transient aging in fractional brownian and langevin equation motion stochastic processes driven by stationary fractional gaussian noise that is fractional brownian motion and fractional langevin equation motion are usually considered to be ergodic in the sense that after an algebraic relaxation time and ensemble averages of physical observables coincide recently it was demonstrated that fractional brownian motion and fractional langevin equation motion under external confinement are transiently nonergodic char22 time and ensemble averages behave differently char22 from the moment when the particle starts to sense the confinement here we show that these processes also exhibit transient aging that is physical observables such as the time averaged mean squared displacement depend on the time lag between the initiation of the system at time t 0 and the start of the measurement at the aging time t a in particular it turns out that for fractional langevin equation motion the aging dependence on t a is different between the cases of free and confined motion we obtain explicit analytical expressions for the aged moments of the particle position as well as the time averaged mean squared displacement and present a numerical analysis of this transient aging phenomenon

inverse gaussian and its inverse process as the subordinators of fractional brownian motion in this paper we study the fractional brownian motion fbm time changed by the inverse gaussian ig process and its inverse called the inverse to the inverse gaussian iig process some properties of the time changed processes are similar to those of the classical fbm such as long range dependence however one can also observe different characteristics that are not satisfied by the fbm we study the distributional properties of both subordinators namely ig and iig processes and also that of the fbm time changed by these subordinators we establish also the connections between the subordinated processes considered and the continuous time random walk model for the application part we introduce the simulation procedures for both processes and discuss the estimation schemes for their parameters the effectiveness of these methods is checked for simulated trajectories

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

path integral representation for fractional brownian motion fractional brownian motion fbm is a generalization of the usual brownian motion a path integral representation that has recently been suggested for it is shown to be not for the fbm but for a different generalization of the brownian motion a new path integral representation is given and its measure has fractional derivatives of the path in it the measure shows that the process is gaussian but is in general non markovian even though brownian motion itself is markovian it is shown how the propagator for the motion of free fbm may be evaluated this is somewhat more complex than for the usual path integrals due to the occurrence of fractional derivatives we also find the propagator in the presence of a linear absorption potential and for fbm on a ring

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

bimodality of the interspike interval distributions for subordinated diffusion models of integrate and fire neurons abstract a subordinated langevin process with a random operational time in the form of an inverse strictly increasing levy type subordinator is considered as a generalization of the conventional perfect and leaky integrate and fire neuron models the parent process is given by standard brownian motion the effect of the random activity of synaptic inputs which arises from other neurons forming local and distant networks is modeled via a levy exponent of the subordinator using a first passage time formulation in an external force field we find exact expressions for the laplace transform of the output interspike interval isi density more detailed analysis is presented on the properties of the isi distribution in the case of the levy exponent which corresponds to the truncated double order time fractional diffusion equation for the probability density of the membrane potential particularly it is shown that at some parameter regimes the isi density exhibits a bimodal structure moreover it is demonstrated that the isis regularity is maximized at an intermediate value of the mean input current

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

packing measure of the sample paths of fractional brownian motion let x t t r be a fractional brownian motion of index in rd if 1 0 where s s 1 log log 1 s 1 2 and p x 0 t is the packing measure of x 0 t

long time behavior of solutions of fisher kpp equation with advection and free boundaries we consider fisher kpp equation with advection ut uxx ux f u for x g t h t where g t and h t are two free boundaries satisfying stefan conditions this equation is used to describe the population dynamics in advective environments we study the influence of the advection coefficient on the long time behavior of the solutions we find two parameters c0 and with c0 0 which play key roles in the dynamics here c0 is the minimal speed of the traveling waves of fisher kpp equation more precisely by studying a family of the initial data 0 where is some compactly supported positive function we show that 1 in case 0 c0 there exists 0 such that spreading happens when i e u t 1 locally uniformly in r and vanishing happens when 0 i e g t h t remains bounded and u t 0 uniformly in g t h t 2 in case c0 there exists 0 such that virtual spreading happens when i e u t 0 locally uniformly in g t and u t ct 1 locally uniformly in r for some c c0 vanishing happens when 0 and in the transition case u t o t v c0 t uniformly the latter is a traveling wave with a big head near the free boundary x c0 t and with an infinite long tail on the left 3 in case c0 there exists 0 such that virtual spreading happens when and u t 0 uniformly in g t h t when 0 4 in case vanishing happens for any solution

