spatial pattern formation in external noise theory and simulation spatial pattern formation in fluctuating media is researched analytically from the point of view of the order parameters concept a reaction diffusion system with external noise is considered as a model of such media stochastic equations for unstable mode amplitudes order parameters the dispersion equation for averaged amplitudes of unstable modes and the fokker planck equation for the order parameters are obtained the theory developed makes it possible to analyze different noise induced effects including the variation of boundaries of ordering and disordering phase transitions depending on the parameters of external noise

long time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise stratonovitch s case in this article we prove new results concerning the long time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non random global attractor which consists of two spatially and temporally homogeneous asymptotic states more precisely we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states or that they set out to oscillate between them in the first case we can also determine exactly the corresponding lyapunov exponents in the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one in both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states our methods of proof rest upon the use of a suitable regularization of the brownian motion along with a related wong zakai approximation procedure

spatially periodic stochastic system with infinite globally coupled oscillators in this paper we study a spatially periodic stochastic system with infinite globally coupled oscillators driven by a constant force f with two typical models we show that when f 0 there is a nonequilibrium transition between the state with zero mean field s 0 and the state with nonzero mean field s not equal 0 for model i the transition is not a phase transition while for the model ii it is second order in addition we find that for coupled oscillators driven only by additive noises when f 0 a transport may emerge if the nonzero mean field breaks the symmetry of the systems with varying f a continuous or discontinuous transition between state s 0 and state s 0 will appear the mean field or current sometimes exhibits hysteresis as a function of f

dynamic mean field models h theorem for stochastic processes and basins of attraction of stationary processes abstract stochastic processes of dynamic mean field models are studied in terms of nonlinear fokker planck equations we show that h theorems for single time point distributions can be used to derive h theorems for hierarchies of joint distributions in doing so we prove the convergence of transient stochastic processes to stationary ones for multistable systems we furthermore determine the basins of attraction of these stationary processes our results are illustrated by means of a mean field model exhibiting bistability

external fluctuations in a pattern forming instability the effect of external fluctuations on the formation of spatial patterns is analyzed by means of a stochastic swift hohenberg model with multiplicative space correlated noise numerical simulations in two dimensions show a shift of the bifurcation point controlled by the intensity of the multiplicative noise this shift takes place in the ordering direction i e produces patterns but its magnitude decreases with that of the noise correlation length analytical arguments are presented to explain these facts

option pricing formulas based on a non gaussian stock price model options are financial instruments that depend on the underlying stock we explain their non gaussian fluctuations using the nonextensive thermodynamics parameter q a generalized form of the black scholes bs partial differential equation and some closed form solutions are obtained the standard bs equation q 1 which is used by economists to calculate option prices requires multiple values of the stock volatility known as the volatility smile using q 1 5 which well models the empirical distribution of returns we get a good description of option prices using a single volatility

large time behavior of interface solutions to the heat equation with fisher wright white noise the one dimensional heat equation driven by fisher wright white noise is studied from initial conditions with compact support solutions retain this compact support and die out in finite time there exist interface solutions which change from the value 1 to the value 0 in a finite region the motion of the interface location is shown to approach that of a brownian motion under rescaling solutions with a finite number of interfaces are approximated by a system of annihilating brownian motions

singular invariant tempered distributions on regular prehomogeneous vector spaces revue des espaces vectoriels prehomogenes distributions temperees invariantes distributions temperees avec un parametre meromorphe puissances complexes des invariants relatifs soit p x un polynome homogene irreductible a coefficients reels sur un espace vectoriel v soit s x v p x 0 se decompose en un nombre fini de g 1 orbites avec g 1 g gl v p g x p x on pose v 1 uv 2 u uv e v s la decomposition en composantes connexes on donne une condition necessaire et suffisante pour qu une distribution temperee g 1 invariante a support dans s puisse s obtenir comme une combinaison lineaire finie de coefficients de laurent negatifs de la distribution p x s vd i

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

abelian manna model in three dimensions and below the abelian manna model of self organized criticality is studied on various three dimensional and fractal lattices the exponents for avalanche size duration and area distribution of the model are obtained by using a high accuracy moment analysis together with earlier results on lower dimensional lattices the present results reinforce the notion of universality below the upper critical dimension and allow us to determine the coefficients of an e expansion by rescaling the critical exponents by the lattice dimension and incorporating the random walker dimension a remarkable relation is observed satisfied by both regular and fractal lattices

