chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

self interacting diffusions iii symmetric interactions let m be a compact riemannian manifold a self interacting diffusion on m is a stochastic process solution to dx t dw t x t 1 t t 0 v xs x t ds dt where w t is a brownian vector field on m and v x y v x y a smooth function let t 1 t t 0 xs ds denote the normalized occupation measure of x t we prove that when v is symmetric t converges almost surely to the critical set of a certain nonlinear free energy functional j furthermore j has generically finitely many critical points and t converges almost surely toward a local minimum of j each local minimum has a positive probability to be selected

stochastic bifurcation models we study an ordinary differential equation controlled by a stochastic process we present results on existence and uniqueness of solutions on associated local times trotter and ray knight theorems and on time and direction of bifurcation a relationship with lipschitz approximations to brownian paths is also discussed

persistence of mass in a chemotaxis system with logistic source this paper studies the dynamical properties of the chemotaxis system ut u u v ru u2 x t 0 vt v v u x t 0 under homogeneous neumann boundary conditions in bounded convex domains rn n 1 with positive constants r and numerical simulations but also some rigorous evidence have shown that depending on the relative size of r and in comparison to the well understood case when 0 this problem may exhibit quite a complex solution behavior including unexpected effects such as asymptotic decay of the quantity u within large subdomains of the present work indicates that any such extinction phenomenon if occurring at all necessarily must be of spatially local nature whereas the population as a whole always persists more precisely it is shown that for any nonnegative global classical solution u v of with u 0 one can find m 0 such that u x t dx m for all t 0 the proof is based on an in this context apparently novel analysis of the functional ln u deriving a lower bound for this quantity along a suitable sequence of times by appropriately exploiting a differential inequality for a suitable linear combination of ln u u and v2

continuity of martingales in the brownian excursion filtration given a brownian motionb we consider the filtration xxer where x is defined as the field generated by the excursions ofb belowx in this paper we prove a conjecture of walsh which says that all e martingales are continuous

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

une extension des theoremes de ray et knight sur les temps locaux browniens we introduce two classes of random variablesv such that the brownian local time process at timev is distributed as a 0 or 2 dimensional bessel bridge moreover we obtain new decompositions of the brownian path on the interval 0 v which generalize williams results

critical probabilities in percolation on decorated graphs the authors consider bond percolation on a graph gamma and on a decorated graph gamma g derived from gamma by replacing every edge in gamma by a finite connected two rooted graph g they show that the critical exponent beta is invariant under such decorations and that by varying g one can construct a set of graphs with critical probabilities dense in 0 1

further comments on the one dimensional percolation problem with multineighbour bonds one dimensional site and bond percolation problems with bonds connecting lth nearest neighbours are solved exactly by using an infinite cell to cell renormalisation group transformation both thermal and magnetic critical exponents are obtained through the scaling relations although universality breaks down for site and bond percolation when l or 2 suzuki s weaker universality 1974 still holds

weak limits of perturbed random walks and the equation y_t b_t alpha sup y_s colon s leq t beta inf y sb s colon s leq t let and be real numbers and f c o 0 we study the existence and uniqueness of solutions g of the equation g t f t sup o 8 t g s inf o 8 t g s carmona petit le gall and yor have shown existence or nonexistence and uniqueness for some we settle the remaining cases we study the nearest neighbor walk on the integers which behaves just like fair random walk unless one neighbor has been visited and the other has not when it jumps to the unvisited neighbor with probability p if p 2 3 we show these processes scaled converge to the solution of the equation above for brownian paths with 2p 1 p

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

blow up in a higher dimensional chemotaxis system despite logistic growth restriction abstract we study radially symmetric solutions of a class of chemotaxis systems generalizing the prototype u t u u v u u x t 0 0 v m t u x t 0 in a ball r n with parameters 0 0 0 and 1 and m t 1 u x t d x it is shown that when n 5 and 3 2 1 2 n 2 then there exist initial data such that the smooth local in time solution of blows up in finite time this indicates that even superlinear growth restrictions may be insufficient to rule out a chemotactic collapse as is known to occur in the corresponding system without any proliferation