relaxation time distributions for an anomalously diffusing particle abstract as well known the generalized langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle here we focus attention on some new aspects of the dynamics successively considering the memory kernel the particle s mean velocity and the scattering function all these quantities are studied from a unique angle namely the discussion of the possible existence of a distribution of relaxation times characterizing their time decay although a very popular concept a relaxation time distribution cannot be associated with any time decreasing quantity from a mathematical point of view the decay has to be described by a completely monotonic function technically we use a memory kernel decaying as a mittag leffler function the mittag leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times we show that in the case of a subdiffusive motion relaxation time distributions can be defined for the memory kernel and for the scattering function but not for the particle s mean velocity the situation is opposite in the superdiffusive case

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

symmetric pairs and gorelik elements abstract the second author gave a formula for the elements of the enveloping algebra of a lie superalgebra defined by gorelik under an appropriate unimodularity assumption we show that this formula is a particular case of a formula for the jacobian of the exponential map of a symmetric superspace

spreading speeds of a lotka volterra predator prey system the role of the predator abstract this paper is concerned with the spreading speeds of a modified lotka volterra predator prey system the intention is to describe the propagation mode whereby the two species coinvade a new habitat under some assumptions it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear therefore the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

regularity and asymptotic behavior of nonlinear stefan problems we study the following nonlinear stefan problem left begin aligned u t d delta u g u quad rm for x in omega t t 0 u 0 rm and u t mu nabla x u 2 quad rm for x in gamma t t 0 u 0 x u 0 x quad rm for x in omega 0 end aligned right where omega t subset mathbb r n n geqq 2 is bounded by the free boundary gamma t with omega 0 omega 0 and d are given positive constants the initial function u0 is positive in omega 0 and vanishes on partial omega 0 the class of nonlinear functions g u includes the standard monostable bistable and combustion type nonlinearities we show that the free boundary gamma t is smooth outside the closed convex hull of omega 0 and as t to infty either omega t expands to the entire mathbb r n or it stays bounded moreover in the former case gamma t converges to the unit sphere when normalized and in the latter case u to 0 uniformly when g u au bu 2 we further prove that in the case omega t expands to mathbb r n u to a b as t to infty and the spreading speed of the free boundary converges to a positive constant moreover there exists mu geqq 0 such that omega t expands to mathbb r n exactly when mu mu

the minimal habitat size for spreading in a weak competition system with two free boundaries abstract in this paper we focus on the dynamics for a lotka volterra type weak competition system with two free boundaries where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species respectively in the weak competition case the dynamics of this model can be classified into four cases which forms a spreading vanishing quartering the notion of the minimal habitat size for spreading is introduced to determine if species can always spread some sufficient conditions for spreading and vanishing are established also when spreading occurs some rough estimates for spreading speed and the long time behavior of solutions are established

invasion by an inferior or superior competitor a diffusive competition model with a free boundary in a heterogeneous environment in this study we consider the population dynamics of an invasive species and a resident species which are modeled as a diffusive competition process in a radially symmetric setting with a free boundary we assume that the resident species undergoes diffusion and growth in r n while the invasive species initially exists in a finite ball but invades the environment with a spreading front evolving according to a free boundary when the invasive species is inferior we show that if the resident species is already well established initially then the invader can never invade deep into the underlying habitat thus it dies out before its invading front reaches a certain finite limiting position when the invasive species is superior a spreading vanishing dichotomy holds and sharp criteria for spreading and vanishing with d 1 mu and u 0 as variable factors are obtained where d 1 mu and u 0 are the dispersal rate expansion capacity and initial number of invaders respectively in particular we obtain some rough estimates of the asymptotic spreading speed when spreading occurs c 2014 elsevier inc all rights reserved

