the true self repelling motion we construct and study a continuous real valued random process which is of a new type it is self interacting self repelling but only in a local sense it only feels the self repellance due to its occu pation time measure density in the immediate neighbourhood of the point it is just visiting we focus on the most natural process with these properties that we call true self repelling motion this is the continuous counterpart to the integer valued true self avoiding walk which had been studied among others by the first author one of the striking properties of true self repelling motion is that although the coupleoxt occupation time measure of x at time tu is a continuous markov process x is not driven by a stochastic diaerential equation and is not a semi martingale it turns out for instance that it has a finite variation of order 3 2 which contrasts with the finite quadratic variation of semi martingales one of the key tools in the construction of x is a continuous system of coalescing brownian motions similar to those that have been constructed by arratia a1 a2 we derive various properties of x existence and properties of the occupation time densities ltoxu local variation etc and an identity that shows that the dynamics of x can be very loosely speaking described as follows dxt is equal to the gradient in space of ltoxu in a gener alized sense even though x7 ltoxu is not diaerentiable

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

random walk with memory a reinforced random walk on the d dimensional lattice is considered it is shown that this walk is equivalent to an iterated function system ifs criteria for the existence of limit cycles are given numerical results and conjectures about the quantitative behavior of the walk are stated

critical random walk in random environment on trees we study the behavior of random walk in random environment rwre on trees in the critical case left open in previous work representing the random walk by an electrical network we assume that the ratios of resistances of neighboring edges of a tree are i i d random variables whose logarithms have mean zero and finite variance then the resulting rwre is transient if simple random walk on is transient but not vice versa we obtain general transience criteria for such walks which are sharp for symmetric trees of polynomial growth in order to prove these criteria we establish results on boundary crossing by tree indexed random walks these results rely on comparison inequalities for percolation processes on trees and on some new estimates of boundary crossing probabilities for ordinary mean zero finite variance random walks in one dimension which are of independent interest

attracting edge property for a class of reinforced random walks using martingale techniques and comparison with the generalized urn scheme it is shown that the edge reinforced random walk on a graph of bounded degree with the weight function w k k 1 traverses crosses a random attracting edge at all large times if the graph is a triangle the above result is in agreement with a conjecture of sellke

singularity formation in chemotaxis a conjecture of nagai consider the initial boundary value problem for the system s ut uxx uvx x vt u av on an interval 0 1 for t 0 where a 0 with ux 0 t ux 1 t 0 suppose mu v0 are positive constants the corresponding spatially homogeneous global solution u t mu v t mu a v0 mu a exp at is stable in the sense that if mu v0 are positive constants the corresponding spatially homogeneous solution will be uniformly close to u cdot v cdot we consider in sequence space an approximate system s which is related to s in the following sense the chemotactic term uvx x is replaced by the inverse fourier transform of the finite part of the convolution integral for the fourier transform of uvx x here the finite part of the convolution on the line at a point x of two functions f g is defined as int 0 x f y g y x dy we prove the following if mu a then in every neighborhood of mu v0 there are spatially nonconstant initial data for which the solution of proble

vertex reinforced random walk this paper considers a class of non markovian discrete time random processes on a finite state space 1 d the transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix r which is real symmetric and nonnegative lets i n keep track of the number of visits to statei up to timen and form the fractional occupation vector v n where v i n s i n mathord left vphantom s i n left sum limits j 1 d s j n right right kern nulldelimiterspace left sum limits j 1 d s j n right it is shown thatv n converges to to a set of critical points for the quadratic formh with matrixr and that under nondegeneracy conditions onr there is a finite set of points such that with probability one v n p for somep in the set there may be more than onep in this set for whichp v n p 0 on the other handp v n p 0 wheneverp fails in a strong enough sense to be maximum forh

vertex reinforced random walk on z has finite range a stochastic process called vertex reinforced random walk vrrw is defined in pemantle ann probab 16 1229 1241 we consider this process in the case where the underlying graph is an infinite chain i e process in the case where the underlying graph is an infinite chain i e the one dimensional integer lattice we show that the range is almost surely finite that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points there are always points visited infinitely often but at a set of times of zero density and we show that the number of visits to such a point to time n may be asymptotically n for a dense set of values 0 1 the power law analysis relies on analysis of a related urn model