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

large time behavior of solution to an attraction repulsion chemotaxis system with logistic source in three dimensions abstract this paper studies the attraction repulsion chemotaxis system with logistic source u t u u v u w f u v t v 1 v 1 u w t w 2 w 2 u in a smooth bounded convex domain r 3 subject to nonnegative initial data and homogeneous neumann boundary conditions where i and i i 1 2 are positive parameters and the logistic source function f fulfills f s s s 1 s 0 0 and 1 it is shown that this system possesses a unique global bounded classical solution under the conditions i 1 2 and max 41 2 1 9 2 9 1 41 2 2 furthermore whenever u 0 0 and for any n the solution of the system approaches to the steady state 1 1 1 1 1 1 1 1 2 2 in the norm of l as t

pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments abstract the current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains we first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models next under the same conditions on the coefficients we show that pointwise persistence occurs that is any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough it should be pointed out that in 21 the authors established the persistence of mass for globally defined positive solutions which indicates that any extinction phenomenon if occurring at all necessarily must be spatially local in nature whereas the population as a whole always persists the pointwise persistence proved in the current paper implies that not only the population as a whole persists but also it persists at any location eventually it also implies the existence of strictly positive entire solutions

brownian charges around loops let x t n be a poisson sequence of independent brownian motions in d d 3 let be a compact oriented submanifold of d of dimensiond 2 and volume l let t be the sum of the windings of x s n 0 s t around then t t converges in law towards a cauchy variable of parameter l 2 a similar result is valid when the winding is replaced by the integral of a harmonic 1 form in d

boundedness and finite time collapse in a chemotaxis system with volume filling effect abstract we consider the elliptic parabolic pde system u t u u u v x t 0 0 v m u x t 0 with nonnegative initial data u 0 having mean value m under homogeneous neumann boundary conditions in a smooth bounded domain r n the nonlinearities and are supposed to generalize the prototypes u u 1 p u u u 1 q 1 with p 0 and q r problems of this type arise as simplified models in the theoretical description of chemotaxis phenomena under the influence of the volume filling effect as introduced by painter and hillen k j painter t hillen volume filling and quorum sensing in models for chemosensitive movement can appl math q 10 2002 501 543 it is proved that if p q 2 n then all solutions are global in time and bounded whereas if p q 2 n q 0 and is a ball then there exist solutions that become unbounded in finite time the former result is consistent with the aggregation inhibiting effect of the volume filling mechanism the latter however is shown to imply that if the space dimension is at least three then chemotactic collapse may occur despite the presence of some nonlinearities that supposedly model a volume filling effect in the sense of painter and hillen

some extensions of the arc sine law as partial consequences of the scaling property of brownian motion the scaling property of brownian motion is exploited systematically in order to extend paul levy s arc sine law to brownian motion perturbed by its local time at 0 other important ingredients of the proofs are some ray knight theorems for local times

boundedness in the higher dimensional chemotaxis haptotaxis model with nonlinear diffusion we consider the chemotaxis haptotaxis model ut d u u u v u w u 1 u w x t 0 vt v v u x t 0 wt vw x t 0 in a bounded smooth domain rn n 2 where and are positive parameters and the diffusivity d u is assumed to generalize the prototype d u u 1 with r under zero flux boundary conditions it is shown that for sufficiently smooth initial data u0 v0 w0 and 2n 1 the corresponding initial boundary problem possesses a unique global in time classical solution which is uniformly bounded this paper develops some lp estimate techniques and thereby extends boundedness results in n 3 to arbitrary space dimensions

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

enlacements du mouvement brownien autour des courbes de l espace limit theorems are proved for the winding numbers of a three dimensional brownian motion around certain curves in space in particular the joint asymptotic distribution of the winding numbers around two curves is obtained this joint distribution generalizes the asymptotic law of the winding numbers of a planar brownian motion around two points which has recently been given by pitman and yor the limiting distributions are closely related to the time spent by a linear brownian motion above and below a multiple of its maximum process proofs rely on stochastic calculus for continuous semi martingales

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

global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the parabolic chemotaxis system u t u u v u u 2 x t 0 v t v v u x t 0 in bounded convex domains r n n 1 with smooth boundary it is shown that if the ratio is sufficiently large then the unique nontrivial spatially homogeneous equilibrium given by u v 1 is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data u 0 v 0 such that u 0 0 the above problem possesses a uniquely determined global classical solution u v with u v t 0 u 0 v 0 which satisfies u t 1 l 0 and v t 1 l 0 as t

brownian motion and random walk perturbed at extrema let bt be brownian motion we show there is a unique adapted process xt which satisfies dxt dbt except when xt is at a maximum or a minimum when it receives a push the magnitudes and directions of the pushes being the parameters of the process for some ranges of the parameters this is already known we show that if a random walk close to bt is perturbed properly its paths are close to those of xt