population dynamics under the laplace assumption in this paper we describe a generic approach to modelling dynamics in neuronal populations this approach models a full density on the states of neuronal populations but finesses this high dimensional problem by re formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered c f the method of moments the particular form for the population density we adopt is a gaussian density c f the laplace assumption this means population dynamics are described by equations governing the evolution of the population s mean and covariance we derive these equations from the fokker planck formalism and illustrate their application to a conductance based model of neuronal exchanges one interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural mass model which rests only on the populations mean this enables us to compare equivalent mean field and neural mass models of the same populations and evaluate quantitatively the contribution of population variance to the expected dynamics the mean field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work

mapping multiplicative to additive noise the langevin formulation of a number of well known stochastic processes involves multiplicative noise in this work we present a systematic mapping of a process with multiplicative noise to a related process with additive noise which may often be easier to analyse the mapping is easily understood in the example of the branching process in a second example we study the random neighbour or infinite range contact process which is mapped to an ornstein uhlenbeck process with absorbing wall the present work might shed some light on absorbing state phase transitions in general such as the role of conditional expectation values and finite size scaling and elucidate the meaning of the noise amplitude while we focus on the physical interpretation of the mapping we also provide a mathematical derivation

macroscopic limit cycle via pure noise induced phase transitions bistability generated via a pure noise induced phase transition is reexamined from the view of bifurcations in macroscopic cumulant dynamics it allows an analytical study of the phase diagram in more general cases than previous methods in addition using this approach we investigate spatially extended systems with two degrees of freedom per site for this system the analytic solution of the stationary fokker planck equation is not available and a standard mean field approach cannot be used to find noise induced phase transitions a different approach based on cumulant dynamics predicts a noise induced phase transition through a hopf bifurcation leading to a macroscopic limit cycle motion which is confirmed by numerical simulation

population dynamics variance and the sigmoid activation function this paper demonstrates how the sigmoid activation function of neural mass models can be understood in terms of the variance or dispersion of neuronal states we use this relationship to estimate the probability density on hidden neuronal states using non invasive electrophysiological eeg measures and dynamic casual modelling the importance of implicit variance in neuronal states for neural mass models of cortical dynamics is illustrated using both synthetic data and real eeg measurements of sensory evoked responses

stochastic resonance between dissipative structures in a bistable noise sustained dynamics we study an extended system that without noise shows a monostable dynamics but when submitted to an adequate multiplicative noise an effective bistable dynamics arises the stochastic resonance between the attractors of the noise sustained dynamicsis investigated theoretically in terms of a two state approximation the knowledge of the exact nonequilibrium potential allows us to obtain the output signal to noise ratio its maximum is predicted in the symmetric case for which both attractors have the same nonequilibrium potential value

nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators a mean field model of nonlinearly coupled oscillators with randomly distributed frequencies and subject to independent external white noises is analyzed in the thermodynamic limit when the frequency distribution isbimodal new results include subcritical spontaneous stationary synchronization of the oscillators supercritical time periodic synchronization bistability and hysteretic phenomena bifurcating synchronized states are asymptotically constructed near bifurcation values of the coupling strength and theirnonlinear stability properties ascertained

numerical method for the nonlinear fokker planck equation a practical method based on distributed approximating functionals dafs is proposed for numerically solving a general class of nonlinear time dependent fokker planck equations the method relies on a numerical scheme that couples the usual path integral concept to the daf idea the high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear fokker planck equation and the method is compared with the accurate k point stirling interpolation formula finite difference method the approach is also used successfully to solve a nonlinear self consistent dynamic mean field problem for which both the cumulant expansion and scaling theory have been found by drozdov and morillo phys rev e 54 931 1996 to be inadequate to describe the occurrence of a long lived transient bimodality the standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

multivariate ornstein uhlenbeck processes with mean field dependent coefficients application to postural sway we study the transient and stationary behavior of many particle systems in terms of multivariate ornstein uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields mean field approximations of this kind of system are derived in terms of fokker planck equations in such systems multiple stationary solutions as well as bifurcations of stationary solutions may occur in addition strictly monotonically decreasing steady state autocorrelation functions that decay faster than exponential functions are found which are used to describe the erratic motion of the center of pressure during quiet standing

comprehensive study of phase transitions in relaxational systems with field dependent coefficients we present a comprehensive study of phase transitions in single field systems that relax to a nonequilibrium global steady state the mechanism we focus on is not the so called stratonovich drift combined with collective effects but is instead similar to the one associated with noise induced transitions in the manner of horsthemke lefever in zero dimensional systems as a consequence the noise interpretation e g ito vs stratonovich merely shifts the phase boundaries with the help of a mean field approximation we present a broad qualitative picture of the various phase diagrams that can be found in these systems to complement the theoretical analysis we present numerical simulations that confirm the findings of the mean field theory

