the universal relation between scaling exponents in first passage percolation it has been conjectured in numerous physics papers that in ordinary rst passage percolation on integer lattices the uctuation exponent and the wandering exponent are related through the universal relation 2 1 irrespective of the dimension this is sometimes called the kpz relation between the two exponents this article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense

fast wave computation via fourier integral operators this paper presents a numerical method for time upscaling wave equations i e performing time steps not limited by the courant friedrichs lewy cfl condition the proposed method leverages recent work on fast algorithms for pseudodierential and fourier integral operators fio this algorithmic approach is not asymptotic it is shown how to construct an exact fio propagator by 1 solving hamilton jacobi equations for the phases and 2 sampling rows and columns of low rank matrices at random for the amplitudes the setting of interest is that of scalar waves in two dimensional smooth periodic media of class c 1 over the torus where the bandlimit n of the waves goes to innity in this setting it is demonstrated that the algorithmic complexity for solving the wave equation to xed time t 1 can be as low as o n 2 logn with controlled accuracy numerical experiments show that the time complexity can be lower than that of a spectral method in certain situations of physical interest

random walks on graphs ideas techniques and results random walks on graphs are widely used in all sciences to describe a great variety of phenomena where dynamical random processes are affected by topology in recent years relevant mathematical results have been obtained in this field and new ideas have been introduced which can be fruitfully extended to different areas and disciplines here we aim at giving a brief but comprehensive perspective of these progresses with a particular emphasis on physical aspects

hardy space estimates for the wave equation on compact lie groups abstract let l be the laplacian in this paper we prove that on a compact lie group g of dimension n the multiplier operator e i s l 1 l 2 s 0 1 extends to a bounded operator on the hardy space h p g 0 p if and only if 1 p 1 2 n 1 the result is an analogue of a well known theorem in euclidean space

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

a simplified proof of the relation between scaling exponents in first passage percolation in a recent breakthrough work chatterjee 5 proved a long standing conjecture that relates the transversal exponentand the fluctuation exponentin first passage percolation on z d the purpose of this paper is to introduce a new and intuitive idea that replaces the main argument of 5 and gives an alternative proof of this relation specifically we show that under the assumption that exponents defined in 5 exist one has the relation 2 1 one advantage of our argument is that it does not require the nearly gamma assumption of 5

wave atoms and time upscaling of wave equations we present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying two dimensional time independent periodic media the method consists in representing the time dependent green s function in wave atoms a tight frame of multiscale directional wave packets obeying a precise parabolic balance between oscillations and support size namely wavelength diameter 2 wave atoms offer a uniquely structured representation of the green s function in the sense that the resulting matrix is universally sparse over the class of c coefficients even for large times the matrix has a natural low rank block structure after separation of the spatial indices the parabolic scaling is essential for these properties to hold as a result it becomes realistic to accurately build the full matrix exponential in the wave atom frame using repeated squaring up to some time typically of the form delta t sim sqrt delta x which is bigger than the standard cfl timestep once the expensive precomputation of the green s function has been carried out it can be used to perform unusually large upscaled cheap time steps the algorithm is relatively simple in that it does not require an underlying geometric optics solver we prove accuracy and complexity results based on a priori estimates of sparsity and separation ranks on a n by n grid the expensive precomputation takes somewhere between o n 3log n and o n 4log n steps depending on the separability of the acoustic medium the complexity of upscaled timestepping however beats the o n 3log n bottleneck of pseudospectral methods on an n by n grid for a wide range of physically relevant situations in particular we show that a naive version of the wave atom algorithm provably runs in o n 2 operations for arbitrarily small but for the final algorithm we had to slightly increase the exponent in order to reduce the large constant as a result we get estimates between o n 2 5 log n and o n 3 log n for upscaled timestepping we also show several numerical examples in practice the current wave atom solver becomes competitive over a pseudospectral method in regimes where the wave equation should be solved hundreds of times with different initial conditions as in reflection seismology in academic examples of accurate propagation of bandlimited wavefronts if the precomputation step is factored out then the wave atom solver is indeed faster than a pseudospectral method by a factor of about 3 5 at n 512 and a factor 10 20 at n 1024 for the same accuracy very similar gains are obtained in comparison versus a finite difference method

