excursions of dual processes there has been considerable interest in the excursions of a markov process in recent years perhaps the most important general result is that of maisonneuve 13 in which the existence of an exit system is established this general result was applied to the study of specific excursions by getoor in 5 then maisonneuve recast and generalized these results in 14 where he showed how to discuss the excursion straddling a certain class of stopping times t these included the excursion straddling a fixed time and the first excursion whose length exceeds a given value which were discussed in 5 the purpose of this paper is to study excursions for processes which satisfy duality conditions a start in this direction was made in section 10 of 5 under assumptions that were much too strong it turns out that the appropriate duality hypotheses are stronger than those of classical duality as described in i we assume the existence of dual transition densities rather than the weaker assumption of dual potential kernel densities our precise assumptions are described in section 2 and especially in section 3 in particular it is shown in these sections that a systematic use of space time processes obviates the need for any regularity assumptions on the density over and above the chapman kolmogorov equation section 4 summarizes the special properties of additive functionals under our hypotheses that are needed in later sections in theorem 4 5 we show that if m is a closed homogeneous optional set then under duality assumptions there exists a borel set f c e e such that me3 0 and t 0 xt xt f are indistinguishable in section 5 we begin the discussion of excursions from a closed homogeneous optional set m with r i n f t 0 t c m if f e e corresponds to m as above r inf t xt art f almost surely especially important is 5 17 which gives the joint distribution of r x r and x r this is obtained by applying a reversal argument to the excursion

sums and products of two quadratic matrices abstract necessary and sufficient conditions are given for a complex matrix t to be the sum of an idempotent and a square zero matrix these conditions are expressed in terms of 1 the block structure of t under similarity as well as 2 the elementary divisor structure of t this leads to a complete characterization of which complex matrices are a sum of two quadratic matrices also using a similar scheme we characterize when a matrix is the product of an involution and a unipotent matrix of index 2

palm distributions and local approximation of regenerative processes for a regenerative process x with associated local time random measure the distribution relative to some fixed times has a simple description in terms of the excursion kernel r and the palm distributions qt with respect to under suitable regularity conditions on the qt admit versions with strong continuity and regularity properties using the latter versions we may establish some local invariance and mixing properties of x and show that the multivariate palm distributions can be approximated by ordinary conditional distributions

lifschitz tail for alloy type models driven by the fractional laplacian abstract we establish precise asymptotics near zero of the integrated density of states for the random schrodinger operators 2 v in l 2 r d for the full range of 0 2 and a fairly large class of random nonnegative alloy type potentials v the ids exhibits the lifschitz tail singularity we prove the existence of the limit lim 0 d ln n c d d d with c 0 the constant c is finite if and only if the common distribution of the lattice random variables charges 0 in this case the constant c is expressed explicitly in terms of this distribution in the limit formula d denotes the dirichlet ground state eigenvalue of the operator 2 in the unit ball in r d and d is the volume of this ball

factorization of matrices into quadratic ones iii abstract it is shown that every complex n n matrix t is the product of four quadratic matrices moreover if t is invertible then the number of required quadratic matrices can be reduced to three and three is sharp for n 4 complex matrices which are expressible as the product of an idempotent and an invertible quadratic matrix are also characterized

improved bounds for the union of locally fat objects in the plane we show that for any gamma 0 the combinatorial complexity of the union of n locally gamma fat objects of constant complexity in the plane is frac n gamma 4 2 o log n for the special case of gamma fat triangles the bound improves to o n log n frac n gamma log 2 frac 1 gamma

absolute continuity of symmetric markov processes we study girsanov s theorem in the context of symmetric markov processes extending earlier work of fukushima takeda and fitzsimmons on girsanov transformations of gradient type we investigate the most general girsanov transformation leading to another symmetric markov process this investigation requires an extension of the forward backward martingale method of lyons zheng to cover the case of processes with jumps

on harnack inequality for α stable ornstein uhlenbeck processes we consider the stable ornstein uhlenbeck process in mathbb r d with the generator l delta alpha 2 lambda x cdot nabla x we show that if 2 1 or 1 d the harnack inequality holds for 1 d we construct a counterexample that shows that the harnack inequality does not hold