on a general link between anomalous diffusion and nonextensivity stochastic processes of systems with composable entropy measures are studied the processes are described by fokker planck equations which are nonlinear with respect to their probability densities and whose diffusion coefficients are derived from the entropy measures in questions in nonextensive extensive systems anomalous normal diffusion is found

hyperscaling in the domany kinzel cellular automaton an apparent violation of hyperscaling at the endpoint of the critical line in the domany kinzel stochastic cellular automaton finds an elementary resolution upon noting that the order parameter is discontinuous at this point we derive a hyperscaling relation for such transitions and discuss applications to related examples

nonequilibrium phase transitions induced by multiplicative noise we review a mean field analysis and give the details of a correlation function approach for spatially distributed systems subject to multiplicative noise white in space and time we confirm the existence of a pure noise induced reentrant nonequilibrium phase transition in the model introduced in c van den broeck et al phys rev lett 73 3395 1994 give an intuitive explanation of its origin and present extensive simulations in dimension d52 the observed critical properties are compatible with those of the ising universality class s1063 651x 97 08704 7

additive noise induced turing transitions in spatial systems with application to neural fields and the swift hohenberg equation this work studies the spatio temporal dynamics of a generic integral differential equation subject to additive random fluctuations it introduces a combination of the stochastic center manifold approach for stochastic differential equations and the adiabatic elimination for fokker planck equations and studies analytically the systems stability near turing bifurcations in addition two types of fluctuation are studied namely fluctuations uncorrelated in space and time and global fluctuations which are constant in space but uncorrelated in time we show that the global fluctuations shift the turing bifurcation threshold this shift is proportional to the fluctuation variance applications to a neural field equation and the swift hohenberg equation reveal the shift of the bifurcation to larger control parameters which represents a stabilization of the system all analytical results are confirmed by numerical simulations of the occurring mode equations and the full stochastic integral differential equation to gain some insight into experimental manifestations the sum of uncorrelated and global additive fluctuations is studied numerically and the analytical results on global fluctuations are confirmed qualitatively

stochastic resonance in a mean field model of cooperative behavior we study the long time response of a stochastic system formed by very many interacting subsystems coupled by a mean field interaction and subject to a time periodic external field in the absence of a driving field the system shows an order disorder phase transition and its time evolution is well described by a fokker planck equation which is nonlinear in the probability density we carry out an analysis of the dynamics in the case of a weak driving field by means of a perturbation analysis linear response theory the response of the system is then given in terms of a generalized susceptibility its evaluation shows that the phenomenon of stochastic resonance typical of driven bistable systems is greatly enhanced by the dynamical feedback induced by mean field coupling

d modules on representations of capelli type abstract let g v be an irreducible multiplicity free finite dimensional representation of a connected reductive complex group g as classified by v g kac 17 and g its derived subgroup denote by g the lie algebra of g and u g its universal enveloping algebra assume that there exists a polynomial f generating the algebra of g invariant polynomials on v c v g c f and such that f c v g such representations are said to be of capelli type if the algebra of g invariant differential operators is the image of the center of u g under the differential of the g action they fall into eight cases given by r howe and t umeda 14 five infinite families and three exceptional examples we prove that the category of regular holonomic d v modules invariant under the action of g is equivalent to the category of graded modules of finite type over a suitable algebra a except for few special cases indeed the levasseur s conjecture 28 conjecture 5 17 p 508 fails in these cases because of the disconnectedness of the stabilizers of some smaller orbits

comparison methods for a class of function valued stochastic partial differential equations a comparison theorem is derived for a class of function valued stochastic partial differential equations spde s with lipschitz coefficients driven by cylindrical and regular hilbert space valued brownian motions moreover we obtain necessary and sufficient conditions for the positivity of the mild solutions of the spde s where the sufficiency follows from the comparison theorem thereby it is e g possible to identify a class of spde s which can serve as stochastic space time models for the density of particles as a consequence we can construct unique mild solutions of spde s on the cone of positive functions with non lipschitz drift parts including the case of arbitrary polynomialsr x withr o o and leading negative coefficient

nonlinear fokker planck equations whose stationary solutions make entropy like functionals stationary abstract the present study extends the correspondence principle of martinez et al that establishes a link between nonlinear fokker planck equations nlfpes and the variational principle approach of the theory of canonical ensembles by virtue of the extended correspondence principle we reobtain results of kaniadakis and quarati for bose and fermi systems and find relations similar to those derived by plastino and plastino and recently by borland concerning the entropy of nonextensive thermostatistics introduced in physics by tsallis moreover we propose nlfpes related to the renyi entropy and to the entropy proposed by sharma and mittal the latter comprises tsallis entropy and the renyi entropy