some exact results on bond percolation we present some exact results on bond percolation we derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice by l bonds connecting the same adjacent vertices thereby yielding the lattice l this relation is used to calculate the bond percolation threshold on l we show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d 2 but changes it on a one dimensional lattice and quasi one dimensional infinite length strips we also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the n limits of several families of n vertex graphs finally we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as n

the strong law of large numbers for a brownian polymer we prove the strong law of large numbers for a continuous time version of reinforced random walk this extends previous results of durrett and rogers

a new approach toward stabilization in a two species chemotaxis model with logistic source abstract this paper aims at providing an alternative approach to study global dynamic properties for a two species chemotaxis model with the main novelty being that both populations mutually compete with the other on account of the lotka volterra dynamics more precisely we consider the following neumann initial boundary value problem u t d 1 u 1 u w 1 u 1 u a 1 v x t 0 v t d 2 v 2 v w 2 v 1 a 2 u v x t 0 0 d 3 w w u v x t 0 in a bounded domain r n n 1 with smooth boundary where d 1 d 2 d 3 1 2 1 2 a 1 a 2 are positive constants when a 1 0 1 and a 2 0 1 it is shown that under some explicit largeness assumptions on the logistic growth coefficients 1 and 2 the corresponding neumann initial boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state u v w of above system in the large time limit the respective decay rate of this convergence is shown to be exponential when a 1 1 and a 2 0 1 if 2 is suitable large for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 0 and v 0 0 the globally bounded solution of above system will stabilize toward 0 1 1 as t in algebraic

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

continuous percolation in one dimension the continuous percolation problem for equal disc size in a one dimensional space is treated exact expressions are given for the relevant functions and the critical exponents it is shown that with an appropriate selection of scaling fields this problem as well as the further neighbour discrete problem belongs to the same universality class as the more common nearest neighbour one dimensional discrete problem

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

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

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

sub logistic source can prevent blow up in the 2d minimal keller segel chemotaxis system it is well known that the neumann initial boundary value problem for the minimal keller segel chemotaxis system in a 2d bounded smooth domain has no blow ups for any presence of logistic source of cell kinetics here for a large class of cell kinetics including sub logistic sources we find an explicit condition involving the chemotactic strength the asymptotic damping rate and the initial mass of cells to ensure the uniform in time boundedness for the corresponding 2d neumann initial boundary value problem our finding in particular shows that sub logistic source can prevent blow up in 2d indicating that logistic damping is not the weakest damping to guarantee boundedness for the 2d keller segel minimal chemotaxis model

boundedness in a quasilinear parabolic parabolic keller segel system with subcritical sensitivity abstract we consider the quasilinear parabolic parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a convex smooth bounded domain r n with n 1 it is proved that if s u d u c u with 2 n and some constant c 0 for all u 1 then the classical solutions to the above system are uniformly in time bounded provided that d u satisfies some technical conditions such as algebraic upper and lower growth resp decay estimates as u this boundedness result is optimal according to a recent result by the second author winkler 2010 27 which says that if s u d u c u for u 1 with c 0 and some 2 n n 2 then for each mass m 0 there exist blow up solutions with mass u 0 m in addition this paper also proves a general boundedness result for quasilinear non uniformly parabolic equations by modifying the iterative technique of moser alikakos alikakos 1979 1

boundedness in a fully parabolic chemotaxis system with singular sensitivity abstract this paper deals with a fully parabolic chemotaxis system u t u u v v v t v v u with singular sensitivity v 0 on a bounded domain r n n 2 the main result solves the open problem of uniform in time boundedness of solutions for 2 n which was conjectured by winkler 16

fine mesh limit of the vrjp in dimension one and bass burdzy flow we introduce a continuous space limit of the vertex reinforced jump process vrjp in dimension one which we call linearly reinforced motion lrm on mathbb r it is constructed out of a convergent bass burdzy flow the proof goes through the representation of the vrjp as a mixture of markov jump processes as a by product this gives a representation in terms of a mixture of diffusions of the lrm and of the bass burdzy flow itself we also show that our continuous space limit can be obtained out of the edge reinforced random walk errw since the errw and the vrjp are known to be closely related compared to the discrete space processes the lrm has an additional symmetry in the initial local times initial occupation profile changing them amounts to a deterministic change of the space and time scales

doubly perturbed jump diffusion processes in this paper we study the existence and uniqueness of the solution to a class of doubly perturbed jump diffusion processes with non lipschitz coefficients some well known results are improved and generalized