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

the greatest convex minorant of brownian motion meander and bridge this article contains both a point process and a sequential description of the greatest convex minorant of brownian motion on a finite interval we use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the brownian motion meets its minorant the equivalence of these descriptions is non trivial which leads to many interesting identities between quantities derived from our analysis the sequential description can be viewed as a markov chain for which we derive some fundamental properties

size biased sampling of poisson point processes and excursions some general formulae are obtained for size biased sampling from a poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function these formulae explain why in certain cases gamma and stable the size biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking residual allocation scheme defined by independent beta random variables an application is made to length biased sampling of excursions of a markov process away from a recurrent point of its statespace with emphasis on the brownian and bessel cases when the associated inverse local time is a stable subordinator results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by brownian motion

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

estimates of gradient perturbation series abstract we give upper and lower bounds of perturbation series for transition densities corresponding to additive gradient perturbations satisfying certain space time integrability conditions

product of invertible operators of quadratic type abstract suppose all invertible quadratic operators t are assumed to satisfy t 2 bt i 0 we show that a complex matrix t is the product of two invertible quadratic matrices if and only if t is similar to a matrix of the form d d 1 i n i m i 1 j i 2 i 2 i 2 i xy 2 i yx 1 where b b 2 4 2 0 1 and gb 2 are not eigenvalues of d n xy and yx are nilpotent and each j i is a nilpotent jordan block of even size also we show that an n n matrix t is the product of finitely many invertible quadratic matrices if and only if det t m where m is an integer for n odd and is an even integer for n even on the other hand for operators on an infinite dimensional hilbert space we characterize the products of two and four invertible quadratic operators among normal operators and show that every invertible operator is the product of six invertible quadratic operators

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

markovian bridges weak continuity and pathwise constructions a markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a markov process given its terminal value as such markovian bridges admit a natural parameterization in terms of the state space of the process in the context of feller processes with continuous transition densities we construct by weak convergence considerations the only versions of markovian bridges which are weakly continuous with respect to their parameter we use this weakly continuous construction to provide an extension of the strong markov property in which the flow of time is reversed in the context of self similar feller process the last result is shown to be useful in the construction of markovian bridges out of the trajectories of the original process

small size eps nets for axis parallel rectangles and boxes we show the existence of varepsilon nets of size o left frac 1 varepsilon log log frac 1 varepsilon right for planar point sets and axis parallel rectangular ranges the same bound holds for points in the plane and fat triangular ranges and for point sets in boldsymbol r 3 and axis parallel boxes these are the first known nontrivial bounds for these range spaces our technique also yields improved bounds on the size of varepsilon nets in the more general context considered by clarkson and varadarajan for example we show the existence of varepsilon nets of size o left frac 1 varepsilon log log log frac 1 varepsilon right for the dual range space of fat regions and planar point sets where the regions are the ground objects and the ranges are subsets stabbed by points plugging our bounds into the technique of bronnimann and goodrich or of even rawitz and shahar we obtain improved approximation factors computable in expected polynomial time by a randomized algorithm for the hitting set or the set cover problems associated with the corresponding range spaces

geometric packing under nonuniform constraints we study the problem of discrete geometric packing here given weighted regions say in the plane and points with capacities one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity we provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings using this framework we get a flotilla of results on this problem and also on its dual where one wants to pick a maximum weight subset of the points when the regions have capacities for example for the case of fat triangles of similar size we show an o 1 approximation and prove that no ptas is possible