stability analysis of stationary states of mean field models described by fokker planck equations abstract so far the stability of stationary solutions of mean field models described by fokker planck equations has only been determined analytically in some special cases following two earlier studies j korean phys soc 40 2002 1037 prog theoret phys suppl 150 2003 48 we discuss a stability analysis for a large class of mean field models based on fokker planck equations to this end we use linear stability analysis in addition to the well known approaches by means of transcendent equations and lyapunov s direct method we demonstrate that all three methods yield consistent results for systems that exhibit free energy functionals e g equilibrium systems we show that the simple transcendent equation analysis fails for systems that do not exhibit free energy functionals e g nonequilibrium systems and show how to solve this problem by means of a more sophisticated transcendent equation analysis furthermore we propose a norm for the perturbations of stationary states and illustrate some of our results by a model that exhibits a reentrant noise induced phase transition

airy distribution function from the area under a brownian excursion to the maximal height of fluctuating interfaces the airy distribution function describes the probability distribution of the area under a brownian excursion over a unit interval surprisingly this function has appeared in a number of seemingly unrelated problems mostly in computer science and graph theory in this paper we show that this distribution function also appears in a rather well studied physical system namely the fluctuating interfaces we present an exact solution for the distribution p h m l of the maximal height h m measured with respect to the average spatial height in the steady state of a fluctuating interface in a one dimensional system of size l with both periodic and free boundary conditions for the periodic case we show that p h m l l 1 2f h m l 1 2 for all l 0 where the function f x is the airy distribution function this result is valid for both the edwards wilkinson ew and the kardar parisi zhang interfaces for the free boundary case the same scaling holds p h m l l 1 2f h m l 1 2 but the scaling function f x is different from that of the periodic case we compute this scaling function explicitly for the ew interface and call it the f airy distribution function numerical simulations are in excellent agreement with our analytical results our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables some of these results were announced in a recent letter s n majumdar and a comtet phys rev lett 92 225501 2004

mean field limit of systems with multiplicative noise a detailed study of the mean field solution of langevin equations with multiplicative noise is presented three different regimes depending on noise intensity weak intermediate and strong noise are identified by performing a self consistent calculation on a fully connected lattice the most interesting strong noise regime is shown to be intrinsically unstable with respect to the inclusion of fluctuations as a ginzburg criterion shows on the other hand the self consistent approach is shown to be valid only in the thermodynamic limit while for finite systems the critical behavior is found to be different in this last case the self consistent field itself is broadly distributed rather than taking a well defined mean value its fluctuations described by an effective zero dimensional multiplicative noise equation govern the critical properties these findings are obtained analytically for a fully connected graph and verified numerically both on fully connected graphs and on random regular networks the results presented here shed some doubt on what is the validity and meaning of a standard mean field approach in systems with multiplicative noise in finite dimensions where each site does not see an infinite number of neighbors but a finite one the implications of all this on the existence of a finite upper critical dimension for multiplicative noise and kardar parisi zhang problems are briefly discussed

the boundary of the numerical range of matrix polynomials abstract some algebraic properties of the sharp points of the numerical range of matrix polynomials are the main subject of this paper we also consider isolated points of the numerical range and the location of the numerical range in a circular annulus

constructing a homogeneous lti descriptor system with desired properties using perturbation theory in the literature regarding linear systems and mathematical control theory several different techniques have been developed for obtaining the solution of homogeneous linear time invariant lti descriptor differential systems in this article applying the complex weierstrass canonical form we investigate the conditions under which a descriptor system with a specific structure and desired properties is being constructed using perturbation theory our approach is very general and as an example a stable homogeneous lti descriptor system is designed thus a proportional and derivative controller can be used such as the case where a family of perturbed pencils is defined and the solutions of the initial and the relative perturbed systems are close with respect to a frobenius distance a step algorithm and an illustrative example are also presented to illustrate the results of this article

stochastic order parameter equation of isometric force production revealed by drift diffusion estimates we address two questions that are central to understanding human motor control variability what kind of dynamical components contribute to motor control variability i e deterministic and or random ones and how are those components structured to this end we derive a stochastic order parameter equation for isometric force production from experimental data using drift diffusion estimates we show that the force variability increases with the required force output because of a decrease of deterministic stability and an accompanying increase of noise intensity a structural analysis reveals that the deterministic component consists of a linear control loop while the random component involves a noise source that scales with force output in addition we present evidence for the existence of a subject independent overall noise level of human isometric force production