global boundedness in a three dimensional chemotaxis haptotaxis model abstract this paper studies the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r 3 with zero flux boundary conditions where the parameters and are positive it is shown that the corresponding initial boundary problem possesses a unique classical solution which is global in time and bounded under the explicit condition 11 2 2 w 0 l 2 2 37 2 4 2 w 0 l 2 and suitable regularity assumptions on the initial data u 0 v 0 w 0

failure of hyperscaling in one dimensional long range bond percolation the authors introduce a result which allows one to deduce the cluster number exponent 2 alpha in long range one dimensional percolation problems this is used to show that in the long range bond problem hyperscaling fails when the bond length exceeds 6

boundedness vs blow up in a chemotaxis system abstract we determine the critical blow up exponent for a keller segel type chemotaxis model where the chemotactic sensitivity equals some nonlinear function of the particle density assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution furthermore we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails

large time behavior of solution to a fully parabolic chemotaxis haptotaxis model in higher dimensions abstract this paper deals with the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r n with zero flux boundary conditions where and are positive parameters it is shown that if is suitably large then for all sufficiently smooth initial data the associated initial boundary value problem possesses a unique global in time classical solution that is bounded in 0 and if the initial data w 0 is small w becomes asymptotically negligible moreover we prove that when domain is convex 1 1 0 is globally asymptotically stable provided that u 0 0 and thereby extends the result of hillen et al 2013 18 to the higher space dimensions

global existence of classical solutions to a combined chemotaxis haptotaxis model with logistic source abstract this paper deals with a mathematical model of cancer invasion of tissue the model consists of a system of reaction diffusion taxis partial differential equations describing interactions between cancer cells matrix degrading enzymes and the host tissue in two space dimensions we prove global existence and uniqueness of classical solutions to this model for any 0 where is the logistic growth rate of cancer cells the crucial point of proof is to raise the regularity estimate of a solution from l 1 to l 3 0 t where r 2 is some bounded domain and t 0 is some constant this paper develops new estimate techniques and improves greatly our previous results y tao m wang global solution for a chemotactic haptotactic model of cancer invasion nonlinearity 21 2008 2221 2238 in 2 dimensions

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

self attracting diffusions case of the constant interaction in this paper we study a self attracting diffusion in the case of a constant self attraction and for dimension larger than two we prove that this process converges almost surely

superdiffusive bounds on self repellent brownian polymers and diffusion in the curl of the gaussian free field in d 2 we consider two models of random diffusion in random environment in two dimensions the first example is the self repelling brownian polymer this describes a diffusion pushed by the negative gradient of its own occupation time measure local time the second example is a diffusion in a fixed random environment given by the curl of massless gaussian free field

trees under attack a ray knight representation of feller s branching diffusion with logistic growth we obtain a representation of feller s branching diffusion with logistic growth in terms of the local times of a reflected brownian motion h with a drift that is affine linear in the local time accumulated by h at its current level as in the classical ray knight representation the excursions of h are the exploration paths of the trees of descendants of the ancestors at time t 0 and the local time of h at height t measures the population size at time t we cope with the dependence in the reproduction by introducing a pecking order of individuals an individual explored at time s and living at time t h s is prone to be killed by any of its contemporaneans that have been explored so far the proof of our main result relies on approximating h with a sequence of harris paths h n which figure in a ray knight representation of the total mass of a branching particle system we obtain a suitable joint convergence of h n together with its local times and with the girsanov densities that introduce the dependence in the reproduction

on convergence to equilibria for the keller segel chemotaxis model we show that any global in time bounded solution to the keller segel chemotaxis model converges to a single equilibrium as time tends to infinity the proof is based on a generalized version of the lojasiewicz simon theorem

etude asymptotique des enlacements du mouvement brownien autour des droites de l espace the existence of a joint asymptotic distribution for the windings of a three dimensional brownian motion around a finite number of straight lines is obtained this complements the recent studies by pitman yor and the authors of the joint asymptotic distribution for the windings of planar brownian motion around a finite number of points

doubly perturbed neutral stochastic functional equations in this paper we prove the existence and uniqueness of the solution to a class of doubly perturbed neutral stochastic functional equations dpnsfes in short under some non lipschitz conditions the solution is constructed by successive approximation furthermore we give the continuous dependence of the solution on the initial value by means of the corollary of bihari inequality

