a feynman kac ito formula for magnetic schrodinger operators on graphs in this paper we prove a feynman kac ito formula for magnetic schrodinger operators on arbitrary weighted graphs to do so we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation on the operator side we identify a very general class of potentials that allows the definition of magnetic schrodinger operators on the probabilistic side we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials apart from linking the world of discrete magnetic operators with the probabilistic world through the feynman kac ito formula the insights from this paper gained on both sides should be of an independent interest as applications of the feynman kac ito formula we prove a kato inequality a golden thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials

population growth in random media ii wavefront propagation in this second part of a two part presentation we continue with the model introduced in part i in this part the initial configuration has one particle at each site to the left of 0 and no particle elsewhere the expected number of particles observed at a site moving at speed has an exponential growth rate speed growth rate that is computed explicitly the result reveals two characteristic wavefront speeds 1 the speed of the front of zero growth rightmost particle and 2 the speed of the front of maximal growth the latter speed exhibits a phase transition changing from zero to positive as the drift in the migration crosses a threshold the qualitative shape of the growth rate as a function of changes as well in particular below the threshold there appears a linear piece which corresponds to the system exhibiting an intermittency effect

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

random walks among time increasing conductances heat kernel estimates for any graph having a suitable uniform poincare inequality and volume growth regularity we establish two sided gaussian transition density estimates and parabolic harnack inequality for constant speed continuous time random walks evolving via time varying uniformly elliptic conductances provided the vertex conductances i e reversing measures increase in time such transition density upper bounds apply for discrete time uniformly lazy walks with the matching lower bounds holding once the parabolic harnack inequality is proved

sample path large deviations for super brownian motion we derive two large deviation principles of freidlin wentzell type for rescaled super brownian motion for one of the appearing rate functions an integral representation is given and interpreted as kakutani hellinger energy as a tool we develop estimates for the laplace functionals of historical super brownian motion and certain maximal inequalities also it is shown that the holder norm of index 1 2 of the processt f x t possesses some finite exponential moments provided the functionf is smooth

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

simple random walk on long range percolation clusters ii scaling limits supported in part at the technion by a landau fellowship supported in part by an alfred sloan fellowship in mathematics

discrete approximations to reflected brownian motion in this paper we investigate three discrete or semi discrete approximation schemes for reflected brownian motion on bounded euclidean domains for a class of bounded domains d in r n that includes all bounded lipschitz domains and the von koch snowflake domain we show that the laws of both discrete and continuous time simple random walks on d 2 k z n moving at the rate 2 2k with stationary initial distribution converge weakly in the space d 0 1 r n equipped with the skorokhod topology to the law of the stationary reflected brownian motion on d we further show that the following myopic conditioning algorithm generates in the limit a reflected brownian motion on any bounded domain d for every integer k 1 let x k j2 k j 0 1 2 be a discrete time markov chain with one step transition probabilities being the same as those for the brownian motion in d conditioned not to exit d before time 2 k we prove that the laws of x k converge to that of the reflected brownian motion on d these approximation schemes give not only new ways of constructing reflected brownian motion but also implementable algorithms to simulate reflected brownian motion

mosco convergence of nonlocal to local quadratic forms abstract we study sequences of nonlocal quadratic forms and function spaces that are related to markov jump processes in bounded domains with a lipschitz boundary our aim is to show the convergence of these forms to local quadratic forms of gradient type under suitable conditions we establish the convergence in the sense of mosco our framework allows bounded and unbounded nonlocal operators to be studied at the same time moreover we prove that smooth functions with compact support are dense in the nonlocal function spaces under consideration

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

a generalized conservation property for the heat semigroup on weighted manifolds in this text we study a generalized conservation property for the heat semigroup generated by a schrodinger operator with nonnegative potential on a weighted manifold we establish khasminskii s criterion for the generalized conservation property and discuss several applications

discontinuity of the percolation density in one dimensional 1 x y 2 percolation models we consider one dimensional percolation models for which the occupation probability of a bond kx y has a slow power decay as a function of the bond s length for independent models and with suitable reformulations also for more general classes of models it is shown that i no percolation is possible if for short bondskx y p 1 this dichotomy resembles one for the magnetization in 1 x y 2 ising models which was first proposed by thouless and further supported by the renormalization group flow equations of anderson yuval and hamann the proofs of the above percolation phenomena involve rigorous renormalization type arguments of a different sort

on the continuity of the critical value for long range percolation in the exponential case we show that for a long range percolation model with exponentially decaying connections the limit of critical values of any sequence of long range percolation models approaching the original model from below is the critical value for the original long range percolation model as an interesting corollary this implies that if a long range percolation model with exponential connections is supercritical then it still percolates even if all long bonds are removed we also show that the percolation probability is continuous in a certain sense in the supercritical regime for long range percolation models with exponential connections

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

quenched invariance principle for simple random walk on percolation clusters we consider the simple random walk on the unique infinite cluster of super critical bond percolation in d with d 2 we prove that for almost every percolation configuration the path distribution of the walk converges weakly to that of non degenerate isotropic brownian motion our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square integrable martingale the size of the deformation expressed by the so called corrector is estimated by means of ergodicity arguments