birth times death times and time substitutions in markov chains given a markov chain xn n 0 random times are studied which are birth times or death times in the sense that the post and pre processes are independent given the present x 1 x at time and the conditional post process birth times or the conditional pre process death times is again markovian the main result for birth times characterizes all time substitutions through homogeneous random sets with the property that all points in the set are birth times the main result for death times is the dual of this and appears as the birth time theorem with the direction of time reversed

splitting at backward times in regenerative sets by a backward time is meant a random time which only depends on the future in the same sense as a stopping time only depends on the past we show that backward times taking values in a regenerative set m split m into conditionally independent subsets the conditional distributions of the past may further be identified with the palm distributions p t with respect to the local time random measure xi of m both a e e xi and wherever p t has a continuous version continuity of p t occurs essentially where e xi has a continuous density and the latter continuity set may be described rather precisely in terms of the growth rate and regularity properties of the levy measure of m

characterizations and embedding properties in exchangeability our key result is the characterization of exchangeable sequences as being strongly stationary i e invariant in distribution under stopping time shifts from this we prove homogeneity characterizations of pure and mixed markov chains these results carry over to continuous time processes and random sets and on a whole our theory provides a unified approach to exchangeability the key result above is closely related to dacunha castelle s embedding characterization of exchangeability which is partially extended here to processes on 0 1 in the other direction we prove that a previsible sample from a finite or infinite exchangeable sequence x may be embedded into a copy of x we finally establish some uniqueness results for exponentially and uniformly killed exchangeable random sets

arcsine laws and interval partitions derived from a stable subordinator levy discovered that the fraction of time a standard one dimensional brownian motion b spends positive before time t has arcsine distribution both for a fixed time when b t 0 almost surely and for t an inverse local time when b t 0 almost surely this identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of b away from 0 similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator for instance a bessel process of dimension d for 0 d 2

sums of idempotent matrices abstract we show that any complex square matrix t is a sum of finitely many idempotent matrices if and only if trt is an integer and trt rank t moreover in this case the idempotents may be chosen such that each has rank one and has range contained in that of t we also consider the problem of the minimum number of idempotents needed to sum to t and obtain some partial results

tight lower bounds for the size of epsilon nets reference epfl article 190901view record in web of science record created on 2013 12 09 modified on 2017 05 12

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

products of unipotent matrices of index 2 abstract we show that an n n complex matrix t is the product of two unipotent matrices of index 2 if and only if t is similar to a matrix of the form d d 1 i n i mi 1 ji where 0 and 1 are not eigenvalues of d n is nilpotent and each ji is a nilpotent jordan block of even size on the other hand t is the product of finitely many unipotent matrices of index 2 if and only if det t 1 in this case the minimal number of required unipotents is 1 if n 1 3 if n 2 and 4 if n 3

the product of two quadratic matrices abstract let p x x 1 k x where 2 2 and let v be a finite dimensional vector space over the field k a linear mapping m v v is called quadratic if p m 0 we characterize products of two quadratic linear mappings

computing coverage kernels under restricted settings abstract given a set b of d dimensional boxes i e axis aligned hyperrectangles a minimum coverage kernel is a subset of b of minimum size covering the same region as b computing it is np hard but as for many similar np hard problems e g box cover and orthogonal polygon covering the problem becomes solvable in polynomial time under restrictions on b we show that computing minimum coverage kernels remains np hard even when restricting the graph induced by the input to a highly constrained class of graphs alternatively we present two polynomial time approximation algorithms for this problem one deterministic with an approximation ratio within o log n and one randomized with an improved approximation ratio within o lg opt with high probability

on the spectra of some combinations of two generalized quadratic matrices let a and b be two generalized quadratic matrices with respect to idempotent matrices p and q respectively such that a p a p 0 a p p a a b q b q 0 b q q b b p q q p ab ba and a b p q p q a b with c let a b be diagonalizable the relations between the spectrum of the matrix a b and the spectra of some matrices produced from a and b are considered moreover some results on the spectrum of the matrix a b are obtained when a b is not diagonalizable finally some results and examples illustrating the applications of the results in the work are given