series study of the one dimensional true self avoiding walk the true self avoiding walk problem is formulated using a grand canonical approach and exact enumeration methods are used to calculate the average end to end distance for one dimensional true self avoiding walks with up to 21 steps the results are in agreement with a universality picture obtained both from monte carlo simulations and from scaling and crossover arguments the extrapolated value of the end to end distance exponent nu is nu 0 67 or 0 04

a comparative study of interacting random walk models the authors analyse four interacting random walk models from a comparative point of view concentrating on one dimensional versions they consider an interacting random walk model recently introduced by stanley et al 1983 the domb joyce model 1972 and the true self avoiding walk of amit et al 1983 in addition they introduce a model based on the weighting of turning points whose properties are related to those of the ising spin chain fixed points and their stability are identified for both attractive and repulsive correlations in all four cases they compare the mechanisms involved in each of these models and comment on the role played in each by its particular form of correlation between steps the different universal behaviours are shown to arise from competition between short range and long range and particularly cumulative memory effects and between local and global normalisation conditions

global existence and asymptotic behavior of classical solutions to a parabolic elliptic chemotaxis system with logistic source on rn in the current paper we consider the following parabolic elliptic semilinear keller segel model on rnrn ut u u v au bu2 x rn t 0 0 i v u x rn t 0 where 0 a 0 b 0 are constant real numbers and n is a positive integer we first prove the local existence and uniqueness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 with u x 0 u0 u0 x u x 0 u0 u0 x for various initial functions u0 x u0 x next under some conditions on the constants a b a b and the dimension n we prove the global existence and boundedness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 for given initial functions u0 x u0 x finally we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions under some conditions on the constants a b a b and the dimension n we show that for every strictly positive initial function u0 u0 limt supx rn u x t u0 ab v x t u0 ab 0 and that for every nonnegative initial function u0 u0 with non empty and compact support supp u0 supp u0 there are 0 clow u0 cup u0 such that limt sup x ct u x t u0 ab v x t u0 ab 0 0 c clow u0 and limt sup x ct u x t u0 v x t u0 0 c cup u0

eventual smoothness and asymptotics in a three dimensional chemotaxis system with logistic source abstract we prove existence of global weak solutions to the chemotaxis system u t u u v u u 2 v t v v u under homogeneous neumann boundary conditions in a smooth bounded convex domain r n for arbitrarily small values of 0 additionally we show that in the three dimensional setting after some time these solutions become classical solutions provided that is not too large in this case we also consider their large time behaviour we prove decay if 0 and the existence of an absorbing set if 0 is sufficiently small

convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source abstract we study the global attractors to the chemotaxis system with logistic source u t u u v a u b u 2 v t v v u in r subject to the homogeneous neumann boundary conditions where smooth bounded domain r n with b 0 a r and 0 1 for the parabolic elliptic case with 0 and n 3 we obtain that the positive constant equilibrium a b a b is a global attractor if a 0 and b max n 2 n a 4 under the assumption n 3 it is proved that for either the parabolic elliptic case with 0 a 0 b max 3 a 4 or the parabolic parabolic case with 1 a 0 b a 4 large enough the system admits the positive constant equilibrium a b a b as a global attractor while the trivial equilibrium 0 0 is a global attractor if a 0 and b 0 it is pointed out that here the convergence rates are established for all of them the results of the paper mainly rely on parabolic regularity theory and lyapunov functionals carefully constructed

site decorations and critical exponents in percolation the authors introduce and discuss a family of site decorations which change local lattice features such as coordination number and planarity but which may be shown explicitly to preserve the usual percolation exponents even in a continuum limit the continuum limit is shown to be equivalent to cell percolation where sites are randomly and independently distributed in space but where connectivity is determined by a cell structure conjectured amplitude relationships are verified within lattice families without recourse to scaling arguments

the sde solved by local times of a brownian excursion or bridge derived from the height profile of a random tree or forest let b be a standard one dimensional brownian motion started at 0 let l t v b be the occupation density of b at level v up to time t the distribution of the process of local times l t v b v 0 conditionally given b 0 and l t 0 b l is shown to be that of the unique strong solution x of the ito sde dx v 4 x 2 a t 0 x v du 1 dv 2 x v dbv on the interval 0 v t x where v t x inf v f v 0 x u du t and x v 0 for all v v t x this conditioned form of the ray knight description of brownian local times arises from study of the asymptotic distribution as n and 2k n l of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points the sde is the continuous analog of a simple description of a galton watson branching process conditioned on its total progeny for l 0 corresponding to asymptotics of a uniform random tree the sde gives a description of the process of local times of a brownian excursion which is equivalent to jeulin s description of these local times as a time change of twice a brownian excursion another corollary is the biane yor description of the local times of a reflecting brownian ridge as a time changed reversal of twice a brownian meander of the same length