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

on a lotka volterra competition diffusion advection model in general heterogeneous environments abstract we consider a two species lotka volterra competition diffusion advection system which has the combined effect of spatial dispersal advection and spatial variations of resource on the population firstly the existence and global asymptotic stability of the positive equilibrium solution for the logistic model are obtained with the assumption of spatially heterogeneous dispersal advection and resource distribution secondly it is shown that when the diffusion coefficients resource functions and competition rates are all spatially heterogeneous the positive equilibrium solution is globally asymptotically stable when it exists

asymptotic spreading in a lotka volterra predator prey system abstract this paper is concerned with the asymptotic spreading of a lotka volterra predator prey system with self limit effect in which the intrinsic growth rates of both species are positive the spreading speeds of both species are estimated which implies the role of nonlinearities in asymptotic spreading biologically our results confirm that the prey may contribute to the asymptotic spreading of the predator

the diffusive logistic equation with a free boundary and sign changing coefficient abstract this short paper concerns a diffusive logistic equation with a free boundary and sign changing coefficient which is formulated to study the spread of an invasive species where the free boundary represents the expanding front a spreading vanishing dichotomy is derived namely the species either successfully spreads to the right half space as time t and survives persists in the new environment or it fails to establish itself and will extinct in the long run the sharp criteria for spreading and vanishing are also obtained when spreading happens we estimate the asymptotic spreading speed of the free boundary

a diffusive logistic equation with a free boundary and sign changing coefficient in time periodic environment abstract this paper concerns a diffusive logistic equation with a free boundary and sign changing intrinsic growth rate in heterogeneous time periodic environment in which the variable intrinsic growth rate may be very negative in a suitable large region see conditions h1 h2 4 3 such a model can be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front in the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound this problem has been studied by du guo peng 11 they established a spreading vanishing dichotomy the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed in the present paper we show that the above results are retained for our problem

stochastic resonance in a linear fractional langevin equation the fractional langevin equation is derived from the generalized langevin equation driven by the additive fractional gaussian noise we investigate the stochastic resonance sr phenomenon in the underdamped linear fractional langevin equation under the external periodic force and multiplicative symmetric dichotomous noise applying the shapiro loginov formula and the laplace transform technique we obtain the exact expressions of the amplitude and signal to noise ratio snr of the system by studying the impacts of the driving frequency and the noise parameters we find the non monotonic behaviors of the output amplitude and snr the results indicate that the bona fide sr conventional sr and the wide sense of sr phenomena occur in the proposed linear fractional system

symmetric lie superalgebras and deformed quantum calogero moser problems abstract the representation theory of symmetric lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum calogero moser systems in the special case of symmetric pair g gl n 2 m k osp n 2 m we establish a natural bijection between projective covers of spherically typical irreducible g modules and the finite dimensional generalised eigenspaces of the algebra of calogero moser integrals d n m acting on the corresponding laurent quasi invariants a n m

note on a two species competition diffusion model with two free boundaries abstract in guo and wu 2015 and wu 2015 the authors studied a two species competition diffusion model with two free boundaries these two free boundaries describing the spreading fronts of two competing species respectively may intersect each other as time evolves the existence uniqueness and long time behavior of global solution have been established in this note we discuss the conditions for spreading and vanishing and more accurate limits of u v as t when spreading occurs some new results and simpler proofs will be provided