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

mathematical models of tumour angiogenesis we first study a parabolic ode system modelling tumour growth proposed by othmer and stevens aggregation blowup and collapse the abc s of taxis in reinforced random walks siam j appl math 57 4 1997 1044 1081 according to levine and sleeman a system of reaction and diffusion equations arising in the theory of reinforced random walks siam j appl math 57 3 1997 683 730 we reduced it to a hyperbolic equation and showed the existence of collapse in a kubo t suzuki asymptotic behavior of the solution to a parabolic ode system modeling tumour growth differential integral equations 17 2004 721 736 we also deal with the system in case the reduced equation is elliptic and show the existence of collapse analogously next we apply the above result to another model proposed by anderson and chaplain arising from tumour angiogenesis and show the existence of collapse further we investigate a contact point between these two models and a common property to them

limit distributions of directionally reinforced random walks abstract as a mathematical model for time and space correlations in ocean surface wave fields mauldin monticino and von weizsacker 1996 adv in math 117 239 252 introduced directionally reinforced random walks and analyzed their recurrence properties this paper develops limiting properties for these random walks inrd in particular it is shown that the limit distribution is a brownian motion if the time between changes of directions has a finite second moment and is a stable process if the time between changes of directions is in a domain of attraction of a stable law the latter gives an affirmative answer to an open question posed by mauldinet al the strong law of large numbers and strong approximations are also discussed

edge reinforced random walk on a ladder we prove that the edge reinforced random walk on the ladder z x 1 2 with initial weights a 3 4 is recurrent the proof uses a known representation of the edge reinforced random walk on a finite piece of the ladder as a random walk in a random environment this environment is given by a marginal of a multicomponent gibbsian process a transfer operator technique and entropy estimates from statistical mechanics are used to analyze this gibbsian process furthermore we prove spatially exponentially fast decreasing bounds for normalized local times of the edge reinforced random walk on a finite piece of the ladder uniformly in the size of the finite piece

limit theorems for reinforced random walks on certain trees consider a linearly edge reinforced random walk defined on the b ary tree b 70 we prove the strong law of large numbers for the distance of this process from the root we give a sufficient condition for this strong law to hold for general edge reinforced random walks and random walks in a random environment we also provide a central limit theorem

vertex reinforced jump processes on trees and finite graphs we study the continuous time process on the vertices of the b ary tree which jumps to each nearest neighbor vertex at the rate of the time already spent at that vertex times plus 1 where is a positive constant we show that for fixed b 1 if is large enough the process is transient and if is close enough to zero it is recurrent related results for some other graphs and trees are also proved

konig chain for compact matrix sets abstract a topological proof via the generalized gelfand spectral radius formula is given of the fact that every compact set of complex n n matrices admits a konig chain

excited brownian motions as limits of excited random walks we obtain the convergence in law of a sequence of excited also called cookies random walks toward an excited brownian motion this last process is a continuous semi martingale whose drift is a function say of its local time it was introduced by norris rogers and williams as a simplified version of brownian polymers and then recently further studied by the authors to get our results we need to renormalize together the sequence of cookies the time and the space in a convenient way the proof follows a general approach already taken by toth and his coauthors in multiple occasions which goes through ray knight type results namely we first prove when is bounded and lipschitz that the convergence holds at the level of the local time processes this is done via a careful study of the transition kernel of an auxiliary markov chain which describes the local time at a given level then we prove a tightness result and deduce the convergence at the level of the full processes

existence and non existence of global solutions to initial boundary value problems for nonlinear evolution equations with strong dissipation abstract the main purpose in this paper is to investigate existence and non existence of global solutions of the initial dirichlet boundary value problem for evolution equations with strong dissipation many authors studied classes consisting of such types of equations for which initial boundary value problems possess global solutions for this purpose we consider a related problem and seek global solutions and blow up solutions of it depending on whether it belongs to such classes or not