first encounters on combs we consider two random walkers embedded in a finite two dimension comb and we study the mean first encounter time mfet evidencing mainly numerically different scalings with the linear size of the underlying network according to the initial position of the walkers if one of the two players is not allowed to move then the first encounter problem can be recast into a first passage problem mfpt for which we also obtain exact results for different initial configurations by comparing mfet and mfpt we are able to figure out possible search strategies and in particular we show that letting one player be fixed can be convenient to speed up the search as long as we can finely control the initial setting while for a random setting on average letting one player rest would slow down the search

a local to global boundedness argument and fourier integral operators abstract we give a criterion for the global boundedness of integral operators which are known to be locally bounded as an application we discuss the global l p boundedness for a class of fourier integral operators while the local l p boundedness of fourier integral operators is known from the work of seeger sogge and stein 26 not so many results are available for the global boundedness on l p r n we give several natural sufficient conditions for them

mesoscopic description of random walks on combs combs are a simple caricature of various types of natural branched structures which belong to the category of loopless graphs and consist of a backbone and branches we study continuous time random walks on combs and present a generic method to obtain their transport properties the random walk along the branches may be biased and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone we analyze the overall diffusion properties along the backbone and find normal diffusion anomalous diffusion and stochastic localization diffusion failure respectively depending on the characteristics of the continuous time random walk along the branches and compare our analytical results with stochastic simulations

exact calculations of first passage quantities on recursive networks we present general methods to exactly calculate mean first passage quantities on self similar networks defined recursively in particular we calculate the mean first passage time and the splitting probabilities associated to a source and one or several targets averaged quantities over a given set of sources e g same connectivity nodes are also derived the exact estimate of such quantities highlights the dependency of first passage processes with respect to the source target distance which has recently revealed to be a key parameter in characterizing transport in complex media we explicitly perform calculations for different classes of recursive networks finitely ramified fractals scale free trans fractals nonfractals mixtures between fractals and nonfractals nondecimable hierarchical graphs of arbitrary size our approach unifies and significantly extends the available results in the field

estimates for rough fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators we study the boundedness of rough fourier integral and pseudodifferential operators defined by general rough hormander class amplitudes on banach and quasi banach lplp spaces thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for hormander class bilinear pseudodifferential operators and certain classes of bilinear as well as multilinear fourier integral operators for these classes of amplitudes the boundedness of the aforementioned fourier integral operators turn out to be sharp furthermore we also obtain results for rough multilinear operators

wave front sets local smoothing and bourgain s circular maximal theorem the purpose of this paper is to improve certain known regularity results for the wave equation and to give a simple proof of bourgain s circular maximal theorem 1 we use easy wave front analysis along with techniques previously used in proofs of the carleson sj6lin theorem see 3 5 7 and in the proof of sharp regularity properties of fourier integral operators 13 the circular means operators are defined by

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

superdiffusive behavior of two dimensional brownian motion in a poissonian potential we consider d dimensional brownian motion in a truncated poissonian potential conditioned to reach a remote location if brownian motion starts at the origin and ends in an hyperplane at distance l from the origin the transverse fluctuation of the path is expected to be of order l we are interested in a lower bound for we first show that 1 2 in dimensions d 2 and then we prove superdiffusive behavior for d 2 resulting in 3 5

information dimension of stochastic processes on networks relating entropy production to spectral properties we consider discrete stochastic processes modeled by classical master equations on networks the temporal growth of the lack of information about the system is captured by its non equilibrium entropy defined via the transition probabilities between different nodes of the network we derive a relation between the entropy and the spectrum of the master equation s transfer matrix our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling in analogy to chaos theory the proportionality factor is called stochastic information dimension and gives a global characterization of the dynamics on the network these general results are corroborated by examples of regular and of fractal networks

a seeger sogge stein theorem for bilinear fourier integral operators abstract we establish the regularity of bilinear fourier integral operators with bilinear amplitudes in s 1 0 m n 2 and non degenerate phase functions from l p l q l r under the assumptions that m n 1 1 p 1 2 1 q 1 2 and 1 p 1 q 1 r this is a bilinear version of the classical theorem of seeger sogge stein concerning the l p boundedness of linear fourier integral operators moreover our result goes beyond the aforementioned theorem in that it also includes the case of quasi banach target spaces