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

stability and lack of memory of the returns of the hang seng index abstract in this paper we show that the logarithmic returns of the hang seng index from january 2 1987 to november 14 2005 statistically resemble a sequence of independent identically distributed levy stable random variables this is in stark contrast to xiu and jin 2007 39 where long memory farima processes with gaussian noise were suggested as well fitted to the data the lack of memory is checked by using lo s modified r s statistic and a new method of estimation of the memory parameter d which applies the notion of empirical mean squared displacement in order to test stability of the data we employ several statistical tests based on the empirical distribution function finally we also show that the returns possess no conditional heteroscedasticity property thus excluding the arch garch family of processes as possible underlying models

fokker planck description for a linear delayed langevin equation with additive gaussian noise we construct an equivalent probability description of linear multi delay langevin equations subject to additive gaussian white noise by exploiting the time convolutionless transform and a time variable transformation we are able to write a fokker planck equation fpe for the 1 time and for the 2 time probability distributions valid irrespective of the regime of stability of the langevin equations we solve exactly the derived fpes and analyze the aging dynamics by studying analytically the conditional probability distribution we discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non markov process as both preparation and observation times have bearing on its dynamics as our analytic procedure can also be applied to linear langevin equations with memory kernels we compare the non markov dynamics of a one delay system with that of a generalized langevin equation with an exponential as well as a power law memory application to a generalization of the green kubo formula is also presented

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

superbosonization of invariant random matrix ensembles superbosonization is a new variant of the method of commuting and anti commuting variables as used in studying random matrix models of disordered and chaotic quantum systems we here give a concise mathematical exposition of the key formulas of superbosonization conceived by analogy with the bosonization technique for dirac fermions the new method differs from the traditional one in that the superbosonization field is dual to the usual hubbard stratonovich field the present paper addresses invariant random matrix ensembles with symmetry group un on or uspn giving precise definitions and conditions of validity in each case the method is illustrated at the example of wegner s n orbital model superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non gaussian and or non invariant type

fractional brownian motion time changed by gamma and inverse gamma process abstract many real time series exhibit behavior adequate to long range dependent data additionally very often these time series have constant time periods and also have characteristics similar to gaussian processes although they are not gaussian therefore there is need to consider new classes of systems to model these kinds of empirical behavior motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena both of them are constructed as the superposition of fractional brownian motion fbm and other process in the first case the internal process which plays role of the time is the gamma process while in the second case the internal process is its inverse we present in detail their main properties paying main attention to the long range dependence property moreover we show how to simulate these processes and estimate their parameters we propose to use a novel method based on rescaled modified cumulative distribution function for estimation of parameters of the second considered process this method is very useful in description of rounded data like waiting times of subordinated processes delayed by inverse subordinators by using the monte carlo method we show the effectiveness of proposed estimation procedures finally we present the applications of proposed models to real time series

the speed of propagation for kpp type problems ii general domains this paper is devoted to nonlinear propagation phenomena in general unbounded domains of rn for reaction diffusion equations with kolmogorov petrovsky piskunov kpp type nonlinearities this article is the second in a series of two and it is the follow up of the paper 7 which dealt which the case of periodic domains this paper is concerned with general domains and we give various definitions of the spreading speeds at large times for solutions with compactly supported initial data we study the relationships between these new notions and analyze their dependence on the geometry of the domain and on the initial condition some a priori bounds are proved for large classes of domains the case of exterior domains is also discussed in detail lastly some domains which are very thin at infinity and for which the spreading speeds are infinite are exhibited the construction is based on some new heat kernel estimates in such domains

asymptotic spreading in heterogeneous diffusive excitable media abstract we establish propagation and spreading properties for nonnegative solutions of nonhomogeneous reaction diffusion equations of the type t u a t x u q t x u f t x u with compactly supported initial conditions at t 0 here a q f have a general dependence in t r and x r n we establish properties of families of propagation sets which are defined as families of subsets s t t 0 of r n such that lim inf t inf x s t u t x 0 the aim is to characterize such families as sharply as possible in particular we give some conditions under which 1 a given path t t 0 where t r n forms a family of propagation sets or 2 one can find such a family with s t x r n x r t and lim t r t this second property is called here complete spreading furthermore in the case q 0 and inf t x r r n f u t x 0 0 as well as under some more general assumptions we show that there is a positive spreading speed that is r t can be chosen so that lim inf t r t t 0 in the general case we also show the existence of an explicit upper bound c 0 such that lim sup t r t t c on the other hand we provide explicit examples of reaction diffusion equations such that for an arbitrary e 0 any family of propagation sets s t t 0 has to satisfy s t x r n x e t for large t in connection with spreading properties we derive some new uniqueness results for the entire solutions of this type of equations lastly in the case of space time periodic media we develop a new approach to characterize the largest propagation sets in terms of eigenvalues associated with the linearized equation in the neighborhood of zero

