f 2 our expansion is in powers of 82 with the nth coefficient related to n fold self intersections of our stable process 1 introduction in this paper we will develop asymptotic expansions for the area of the neighborhood of x the symmetric stable process of order f3 1 f3 2 in r2 our asymptotic expansion as 8 0 is in powers 82 n with coefficients yn which are random variables related to n fold self intersections of x the 8 neighborhood of x known as the stable sausage is defined as

generalized ornstein uhlenbeck processes and associated self similar processes we consider three types of generalized ornstein uhlenbeck processes the stationary process obtained from the lamperti transformation of fractional brownian motion the process with stretched exponential covariance and the process obtained from the solution of the fractional langevin equation these stationary gaussian processes have many common properties such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency limit in addition the generalized ornstein uhlenbeck processes can be shown to be local stationary representations of fractional brownian motion two new self similar gaussian processes in addition to fractional brownian motion are obtained by applying the inverse lamperti transformation to the generalized ornstein uhlenbeck processes we study some of the properties of these self similar processes such as the long range dependence we give a simulation of their sample paths based on numerical karhunan loeve expansion

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

an explicit schilder type theorem for super brownian motions like ordinary brownian motion super brownian motion a central object in the theory of superprocesses is a universal object arising in a variety of settings schilder type theorems and cramer type theorems are two of the major topics for large deviation theory a schilder type which is also a cramer type sample large deviation for super brownian motions with a good rate function represented by a variation formula was established in 1993 and 1994 since then there have been very valuable contributions for giving an affirmative answer to the question of whether this sample large deviation holds with an explicit good rate function in this paper thanks to previous results on this issue and the brownian snake we establish such a large deviation for nonzero finite initial measures 2010 wiley periodicals inc

characterizations of almost surely continuous p stable random fourier series and strongly stationary processes on etend le probleme de dudley fernique aux processus p stables fortement stationnaires 1 p 2

local nondeterminism and local times for stable processes our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time the approach is through the l p representation for such processes we develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process local nondeterminism is defined for stable processes and shown to be equivalent to locally approximately independent increments sufficient conditions for several classes of stable processes to be local nondeterministic are given these ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times the results extend most gaussian results to their stable analogs

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

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

stationarity conservation laws for fractional differential equations with variable coefficients in this paper we study linear fractional differential equations with variable coefficients it is shown that by assuming some conditions for the coefficients the stationarity conservation laws can be derived the area where these are valid is restricted by the asymptotic properties of solutions of the respective equation applications of the proposed procedure include the fractional fokker planck equation in 1 1 and d 1 dimensional space and the fractional klein kramers equation

fokker planck equation and path integral representation of the fractional ornstein uhlenbeck process with two indices this paper considers the fokker planck equation and path integral formulation of the fractional ornstein uhlenbeck process parametrized by two indices the effective fokker planck equation of this process is derived from the associated fractional langevin equation the path integral representation of the process is constructed and the basic quantities are evaluated

weighted procrustes problems abstract let h be a hilbert space l h the algebra of bounded linear operators on h and w l h a positive operator such that w 1 2 is in the p schatten class for some 1 p given a l h with closed range and b l h we study the following weighted approximation problem analyze the existence of m i n x l h a x b p w where x p w w 1 2 x p in this paper we prove that the existence of this minimum is equivalent to a compatibility condition between r b and r a involving the weight w and we characterize the operators which minimize this problem as w inverses of a in r b

global minimum and orthogonality in c1 classes abstract in this paper we characterize the global minimum of an arbitrary function defined on a banach space in terms of a new concept of derivatives adapted for our case from a recent work due to d j keckic j operator theory submitted for publication using these results we establish several new characterizations of the global minimum of the map f u r defined by f x x 1 where u c 1 is a map defined by x s x and b h b h is a linear map s c1 and u x b h x c 1 further we apply these results to characterize the operators which are orthogonal to the range of elementary operators

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

on the second moment of the number of crossings by a stationary gaussian process cram er and leadbetter introduced in 1967 the sucien t condition z 0 r 00 s r 00 0 s ds 0 to have a nite variance of the number of zeros of a centered stationary gaussian process with twice dieren tiable covariance function r this condition is known as the geman condition since geman proved in 1972 that it was also a necessary condition up to now no such criterion was known for counts of crossings of a level other than the mean this paper shows that the geman condition is still sucien t and necessary to have a nite variance of the number of any xed level crossings for the generalization to the number of a curve crossings a condition on the curve has to be added to the geman condition

large deviations for the three dimensional super brownian motion let t dx denote a three dimensional super brownian motion with deterministic initial state 0 dx dx the lebesgue measure let v r 3 r be holder continuous with compact support not identically zero and such that r 3v x dx 0 we show that log p 0 t r 3v x s dx ds bt 3 4 is of order t 1 2 as t for b 0 this should be compared with the known result for the case r 3v x dx 0 in that case the normalization bt 3 4 b 0 must be replaced by bt b r 3v x dx in order that the same statement hold true while this result only captures the logarithmic order the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems

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

occupation time large deviations for critical branching brownian motion super brownian motion and related processes we derive a large deviation principle for the occupation time functional acting on functions with zero lebesgue integral for both super brownian motion and critical branching brownian motion in three dimensions our technique based on a moment formula of dynkin allows us to compute the exact rate functions which differ for the two processes obtaining the exact rate function for the super brownian motion solves a conjecture of lee and remillard we also show the corresponding clt and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index in r d with d 2 2 d

polylinear additive functionals of superprocesses abstract letxbe a superdiffusion in a domaineof r d a polylinear additive functional ofxcorresponding to a positive borel function is given by the formula a b b dt 1 dt k e k t 1 z 1 t k z k x t 1 dz 1 x t k dz k by a passage to the limit we extend this definition to a certain class of generalized functions more precisely we associate an additive functionala of xt p with every measure subject to the condition dt 1 dz 1 dt k dz k dt 1 dz 1 dt k dz k emsp q t 1 z 1 t k dz k t 1 z 1 t k dz k for a certain kernelq we prove that if t ti s 0 fori 1 kand for alls then measurea has a s the same property this result is applicable in particular to self intersection local times ofx

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

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