eventual smoothness and stabilization of large data solutions in a three dimensional chemotaxis system with consumption of chemoattractant abstract this paper deals with positive solutions of u t u u v x t 0 v t v u v x t 0 under homogeneous neumann boundary conditions in bounded convex domains r 3 with smooth boundary it is shown that for arbitrarily large initial data this problem admits at least one global weak solution for which there exists t 0 such that u v is bounded and smooth in t moreover it is asserted that such solutions approach spatially constant equilibria in the large time limit

boundedness in quasilinear keller segel systems of parabolic parabolic type on non convex bounded domains abstract this paper deals with the quasilinear fully parabolic keller segel system u t d u u s u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary n n the diffusivity d u is assumed to satisfy some further technical conditions such as algebraic growth and d 0 0 which says that the diffusion is allowed to be not only non degenerate but also degenerate the global in time existence and uniform in time boundedness of solutions are established under the subcritical condition that s u d u k u e for u 0 with 2 n k 0 and e 0 when d 0 0 this paper represents an improvement of tao and winkler 17 because the domain does not necessarily need to be convex in this paper in the case r n and d 0 0 uniform in time boundedness is an open problem left in a previous paper 7 this paper also gives an answer to it in bounded domains

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

perturbed brownian motions we study perturbed brownian motions that can be loosely speaking described as follows they behave exactly as linear brownian motion except when they hit their past maximum or and maximum where they get an extra push we define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain natural class of processes in the case where both perturbations at the maximum and at the minimum are self repelling we show that in fact more is true such a process can almost surely be constructed from brownian paths by a one to one measurable transformation this generalizes some results of carmona petit yor and davis we also derive some fine properties of perturbed brownian motions hausdorff dimension of points of monotonicity for example

random walk with barycentric self interaction we study the asymptotic behaviour of a d dimensional self interacting random walk x n n 1 2 3 which is repelled or attracted by the centre of mass g n n 1 sum i 1 n x i of its previous trajectory the walk s trajectory x 1 x n models a random polymer chain in either poor or good solvent in addition to some natural regularity conditions we assume that the walk has one step mean drift mathbb e x n 1 x n mid x n g n mathbf x approx rho mathbf x beta hat mathbf x for and 0 when 0 we show that x n is transient with a limiting random direction and satisfies a super diffusive law of large numbers n 1 1 x n converges almost surely to some random vector when 0 1 there is sub ballistic rate of escape when 0 and we give almost sure bounds on the norms x n which in the context of the polymer model reveal extended and collapsed phases

energy type estimates and global solvability in a two dimensional chemotaxis haptotaxis model with remodeling of non diffusible attractant this paper deals with the coupled chemotaxis haptotaxis model ut u u v u w u 1 u w x t 0 0 v u v x t 0 wt vw w 1 w u x t 0 which was initially proposed by chaplain and lolas 2006 10 to describe the interactions between cancer cells the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue extracellular matrix here r2 is a bounded domain with smooth boundary and and are positive parameters as compared to previous mathematical studies the novelty here consists of allowing for positive values of reflecting processes of self remodeling of the extracellular matrix under zero flux boundary conditions it is shown that for any sufficiently smooth initial data u0 w0 satisfying first order compatibility conditions the model admits a unique global smooth solution a crucial ingredient in the proof is an energy like inequality which given t 0 yields boundedness of u t in llogl this serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions

lattice decorations and one dimensional percolation it is well known that in long range one dimensional percolation the critical exponents depend on the range a family of decorations is discussed which have range dependent exponents in one dimension although the corresponding decorations in higher dimension preserve the values of the exponents these decorations clarify the unusual nature of one dimensional percolation

pathwise uniqueness for perturbed versions of brownian motion and reflected brownian motion any solution of the functional equation wta bta asup st wsa 1 l w t 0 where b is a brownian motion behaves like a reflected brownian motion except when it attains a new maximum we call it an a per turbed reflected brownian motion similarly any solution of xta bta asup st xsa binft xs t 0 behaves like a brownian motion except when it attains a new maxi mum or minimum we call it an a b doubly perturbed brownian motion we complete some recent investigations by showing that for all permissible values of the parameters a a and b respectively these equations have pathwise unique solutions and these are adapted to the filtration of b