notes on the two dimensional fractional brownian motion we study the two dimensional fractional brownian motion with hurst parameter h 1 2 in particular we show using stochastic calculus that this process admits a skew product decomposition and deduce from this representation some asymptotic properties of the motion

on some free boundary problems of the prey predator model abstract in this paper we investigate some free boundary problems for the lotka volterra type prey predator model in one space dimension the main objective is to understand the asymptotic behavior of the two species prey and predator spreading via a free boundary we prove a spreading vanishing dichotomy namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment or they fail to establish and die out in the long run the long time behavior of solution and criteria for spreading and vanishing are also obtained finally when spreading successfully we provide an estimate to show that the spreading speed if exists cannot be faster than the minimal speed of traveling wavefront solutions for the prey predator model on the whole real line without a free boundary

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

laplace operators of infinite dimensional lie algebras and theta functions abstract until recently the generalized casimir operator constructed by kac kac v g 1974 funct anal appl 8 68 70 has been the only known element of the center of a completion of the enveloping algebra of a kac moody algebra it has been conjectured deodhar v v gabber o kac v g 1982 adv math 45 92 116 however that the image of the harish chandra homomorphism contains all theta functions defined on the interior of the complexified tits cone and hence separates the orbits of the weyl group developing the ideas of feigin and fuchs feigin b l fuchs d b 1983 dokl akad nauk sssr 269 1057 1060 i prove this conjecture another application of this method is the chevalley type restriction theorem for simple finite dimensional lie superalgebras

effect of tumor microenvironmental factors on tumor growth dynamics modeled by correlated colored noises with colored cross correlation the effect of non immunogenic tumor microenvironmental factors on tumor growth dynamics modeled by correlated additive and multiplicative colored noises is investigated using the novikov theorem fox approach and ansatz of hanggi an approximate fokker planck equation for the system is obtained and analytic expression for the steady state distribution pst x is derived based on the numerical results we find that fluctuations of microenvironmental factors within the tumor site with parameter have a diffusive effect on the tumor growth dynamics and the tumor response to the microenvironmental factors with parameter inhibits growth at weak correlation time moreover at increasing correlation time the inhibitive effect of tumor response is suppressed and instead a systematic growth promotion is noticed the result also reveals that the strength of the correlation time has a strong influence on the growth effects exerted by the non immunogenic component of tumor microenvironment on tumor growth

on a reaction diffusion equation with robin and free boundary conditions this paper studies the following problem ut uxx f u 0 0 u 0 t bux 0 t t 0 u h t t 0 h t ux h t t t 0 h 0 h0 u x 0 x 0 x h0 where f is an unbalanced bistable nonlinearity b 0 0 and is a compactly supported c2 function we prove that there exists 0 such that vanishing happens when 0 and u t converges as t to 0 uniformly in 0 h t spreading happens when i e h t c t tends to a constant for some c 0 u t converges to a positive stationary solution locally uniformly in 0 and to a traveling semi wave with speed c near x h t in the transition case when u t v t h2 0 h t tends to 0 as t where t is a maximum point of u t and v is the unique even positive solution of v f v 0 subject to v 0 moreover with respect to b and f t pln t q o 1 for some p 0 and q r or t z for some root z of v z bv z

free boundary problem of a reaction diffusion equation with nonlinear convection term abstract in this paper we consider the free boundary problem of a reaction diffusion equation with nonlinear convection term in one dimensional space our study contains three parts in the first part we establish the existence and uniqueness of global solution in the second part we obtain the spreading vanishing dichotomy and in third part we obtain some estimations of the asymptotic speed of free boundaries when spreading happens