localisation in a growth model with interaction this paper concerns the long term behaviour of a growth model describing a random sequential allocation of particles on a finite cycle graph the model can be regarded as a reinforced urn model with graph based interaction it is motivated by cooperative sequential adsorption where adsorption rates at a site depend on the configuration of existing particles in the neighbourhood of that site our main result is that with probability one the growth process will eventually localise either at a single site or at a pair of neighbouring sites

attracting edge and strongly edge reinforced walks the goal is to show that an edge reinforced random walk on a graph of bounded degree with reinforcement weight functionw taken from a general class of reciprocally summable reinforcement weight functions traverses a random attracting edge at all large times the statement of the main theorem is very close to settling a conjecture of sellke technical report 94 26 1994 purdue univ an important corollary of this main result says that if w is reciprocally summable and nondecreasing the attracting edge exists on any graph of bounded degree with probability 1 another corollary is the main theorem of limic ann probab 31 2003 1615 1654 where the class of weights was restricted to reciprocally summable powers the proof uses martingale and other techniques developed by the authors in separate studies of edge and vertex reinforced walks ann probab 31 2003 1615 1654 ann probab 32 2004 2650 2701 and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics c r acad sci ser i math 330 2000 125 130

the true self avoiding walk with bond repulsion on z limit theorems the true self avoiding walk with bond repulsion is a nearest neighbor random walk on z for which the probability of jumping along a bond of the lattice is proportional to exp g number of previous jumps along that bond first we prove a limit theorem for the distribution of the local time process of this walk using this result later we prove a local limit theorem as a for the distribution of a 2 3 x s a where s a is a random time distributed geometrically with mean e s a 1 e s a 1 a s 0 1 as a by product we also obtain an apparently new identity related to brownian excursions and bessel bridges

continuous time vertex reinforced jump processes we study the continuous time integer valued process which jumps to each of its two nearest neighbors at the rate of one plus the total time the process has previously spent at that neighbor we show that the proportion of the time before t which this process spends at integers j converges to positive random variables v j which sum to one and whose joint distribution is explicitly described we also show

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

self interacting diffusions this paper is concerned with a general class of self interacting diffusions x t t 0 living on a compact riemannian manifold m these are solutions to stochastic differential equations of the form dx t brownian increments drift term depending on x t and t the normalized occupation measure of the process it is proved that the asymptotic behavior of t can be precisely related to the asymptotic behavior of a deterministic dynamical semi flow t t 0 defined on the space of the borel probability measures on m in particular the limit sets of t are proved to be almost surely attractor free sets for these results are applied to several examples of self attracting repelling diffusions on the n sphere for instance in the case of self attracting diffusions our results apply to prove that t can either converge toward the normalized riemannian measure or to a gaussian measure depending on the value of a parameter measuring the strength of the attraction

zeno s walk a random walk with refinements a self modifying random walk on bbb q is derived from an ordinary random walk on the integers by interpolating a new vertex into each edge as it is crossed this process converges almost surely to a random variable which is totally singular with respect to lebesgue measure and which is supported on a subset of bbb r having hausdorff dimension less than 1 which we calculate by a theorem of billingsley by generating function techniques we then calculate the exponential rate of convergence of the process to its limit point which may be taken as a bound for the convergence of the measure in the wasserstein metric we describe how the process may viewed as a random walk on the space of monotone piecewise linear functions where moves are taken by successive compositions with a randomly chosen such function

stability of solutions of chemotaxis equations in reinforced random walks in this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation the system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher or lower concentrations of the chemical substance or by aggregating or dispersing we prove that stationary solutions of the system are asymptotically stable

subdiffusive continuous time random walks with stochastic resetting we analyze two models of subdiffusion with stochastic resetting each of them consists of two parts subdiffusion based on the continuous time random walk scheme and independent resetting events generated uniformly in time according to the poisson point process in the first model the whole process is reset to the initial state whereas in the second model only the position is subject to resets the distinction between these two models arises from the non markovian character of the subdiffusive process we derive exact expressions for the two lowest moments of the full propagator stationary distributions and first hitting time statistics we also show with an example of a constant drift how these models can be generalized to include external forces possible applications to data analysis and modeling of biological systems are also discussed