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

stability of incoherence in a population of coupled oscillators we analyze a mean field model of coupled oscillators with randomly distributed frequencies this system is known to exhibit a transition to collective oscillations for small coupling the system is incoherent with all the oscillators running at their natural frequencies but when the coupling exceeds a certain threshold the system spontaneously synchronizes we obtain the first rigorous stability results for this model by linearizing the fokker planck equation about the incoherent state an unexpected result is that the system has pathological stability properties the incoherent state is unstable above threshold butneutrally stable below threshold we also show that the system is singular in the sense that its stability properties are radically altered by infinitesimal noise

phase separation in binary systems with internal multiplicative noise abstract a study of a phase separation process in stochastic systems with a field dependent kinetic coefficient and an internal multiplicative noise is presented dynamics of spinodal decomposition at early and late stages is investigated by computer simulations where the domain growth law is generalized a mean field approach was carried out in order to obtain the stationary probability bifurcation and phase diagrams displaying reentrant phase transitions we relate our approach to entropy driven phase transitions theory

the stampacchia maximum principle for stochastic partial differential equations and applications abstract stochastic partial differential equations spdes are considered linear and nonlinear for which we establish comparison theorems for the solutions or positivity results a e and a s for suitable data comparison theorems for spdes are available in the literature the originality of our approach is that it is based on the use of truncations following the stampacchia approach to maximum principle we believe that our method which does not rely too much on probability considerations is simpler than the existing approaches and to a certain extent more directly applicable to concrete situations among the applications boundedness results and positivity results are respectively proved for the solutions of a stochastic boussinesq temperature equation and of reaction diffusion equations perturbed by a non lipschitz nonlinear noise stabilization results to a chafee infante equation perturbed by a nonlinear noise are also derived

nonlinear fokker planck equation exhibiting bifurcation phenomena and generalized thermostatistics a nonlinear fokker planck equation exhibiting bifurcation phenomena is proposed within the framework of generalized thermostatistics the nonlinearity responsible for the occurrence of bifurcation of solutions is assumed to be of the form appearing in the standard mean field model a liapunov function is defined that takes the form of free energy involving generalized entropies of tsallis and an h theorem is proved to show that the free energy which is bounded below continues to decrease until the system approaches one of the equilibrium distributions the h theorem ensures instead of uniqueness of the equilibrium distribution global stability of the system in that either one of multisolutions must be approached for large times local stability analysis is conducted and the second order variation of the liapunov function is computed to find its relevant part whose sign governs stability of the equilibrium distribution of the system the case with a bistable potential is investigated as an example of confirming the theory to give the bifurcation diagram displaying the order parameter as a function of the coefficient of the nonlinear diffusion term

noise induced phase separation mean field results we present a study of a phase separation process induced by the presence of spatially correlated multiplicative noise we develop a mean field approach suitable for conserved order parameter systems and use it to obtain the phase diagram of the model mean field results are compared with numerical simulations of the complete model in two dimensions additionally a comparison between the noise driven dynamics of conserved and nonconserved systems is made at the level of the mean field approximation

a langevin approach for the microscopic dynamics of nonlinear fokker planck equations abstract for stochastic processes defined by nonlinear fokker planck equations a microscopic dynamics is proposed in terms of generalized langevin equations these langevin equations can for example be used to model stochastic processes with mean field interactions and random walks related to the generalized thermostatistics suggested by tsallis

systems with multiplicative noise critical behavior from kpz equation and numerics we show that certain critical exponents of systems with multiplicative noise can be obtained from exponents of the kpz equation numerical simulations in 1d confirm this prediction and yield other exponents of the multiplicative noise problem the numerics also verify an earlier prediction of the divergence of the susceptibility over an entire range of control parameter values and show that the exponent governing the divergence in this range varies continuously with control parameter

multivariate markov processes for stochastic systems with delays application to the stochastic gompertz model with delay using the method of steps we describe stochastic processes with delays in terms of markov diffusion processes thus multivariate langevin equations and fokker planck equations are derived for stochastic delay differential equations natural periodic and reflective boundary conditions are discussed both ito and stratonovich calculus are used in particular our fokker planck approach recovers the generalized delay fokker planck equation proposed by guillouzic et al the results obtained are applied to a model for population growth the gompertz model with delay and multiplicative white noise