diffusion and memory effects for stochastic processes and fractional langevin equations we consider the diffusion processes defined by stochastic differential equations when the noise is correlated a functional method based on the dyson expansion for the evolution operator associated to the stochastic continuity equation is proposed to obtain the fokker planck equation after averaging over the stochastic process in the white noise limit the standard result corresponding to the stratonovich interpretation of the non linear langevin equation is recovered when the noise is correlated the averaged operator series cannot be summed unless a family of time dependent operators commutes in the case of a linear equation the constraints are easily worked out the process defined by a linear langevin equation with additive noise is gaussian and the probability density function of its fluctuating component satisfies a fokker planck equation with a time dependent diffusion coefficient the same result holds for a linear langevin equation with a fractional time derivative defined according to caputo 1 in the generic linear or nonlinear case approximate equations for small noise amplitude are obtained for small correlation time the evolution equations further simplify in agreement with some previous alternative derivations the results are illustrated by the linear oscillator with coloured noise and the fractional wiener process where the numerical simulation for the probability density and its moments is compared with the analytical solution

asymptotic spreading of a diffusive competition model with different free boundaries abstract recently the authors of 22 studied a diffusive prey predator model with two different free boundaries they first obtained the existence uniqueness regularity uniform estimates and long time behaviors of global solution and then established the conditions for spreading and vanishing especially when spreading occurs they provided accurate limits of two species as t and gave some estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries motivated by the paper 22 in this paper we discuss the diffusive competition model with two different free boundaries which had been investigated by 7 11 15 21 the main purpose of this paper is to establish much sharper estimates of asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries when spreading occurs furthermore how the solution approaches the semi wave when spreading happens is also described

spherical representations of lie supergroups abstract the classical cartan helgason theorem characterises finite dimensional spherical representations of reductive lie groups in terms of their highest weights we generalise the theorem to the case of a reductive symmetric supergroup pair g k of even type along the way we compute the harish chandra c function of the symmetric superspace g k by way of an application we show that in type a iii a iii all spherical representations are self dual

dynamics for a diffusive prey predator model with different free boundaries abstract to understand the spreading and interaction of prey and predator in this paper we study the dynamics of the diffusive lotka volterra type prey predator model with different free boundaries these two free boundaries which may intersect each other as time evolves are used to describe the spreading of prey and predator we investigate the existence and uniqueness regularity and uniform estimates and long time behaviors of global solution some sufficient conditions for spreading and vanishing are established when spreading occurs we provide the more accurate limits of u v as t and give some estimates of asymptotic spreading speeds of u v and asymptotic speeds of g h some realistic and significant spreading phenomena are found

diffusion of finite sized hard core interacting particles in a one dimensional box tagged particle dynamics we solve a nonequilibrium statistical mechanics problem exactly namely the single file dynamics of n hard core interacting particles the particles cannot pass each other of size delta diffusing in a one dimensional system of finite length l with reflecting boundaries at the ends we obtain an exact expression for the conditional probability density function rho t yt t vertical bar yt 0 that a tagged particle t t 1 n is at position yt at time t given that it at time t 0 was at position yt 0 using a bethe ansatz we obtain the n particle probability density function and by integrating out the coordinates and averaging over initial positions of all particles but particle t we arrive at an exact expression for rho t yt t vertical bar yt 0 in terms of jacobi polynomials or hypergeometric functions going beyond previous studies we consider the asymptotic limit of large n maintaining l finite using a nonstandard asymptotic technique we derive an exact expression for rho t yt t vertical bar yt 0 for a tagged particle located roughly in the middle of the system from which we find that there are three time regimes of interest for finite sized systems a for times much smaller than the collision time t tau coll 1 rho d 2 where rho n l is the particle concentration and d is the diffusion constant for each particle the tagged particle undergoes a normal diffusion b for times much larger than the collision time t tau coll but times smaller than the equilibrium time t tau eq l 2 d we find a single file regime where rho t yt t vertical bar yt 0 is a gaussian with a mean square displacement scaling as t 1 2 and c for times longer than the equilibrium time t tau eq rho t yt t vertical bar yt 0 approaches a polynomial type equilibrium probability density function notably only regimes a and b are found in the previously considered infinite systems