regularity of fourier integral operators with caustics the caustics of fourier integral operators are defined as caustics of the corresponding schwartz kernels lagrangian distributions on x y the caustic set c of the canonical relation is characterized as the set of points where the rank of the projection c x x y is smaller than its maximal value dim x x y 1 we derive the l p y l q x estimates on fourier integral operators with caustics of corank 1 such as caustics of type a m 1 m n for the values of p and q outside of a certain neighborhood of the line of duality q p the l p l q estimates are proved to be caustics insensitive we apply our results to the analysis of the blow up of the estimates on the half wave operator just before the geodesic flow forms caustics

shape theorem lyapounov exponents and large deviations for brownian motion in a poissonian potential we derive a large deviation principle governing the position of a d dimensional brownian motion moving in a poissonian potential the derivation of this large deviation principle and the form of the rate function rely on a result similar to the shape theorem of first passage percolation this result produces certain constants which play in this multidimensional situation a similar role as the lyapounov exponents in the one dimensional case the large deviation principle enables us to investigate the transition of regime which occurs between the small h and the large h case for brownian motion with a constant drift h moving in the same potential 1994 john wiley sons inc

slow encounters of particle pairs in branched structures dipartimento di fisica universita di parma dated may 24 2014 on in nite homogeneous structures two random walkers meet with certainty if and only if thestructure is recurrent i e a single random walker returns to its starting point with probability 1 however on general inhomogeneous structures this property does not hold and although a singlerandom walker will certainly return to its starting point two moving particles may never meet thisstriking property has been shown to hold for instance on in nite combs due to the huge varietyof natural phenomena which can be modeled in terms of encounters between two or more particlesdi using in comb like structures it is fundamental to investigate if and if so to what extent similare ects may take place in nite structures by means of numerical simulations we evidence that indeed even on nite structures the topological inhomogeneity can qualitatively a ect the two particle problem in particular the mean encounter time can be polynomially larger than the timeexpected from the related one particle problem

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

scaling properties of diffusion on comb like structures the authors study the scaling properties of the probability density of random walks on comb like structures they find that in contrast to the first passage time in which several exponents are needed to characterise their distribution the probability density can be described by a single exponent

fourier integral operators on noncompact symmetric spaces of real rank one let x g k be a noncompact symmetric space of real rank one the purpose of this paper is to investigate lp boundedness properties of a certain class of radial fourier integral operators on the space x we will prove that if u is the solution at some fixed time of the natural wave equation on x with initial data f and g and 1 p then u lp x cp f lpbp x 1 g lpbp 1 x we will obtain both the precise behavior of the norm cp and the sharp regularity assumptions on the functions f and g i e the exponent bp that make this inequality possible in the second part of the paper we deal with the analog of e m stein s maximal spherical averages and prove exponential decay estimates of a highly non euclidean nature on the lp norm of supt t 1 f d z where d is a normalized spherical measure

random walks on bundled structures bundled structures bs are discrete structures obtained joining to each point of a base graph a copy of a fiber graph in condensed matter physics bs are used as realistic models for the geometry and dynamics of nontranslationally invariant systems polymers inhomogeneous systems etc we present an analytical solution for the random walk problem on these structures which is possible when we know the solution for base and fiber separately we obtain an expression for the spectral dimension of the bs as a function of the spectral dimensions of its components moreover we discuss some applications of these results concerning anomalous diffusion laws proving the existence of nondisordered structures with logarithmic and sublogarithmic diffusion laws due only to geometric features

distance fluctuations and lyapounov exponents we associate certain translation invariant random metrics on r d to brownian motion evolving in a truncated poissonian potential these metrics behave over large distances in an appropriate sense like certain deterministic norms the so called lyapounov exponents we prove here upper bounds on the size of fluctuations of the metrics around their mean under an additional assumption of rotational invariance we also derive upper bounds on the difference between the mean of the metrics and the lyapounov norms