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

noise driven mechanism for pattern formation tion we modify the coupling term of the model in such a way that a morphological instability can appear we will show by means of numerical simulations and analytic calculations that an increasingly ordered spatial structure develops as the intensity of the fluctuation increases beyond a critical value moreover we show that for sufficiently high noise intensity there is nonmonotonic behavior as a function of the coupling that is there exists an optimal value of the coupling for which the ordered structure exhibits maximum coherence the paper is organized as follows we introduce the model in sec ii and explicitly show that no short time instability can drive pattern formation in this system in sec iii and the appendix we present a modulated mean field theory and the phase diagram of the model obtained from this theory numerical simulations that confirm the qualitative validity of the theoretical results are presented in sec iv finally we summarize our main conclusions in sec v

associating matrix pencils to generalized linear multivariable systems we consider quadruples of matrices e a b c representing generalized linear multivariable systems ex t ax t bu t y t cx t with e a square matrices and b c rectangular matrices we characterize equivalent quadruples by associating matrix pencils to them with respect to the equivalence relation corresponding to standard transformations basis changes for the state control and output spaces state feedback derivative feedback and output injection equivalent quadruples are those whose associated matrix pencils are simultaneously equivalent

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

local martingale and pathwise solutions for an abstract fluids model we establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system the primary application of this abstract framework is to infer the local existence of strong pathwise solutions to the 3d primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise instead of developing our results specifically for the 3d primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations atmosphere oceans coupled atmosphere ocean on the sphere on the beta plane approximation etc and the incompressible navier stokes equations in applications all of the details are given for the beta plane approximation of the oceans equations

invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices the real special linear group of degree n naturally acts on the vector space of n n real symmetric matrices how to determine invariant hyperfunction solutions of invariant linear differential equations with polynomial coefficients on the vector space of n n real symmetric matrices is discussed in this paper we prove that every invariant hyperfunction solution is expressed as a linear combination of laurent expansion coefficients of the complex power of the determinant function with respect to the parameter of the power then the problem is reduced to the determination of laurent expansion coefficients

on population resilience to external perturbations we study a spatially explicit harvesting model in periodic or bounded environments the model is governed by a parabolic equation with a spatially dependent nonlinearity of kolmogorov petrovsky piskunov type and a negative external forcing term delta using sub and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators we obtain existence and nonexistence results as well as results on the number of stationary solutions we also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude in particular we define two critical values delta and delta 2 such that if delta is smaller than delta the population density converges to a significant state which is everywhere above a certain small threshold whereas if delta is larger than delta 2 the population density converges to a remnant state everywhere below this small threshold our results are shown to be useful for studying

phase ordering dynamics with an order parameter dependent mobility the large n limit the effect of an order parameter dependent mobility or kinetic coefficient on the phase ordering dynamics of a system described by an n component vector order parameter is addressed at zero temperature in the large n limit we consider cases in which the mobility or kinetic coefficient vanishes when the magnitude of the order parameter takes its equilibrium value in the large n limit the system is exactly soluble for both conserved and non conserved order parameter in the non conserved case the scaling form for the correlation function and it s fourier transform the structure factor is established with the characteristic length growing as a power of time in the conserved case the structure factor is evaluated and found to exhibit a multi scaling behaviour with two growing length scales differing by a logarithmic factor in both cases the rate of growth of the length scales depends on the manner in which the mobility or kinetic coefficient vanishes as the magnitude of the order parameter approaches its equilibrium value

numerical range of linear pencils abstract consider a linear pencil a b where a and b are n n complex matrices the numerical range of a b is defined as w a b c x a b x 0 for some nonzero x c n in this paper we study the geometrical properties of w a b with emphasis to its boundary an answer to the problem of the numerical approximation of w a b when one of the coefficients a and b is hermitian is presented the numerical range of a matrix on an indefinite inner product space is also considered

local and global existence of smooth solutions for the stochastic euler equations with multiplicative noise we establish the local existence of pathwise solutions for the stochastic euler equations in a three dimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise in the two dimensional case we obtain the global existence of these solutions with additive or linear multiplicative noise finally we show that in the three dimensional case the addition of linear multiplicative noise provides a regularizing effect the global existence of solutions occurs with high probability if the initial data is sufficiently small or if the noise coefficient is sufficiently large

simple unified view of branching process statistics random walks in balanced logarithmic potentials we revisit the problem of deriving the mean field values of avalanche exponents in systems with absorbing states these are well known to coincide with those of unbiased branching processes here we show that for at least four different universality classes directed percolation dynamical percolation the voter model or compact directed percolation class and the manna class of stochastic sandpiles this common result can be obtained by mapping the corresponding langevin equations describing each of them into a random walker confined to the origin by a logarithmic potential we report on the emergence of nonuniversal continuously varying exponent values stemming from the presence of small external driving that might induce avalanche merging that to the best of our knowledge has not been noticed in the past many of the other results derived here appear in the literature as independently derived for individual universality classes or for the branching process itself still we believe that a simple and unified perspective as the one presented here can help 1 clarify the overall picture 2 underline the superuniversality of the behavior as well as the dependence on external driving and 3 avoid the common existing confusion between unbiased branching processes equivalent to a random walker in a balanced logarithmic potential and standard unconfined random walkers