correlations in a generalized elastic model fractional langevin equation approach the generalized elastic model gem provides the evolution equation which governs the stochastic motion of several many body systems in nature such as polymers membranes and growing interfaces on the other hand a probe tracer particle in these systems performs a fractional brownian motion due to the spatial interactions with the other system s components the tracer s anomalous dynamics can be described by a fractional langevin equation fle with a space time correlated noise we demonstrate that the description given in terms of gem coincides with that furnished by the relative fle by showing that the correlation functions of the stochastic field obtained within the fle framework agree with the corresponding quantities calculated from the gem furthermore we show that the fox h function formalism appears to be very convenient to describe the correlation properties within the fle approach

dynamics for the diffusive mutualist model with advection and different free boundaries abstract in this paper we study the diffusive mutualist model with advection and different free boundaries in one space dimension these two free boundaries may intersect each other as time evolves and can be used to describe the spreading of invasive and native species directly we first prove the existence and uniqueness regularity and uniform estimates of global solution then we provide the criteria governing spreading and vanishing at last we investigate long time behaviors and asymptotic spreading speeds of two species and asymptotic speeds of two free boundaries

spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection diffusion equations abstract in this study we consider free boundary problems for nonlinear advection diffusion equations of the form u t u x x u x f u for t 0 g t x h t where x g t and x h t are free boundaries this problem may be used to describe the spreading of a biological or chemical species where the free boundaries represent the expanding fronts when f is a logistic nonlinearity it has been shown that the asymptotic spreading speeds of the two fronts h t and g t are different due to the advection term in this study for monostable bistable and combustion types nonlinearities we give much sharper estimates of the different spreading speeds of the fronts and we also prove that the solution converges to a semi wave in c 2 norm as t when spreading occurs we develop new approaches and extend a previous result to the problem with the advection term

non markovian levy diffusion in nonhomogeneous media we study the diffusion equation with a position dependent power law diffusion coefficient the equation possesses the riesz weyl fractional operator and includes a memory kernel it is solved in the diffusion limit of small wave numbers two kernels are considered in detail the exponential kernel for which the problem resolves itself to the telegrapher s equation and the power law one the resulting distributions have the form of the l evy process for any kernel the renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system

winding angle distributions for random walks and flux lines we study analytically and numerically the winding of a flux line around a columnar defect reflecting and absorbing boundary conditions apply to marginal or repulsive defects respectively in both cases the winding angle distribution decays exponentially for large angles with a decay constant depending only on the boundary condition but not on microscopic features nonuniversal distributions are encountered for chiral defects that preferentially twist the flux line in one direction the resulting asymmetric distributions have decay constants that depend on the degree of chirality in particular strong chirality encourages entanglements and leads to broad distributions we also examine the windings of flux lines in the presence of point impurities random bonds our results suggest that pinning to impurities reduces entanglements leading to a narrow gaussian distribution textcopyright 1996 the american physical society

fractional langevin equation overdamped underdamped and critical behaviors the dynamical phase diagram of the fractional langevin equation is investigated for a harmonically bound particle it is shown that critical exponents mark dynamical transitions in the behavior of the system four different critical exponents are found i ensuremath alpha c 0 402 ifmmode pm else textpm fi 0 002 marks a transition to a nonmonotonic underdamped phase ii ensuremath alpha r 0 441 dots marks a transition to a resonance phase when an external oscillating field drives the system and iii ensuremath alpha ensuremath chi 1 0 527 dots and iv ensuremath alpha ensuremath chi 2 0 707 dots mark transitions to a double peak phase of the loss when such an oscillating field present as a physical explanation we present a cage effect where the medium induces an elastic type of friction phase diagrams describing over and underdamped regimes with or without resonances show behaviors different from normal