influence of weight heterogeneity on random walks in scale free networks many systems are best described by weighted networks in which the weights of the edges are heterogeneous in this paper we focus on random walks in weighted network investigating the impacts of weight heterogeneity on the behavior of random walks we study random walks in a family of weighted scale free tree like networks with power law weight distribution we concentrate on three cases of random walk problems with a trap located at a hub node a leaf adjacent to a hub node and a farthest leaf node from a hub for all these cases we calculate analytically the global mean first passage time gmfpt measuring the efficiency of random walk as well as the leading scaling of gmfpt we find a significant decrease in the dominating scaling of gmfpt compared with the corresponding binary networks in all three random walk problems which implies that weight heterogeneity has a significant influence on random walks in scale free networks

maximal averages over surfaces letmf x supt 0 f t d x denote the maximal operator associated with surface measured on a smooth surfaces we prove that ifsis convex and has finite order contact with its tangent lines then m is bounded onlp rn p 2 if and only ifd x h 1 l1 loc s for all tangent planeshnot passing through the origin letm f x supt 0 f t d x be the maximal operator associated with a nonisotropic dilation tof surface measured we prove that m often behaves far better than m due to a rotational curvature in the time parametert

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

brownian directed polymers in random environment we study the thermodynamics of a continuous model of directed polymers in random environment the environment is given by a space time poisson point process whereas the polymer is defined in terms of the brownian motion we mainly discuss i the normalized partition function its positivity in the limit which characterizes the phase diagram of the model ii the existence of quenched lyapunov exponent its positivity and its agreement with the annealed lyapunov exponent iii the longitudinal fluctuation of the free energy some of its relations with the overlap between replicas and with the transversal fluctuation of the path the model considered here enables us to use stochastic calculus with respect to both brownian motion and poisson process leading to handy formulas for fluctuations analysis and qualitative properties of the phase diagram we also relate our model to some formulation of the kardar parisi zhang equation more precisely the stochastic heat equation our fluctuation results are interpreted as bounds on various exponents and provide a circumstantial evidence of super diffusivity in dimension one we also obtain an almost sure large deviation principle for the polymer measure

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

convergence in lp for feynman path integrals abstract we consider a class of schrodinger equations with time dependent smooth magnetic and electric potentials having a growth at infinity at most linear and quadratic respectively we study the convergence in l p with loss of derivatives 1 p of the time slicing approximations of the corresponding feynman path integral the results are completely sharp and hold for long time where no smoothing effect is available the techniques are based on the decomposition and reconstruction of functions and operators with respect to certain wave packets in phase space

scaling identity for crossing brownian motion in a poissonian potential we consider d dimensional brownian motion in a truncated poissonian potential d 2 if brownian motion starts at the origin and ends in the closed ball with center y and radius 1 then the transverse fluctuation of the path is expected to be of order y whereas the distance fluctuation is of order y physics literature tells us that and should satisfy a scaling identity 2 1 we give here rigorous results for this conjecture

self similar solutions of semilinear wave equation with variable speed of propagation abstract we investigate the issue of existence of the self similar solutions of the generalized tricomi equation in the half space where the equation is hyperbolic we look for the self similar solutions via the cauchy problem an integral transformation suggested in k yagdjian a note on the fundamental solution for the tricomi type equation in the hyperbolic domain j differential equations 206 2004 227 252 is used to represent solutions of the cauchy problem for the linear tricomi type equation in terms of fundamental solutions of the classical wave equation this representation allows us to prove decay estimates for the linear tricomi type equation with a source term obtained in k yagdjian the self similar solutions of the tricomi type equations z angew math phys in press doi 10 1007 s00033 006 5099 2 estimates for the self similar solutions of the linear tricomi type equation are the key tools to prove existence of the self similar solutions

transversal fluctuations for increasing subsequences on the plane consider a realization of a poisson process in 2 with intensity 1 and take a maximal up right path from the origin to n n consisting of line segments between the points where maximal means that it contains as many points as possible the number of points in such a path has fluctuations of order n where 1 3 bdj here we show that typical deviations of a maximal path from the diagonal x y is of order n with 2 3 this is consistent with the scaling identity 2 1 which is believed to hold in many random growth models