on the linear discrepancy model and risky shifts in group behavior a nonlinear fokker planck perspective using a nonlinear fokker planck perspective we re formulate the linear discrepancy model proposed by boster and colleagues that describes the emergence of risky shifts during group decision making analytical expressions for the stationary case are derived and risky shifts are obtained by monte carlo simulations striking similarities with the kuramoto model for group synchronization are pointed out

towards a comprehensive theory of brain activity coupled oscillator systems under external forces abstract recently jirsa et al and haken discussed a theory comprising the brain wave equation proposed by nunez the wilson cowan ermentrout cowan model and hopfield networks this theory was applied to model findings obtained in an experiment that relates brain activity and behavior the so called julliard experiment in previous works of jirsa et al and frank et al the focus was on the brain wave aspect recently we conducted similar experiments the results obtained are modeled in this paper in terms of coupled oscillator systems coupled oscillator systems are considered to represent the wilson cowan ermentrout cowan model aspect of the unifying theory building on a model proposed by haken kelso and bunz and the theory of weakly coupled oscillators established by winfree and by kuramoto we derive a nonlinear fokker planck equation whose stationary solutions mimic the neocortical brain activity observed in our experiments

h theorem for a mean field model describing coupled oscillator systems under external forces this article studies the asymptotic behavior of solutions of fokker planck equations describing mean field approximations of weakly coupled oscillator systems subjected to external forces using an h theorem we show that transient probability densities converge to stationary ones furthermore stability criteria are derived for the stationary solutions of these fokker planck equations the obtained results are applied to a model that combines the haken kelso bunz model and the models of weakly coupled oscillators proposed by winfree and kuramoto the stability criteria based on the h theorem agree with those derived in our earlier analyses

reentrant transition induced by multiplicative noise in the time dependent ginzburg landau model an effect of multiplicative noise in the time dependent ginzburg landau model is reported namely that noise at a relatively low intensity induces a phase transition towards an ordered state whereas strong noise plays a destructive role driving the system back to its disordered state through a reentrant phase transition the phase diagram is calculated analytically using a mean field theory and a more sophisticated approach and is compared with the results from extensive numerical simulations

on quasi linear stochastic partial differential equations we prove existence and uniqueness of the solution of a parabolic spde in one space dimension driven by space time white noise in the case of a measurable drift and a constant diffusion coefficient as well as a comparison theorem

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

on the relation of the drazin inverse and matrix pencil theory methods for the study of generalized linear systems in control theory the methodology of the drazin inverse and matrix pencil theory methods for generalized regular systems are extensively used thus in this paper a relation between the drazin inverse and the weierstrass form of a regular pencil is derived and fully investigated

phase and frequency shifts in a population of phase oscillators we investigate the synchronization behavior of a population of globally coupled phase oscillators with randomly distributed eigenfrequencies a variety of synchronized states is rigorously analyzed by means of the center manifold theorem combined with a nonlinear renormalization procedure this way the different synchronized states are determined explicitly we observe n cluster states where n 1 dots 4 in a synchronized state all oscillators have the same frequency synchronization frequency and mutual phase differences depend on model parameters as well as on the configuration of the synchronized state i e the number of oscillators within each cluster the synchronization frequency may decisively differ from the mean of the eigenfrequencies e g giving rise to frozen states which are synchronized states with vanishing synchronization frequency unstable cluster states are associated with transitions between different synchronization frequencies our approach can easily be extended to a population of oscillators with randomly distributed coupling strengths the different synchronized states are discussed in the context of neural coding

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

nonequilibrium transitions induced by multiplicative noise it is well established that the effects of noise in stochastic equations can be rather surprising and counterintuitive the presence of a noisy term can modify in quite an unexpected way the behavior of the corresponding deterministic noiseless equation a well known example of this is the stochastic resonance in which the response to an external periodically oscillating field of a stochastic system is enhanced by increasing its internal noise amplitude 1 other examples are the resonant activation 2 noise induced spatial patterns 3 the noise induced ordering transition niot 4 6 and the noise induced disordering transition nidt 4 these last two transitions were first analyzed together in an interesting work to the best of our knowledge by van den broeck parrondo and toral 4 vpt hereafter they considered a langevin equation with a monostable deterministic term to which a nontrivial noise is added owing to the combined effect of the noise and the spatial coupling two real phase transitions were shown to appear for small noise amplitudes the system is disordered while by increasing the noise amplitude the system gets ordered in a symmetrybreaking state exhibiting therefore a niot if the noise is increased further the ordered state is destroyed and therefore there is a nidt afterwards different works on niot and nidt appearing in different models and systems have been proposed in the literature 7 9 in particular in 7 a noiseinduced reentrant transition is reported for a time dependent ginzburg landau model with both additive and multiplicative noise in 8 9 first order transitions are shown to be induced by the presence of nontrivial noises an interesting task is that of understanding the nature of this striking phenomena in the simplest possible model understanding whether it comes from an interplay between additive and multiplicative noises or not and determining whether the critical behavior of these transitions can be described by any standard universality class in this direction the following langevin equation has been proposed by vpt as a possible simplest model exhibiting these two noiseinduced transitions