winding number of fractional brownian motion we find the exact winding number distribution of riemann liouville fractional brownian motion for large times in two dimensions using the propagator of a free particle the distribution is similar to the brownian motion case and is of cauchy type in addition we find the winding number distribution of the fractal time process i e the time fractional fokker planck equation in the presence of a finite size winding center

chevalley s restriction theorem for reductive symmetric superpairs let g k be a reductive symmetric superpair of even type i e so that there exists an even cartan subspace a p the restriction map s p k s a w where w w g0 a is the weyl group is injective we determine its image explicitly in particular our theorem applies to the case of a symmetric superpair of group type i e k k k with the flip involution where k is a classical lie superalgebra with a non degenerate invariant even form equivalently a finite dimensional contragredient lie superalgebra thus we obtain a new proof of the generalisation of chevalley s restriction theorem due to sergeev and kac gorelik for general symmetric superpairs the invariants exhibit a new and surprising behaviour we illustrate this phenomenon by a detailed discussion in the example g c q 1 osp 2 2q c endowed with a special involution here the invariant algebra defines a singular algebraic curve

fractional langevin equation we investigate fractional brownian motion with a microscopic random matrix model and introduce a fractional langevin equation we use the latter to study both subdiffusion and superdiffusion of a free particle coupled to a fractal heat bath we further compare fractional brownian motion with the fractal time process the respective mean square displacements of these two forms of anomalous diffusion exhibit the same power law behavior here we show that their lowest moments are actually all identical except the second moment of the velocity this provides a simple criterion that enable us to distinguish these two non markovian processes

different spreading speeds in a weak competition model with two free boundaries abstract in this paper we focus on the spreading behavior of a two species weak competition system with two free boundaries in a spherically symmetric setting this model may describe how two weakly competing species spread into the right empty space when both two species spread successfully and one species cannot keep pace with the other one it can be expected that two species spread asymptotically in a linear fashion but spread with different speeds by the help of a semi wave system and a construction of supersolution and subsolution we establish the precise spreading speeds and population distribution of two weakly competing species which provides a new spreading behavior

systems with nonlocal vs local diffusions and free boundaries abstract we study a class of free boundary problems of ecological models with nonlocal and local diffusions which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal with the free boundary representing the spreading front of the species we first prove the existence uniqueness and regularity of global solution for the classical competition prey predator and mutualist models we show that a spreading vanishing dichotomy holds and establish the criteria of spreading and vanishing and long time behavior of the solution

integration questions related to fractional brownian motion let b h u u be a fractional brownian motion fbm with index h 0 1 and b h be the closure in l 2 of the span sp b h of the increments of fbm b h it is well known that when b h b 1 2 is the usual brownian motion bm an element x b 1 2 can be characterized by a unique function f x l 2 in which case one writes x in an integral form as x f x u db 1 2 u from a different though equivalent perspective the space l 2 forms a class of integrands for the integral on the real line with respect to bm b 1 2 in this work we explore whether a similar characterization of elements of b h can be obtained when h 0 1 2 or h 1 2 1 since it is natural to define the integral of an elementary function f k 1 n f k 1 uk uk 1 by k 1 n f k b h u k 1 b h u k we want the spaces of integrands to contain elementary functions these classes of integrands are inner product spaces if the space of integrands is not complete then it characterizes only a strict subset of b h when 0 h 1 2 by using the moving average representation of fbm b h we construct a complete space of integrands when 1 2 h 1 however an analogous construction leads to a space of integrands which is not complete when 0 h 1 2 or 1 2 h 1 we also consider a number of other spaces of integrands while smaller and henceincomplete they form a natural choice and are convenient to workwith we compare these spaces of integrands to the reproducing kernel hilbert space of fbm