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

phase transition for the once reinforced random walk on mathbb z d like trees in this short paper we consider the once reinforced random walk with reinforcement parameter a on trees with bounded degree which are transient for the simple random walk on each of these trees we prove that there exists an explicit critical parameter a 0 such that the once reinforced random walk is almost surely recurrent if a a 0 and almost surely transient if a a 0 this provides the first examples of phase transition for the once reinforced random walk

on self attracting d dimensional random walks let x t t geq 0 be a symmetric nearest neighbor random walk on mathbb z d with exponential holding times of expectation 1 d starting at the origin for a potential v mathbb z d to 0 infty with finite and nonempty support define transformed path measures by d hat mathbb p t equiv exp t 1 int 0 t int 0 t v x s x t ds dt d mathbb p z t for t 0 where z t is the normalizing constant if d 1 or if the self attraction is sufficiently strong then x t infty has an exponential moment under mathbb p t which is uniformly bounded for t 0 and t in 0 t we also prove that x t t geq 0 under suitable subsequences of hat mathbb p t t 0 behaves for large t asymptotically like a mixture of space inhomogeneous ergodic random walks for special cases like a sufficiently strong dirac type interaction we even prove convergence of the transformed path measures and the law of x t as well as of the law of the empirical measure l t under hat mathbb p t t 0

large deviations in the reinforced random walk model on trees in this paper we consider the linearly reinforced and the once reinforced random walk models in the transient phase on trees we show the large deviations for the upper tails for both models we also show the exponential decay for the lower tail in the once reinforced random walk model however the lower tail is in polynomial decay for the linearly reinforced random walk model

measures for information propagation in boolean networks boolean networks provide a large scale model of gene regulatory and neuronal networks in this paper we study what kinds of boolean networks best propagate and process signals i e information in the presence of stochasticity and noise we first examine two existing approaches that use mutual information and find that these approaches do not capture well the phenomenon studied we propose a new measure for information propagation based on perturbation avalanches in boolean networks and find that the measure is maximized in dynamically critical networks and in subcritical networks if noise is present

reinforced brownian motion a prototype we analyze the brownian motion limit of a prototypical unit step reinforced random walk on the half line a reinfoced random walk is one which changes the weight of any edge or vertex visited to increase the frequency of return visits the generating function for the discrete case is first derived for the joint probability distribution of s n the location of the walker at the n th step and a n the maximum location the walker achieved in n steps then the bulk of the analysis concerns the statistics of the limiting brownian walker and of its environment both parametrized by the amplitude delta of the reinforcement the walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement details depending very much on the value of 1 delta infty

on the transience of processes defined on galton watson trees we introduce a simple technique for proving the transience of certain processes defined on the random tree mathcal g generated by a supercritical branching process we prove the transience for once reinforced random walks on mathcal g that is a generalization of a result of durrett kesten and limic probab theory related fields 122 2002 567 592 moreover we give a new proof for the transience of a family of biased random walks defined on mathcal g other proofs of this fact can be found in ann probab 16 1988 1229 1241 and ann probab 18 1990 931 958 as part of more general results a similar technique is applied to a vertex reinforced jump process a by product of our result is that this process is transient on the 3 ary tree davis and volkov probab theory related fields 128 2004 42 62 proved that a vertex reinforced jump process defined on the b ary tree is transient if b 4 and recurrent if b 1 the case b 2 is still open

a pde model for the dynamics of trail formation by ants abstract armies of ants are known to move in trails these trails are formed by a chemotactic force induced by pheromone secreted by the ants in this paper we develop a mathematical model consisting of two partial differential equations which explain when and how these trails are formed the first equation for the ants includes the chemotaxis effect of pheromone and the dispersion caused by overcrowding the second equation is a reaction diffusion equation for the pheromone concentration the strength of the chemotactic force plays a critical role in the analysis we prove that trails cannot be formed if is small while many trails exist if is large

phase transition in reinforced random walk and rwre on trees a random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future using exchangeability theory the process is shown to be equivalent to a random walk in a random environment rwre that is to say a mixture of markov chains criteria are given to determine whether a rwre is transient or recurrent these criteria apply to show that the reinforced random walk can vary from transient to recurrent depending on the value of an adjustable parameter measuring the strength of the feedback the value of the parameter at the phase transition is calculated