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

isoperimetry and heat kernel decay on percolation clusters we prove that the heat kernel on the infinite bernoulli percolation cluster in zd almost surely decays faster than t d 2 we also derive estimates on the mixing time for the random walk confined to a finite box our approach is based on local isoperimetric inequalities some of the results of this paper were previously announced in the note of mathieu and remy c r acad sci paris ser i math 332 2001 927 931

long range percolation in one dimension the problem of long range percolation in one dimension is proposed the authors consider a one dimensional bond percolation system with bonds connecting an infinite number of neighbours where the occupation probability for the nth nearest neighbour bond pn varies as p1 ns using the transfer matrix method they find that when s 2 only the short range percolation exists namely the system percolates only when p1 1 a transition to long range percolation is found at s 2 where the percolation threshold drops suddenly from the short range value p1c 1 to the long range value p1c 0

convergence of symmetric markov chains on mathbb z d for each n let y n t be a continuous time symmetric markov chain with state space n 1 mathbb z d conditions in terms of the conductances are given for the convergence of the y n t to a symmetric markov process yt on mathbb r d we have weak convergence of y n t t leq t 0 for every t0 and every starting point the limit process y has a continuous part and may also have jumps

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

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

persistence of critical multitype particle and measure branching processes we consider a class of systems of particles ofk types inr d undergoing spatial diffusion and critical multitype branching where the diffusions the particle lifetimes and the branching laws depend on the types we prove persistence criteria for such systems and for their corresponding high density limits known as multitype dawson watanabe processes the main tool is a representation of the palm distributions for a general class of inhomogeneous critical branching particle systems constructed by means of a backward tree

markov chain approximations to nonsymmetric diffusions with bounded coefficients we consider a certain class of nonsymmetric markov chains and obtain heat kernel bounds and parabolic harnack inequalities using the heat kernel estimates we establish a sufficient condition for the family of markov chains to converge to nonsymmetric diffusions as an application we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric markov chains this extends the results by stroock and zheng to the nonsymmetric divergence forms 2012 wiley periodicals inc

extending isotopies of planar continua in this paper we solve the following problem in the affirmative let z be a continuum in the plane c and suppose that h z x 0 1 c is an isotopy starting at the identity can h be extended to an isotopy of the plane we will provide a new characterization of an accessible point in a planar continuum z and use it to show that accessibility of a point is preserved during the isotopy we show next that the isotopy can be extended over small hyperbolic crosscuts which are shown to remain small under the isotopy the proof makes use of the notion of a metric external ray which mimics the notion of a conformal external ray but is easier to control during an isotopy it also relies on the existence of a partition of a hyperbolic simply connected domain u in the sphere into hyperbolically convex subsets which have limited distortion under conformal maps to the unit disk

critical branching in a highly fluctuating random medium a model of one dimensional critical branching superprocess is constructed in a random medium fluctuating both in time and space the medium describes a moving system of point catalysts and branching occurs only in the presence of these catalysts although the medium has an infinite overall density the clumping features of the branching model can be exhibited by rescaling time space and mass by an exactly calculated scaling power which is stronger than in the constant medium case the main technique used is the asymptotic analysis of a generalized diffusion reaction equation in the space time random medium which given the medium prescribes the evolution of the laplace transition functional of the markov branching process

stochastic calculus for symmetric markov processes using time reversal we introduce a stochastic integral for zero energy additive functionals of symmetric markov processes extending earlier work of s nakao various properties of such stochastic integrals are discussed and an ito formula for dirichlet processes is obtained

stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium a particle system in euclidean space is considered where the particles are subject to spatial motion according to a symmetric stable law and to a critical branching law in the domain of attraction of a stable law the branching intensity may be position dependent varying medium or be given by a realization of a random field random medium it is shown that under natural assumptions the hydrodynamic limit fluctuations around the macroscopic flow are the same as those given by the averaged medium the limit being a generalized stable ornstein uhlenbeck process the convergence proof is based on an analysis of a nonlinear integral equation with random coefficients

simple random walk on long range percolation clusters i heat kernel bounds in this paper we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process for any d 1 and for any exponent s in d d 2 wedge 2d giving the rate of decay of the percolation process we show that the return probability decays like t d s d up to logarithmic corrections where t denotes the time the walk is run our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box the bounds and accompanying understanding of the geometry of the cluster play a crucial role in the companion paper crawford and sly in simple randomwalk on long range percolation clusters ii scaling limit 2010 where we establish the scaling limit of the random walk to be stable levy motion

symmetric markov chains on ℤ with unbounded range we consider symmetric markov chains on z d where we do not assume that the conductance between two points must be zero if the points are far apart under a uniform second moment condition on the conductances we obtain upper bounds on the transition probabilities estimates for exit time probabilities and certain lower bounds on the transition probabilities we show that a uniform harnack inequality holds if an additional assumption is made but that without this assumption such an inequality need not hold we establish a central limit theorem giving conditions for a sequence of normalized symmetric markov chains to converge to a diffusion on d corresponding to an elliptic operator in divergence form

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models