distribution of the first return time in fractional brownian motion and its application to the study of on off intermittency herein the term fractional brownian motion is used to refer to a class of random walks with long range correlated steps where the mean square displacement of the walker at large time t is proportional to mathit t 2 mathit h with 0h1 for ordinary brownian motion we obtain h 1 2 let t denote the time at which the random walker starting at the origin first returns to the origin the purpose of this paper is to show that the probability distribution of t scales with t as p t ensuremath sim mathit t mathit h mathrm ensuremath 2 theoretical arguments and numerical simulations are presented to support the result additional issues explored include modification to the power law distribution when the random walk is biased and the application of the result to the characterization of on off intermittency a recently proposed mechanism for bursting

exact time dependent solutions of the renyi fokker planck equation and the fokker planck equations related to the entropies proposed by sharma and mittal abstract exact time dependent solutions representing generalized one dimensional ornstein uhlenbeck processes are derived from nonlinear fokker planck equations and the corresponding diffusion processes are studied the corresponding systems are related to the renyi entropy and entropies proposed by sharma and mittal in the former case normal diffusion is found with a diffusion rate depending on the order of the renyi entropy in the latter case we find anomalous diffusion processes

stability of strong solutions for a model of incompressible two phase flow under thermal fluctuations abstract we consider a model of a two phase flow based on the phase field approach where the fluid bulk velocity obeys the standard navier stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the allen cahn equations possible thermal fluctuations are incorporated through a random forcing term in the allen cahn equation we show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality this fact is used for showing the weak strong uniqueness principle both pathwise and in law

collective synchronization of pulse coupled oscillators and excitable units abstract a phase model is proposed and studied for a population of integrate and fire type oscillators and excitable units with pulsatile coupling the units are subject to external noise and their mutual connection is all to all an evolution equation is derived for the phase distribution in a closed form its steady state analysis in the limit of weak noise shows the possibility of collective bistability the evolution equation is simplified drastically when the coupling as well as noise is sufficiently weak this reduction of equation enables analytical calculation of various quantities among others the stability of the steady distribution is analyzed explicity and the onset of collective oscillations is described in terms of the hopf bifurcation it is also shown that the present model is improved in some respects by introducing absolute refractoriness

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

critical dynamics and fluctuations for a mean field model of cooperative behavior the main objective of this paper is to examine in some detail the dynamics and fluctuations in the critical situation for a simple model exhibiting bistable macroscopic behavior the model under consideration is a dynamic model of a collection of anharmonic oscillators in a two well potential together with an attractive mean field interaction the system is studied in the limit as the number of oscillators goes to infinity the limit is described by a nonlinear partial differential equation and the existence of a phase transition for this limiting system is established the main result deals with the fluctuations at the critical point in the limit as the number of oscillators goes to infinity it is established that these fluctuations are non gaussian and occur at a time scale slower than the noncritical fluctuations the method used is based on the perturbation theory for markov processes developed by papanicolaou stroock and varadhan adapted to the context of probability measure valued processes

the ratio field of values abstract the ratio field of values a generalization of the classical field of values to a pair of n by n matrices is defined and studied primarily from a geometric point of view basic functional properties of the ratio field are developed and used a decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation primary theoretical results include the following it is shown 1 for which denominator matrices the ratio field is always convex 2 certain other cases of convex pairs are given and 3 that at least for n 2 the ratio field obeys a near convexity property that the intersection with any line segment has at most n components generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated it is shown that generally such extensions need not be convex or even simply connected

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2

boundary layer analysis for the stochastic nonlinear reaction diffusion equations abstract singularly perturbed stochastic and deterministic nonlinear reaction diffusion equations are considered we first study the governing problem posed in the channel domain with lateral periodicity and extend the results to general smooth domains introducing corrector functions which correct the boundary values discrepancies we are able to develop the convergence analysis for the analysis we make use of the maximum principle to estimate the corrector functions the stochastic problems also rely on the deterministic corrector functions which lead to simpler computations than those of the stochastic version of the correctors