on the stochastic pendulum with ornstein uhlenbeck noise we study a frictionless pendulum subject to multiplicative random noise because of destructive interference between the angular displacement of the system and the noise term the energy fluctuations are reduced when the noise has a non zero correlation time we derive the long time behaviour of the pendulum in the case of ornstein uhlenbeck noise by a recursive adiabatic elimination procedure an analytical expression for the asymptotic probability distribution function of the energy is obtained and the results agree with numerical simulations lastly we compare our method with other approximation schemes

global regular and singular solutions for a model of gravitating particles the existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of generalized pseudomeasures this approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem

critical finite size corrections for the sherrington kirkpatrick spin glass starting from the effective lagrangian of the sherrington kirkpatrick spin glass near the critical temperature the authors investigate the finite size effects for the free energy and the propagator above and at tc 1 they perform the calculation going up to the three loop diagrams for t to tc the cubic vertex only gives the leading term the inclusion of the quartic vertex gives the following terms the numerical coefficients in these terms are established by the different methods of the partial summation of perturbation series the comparison with numerical simulations is done

anharmonic oscillator driven by additive ornstein uhlenbeck noise we present an analytical study of a nonlinear oscillator subject to an additive ornstein uhlenbeck noise known results are mainly perturbative and are restricted to the large dissipation limit obtained by neglecting the inertial term or to a quasi white noise i e a noise with vanishingly small correlation time here in contrast we study the small dissipation case we retain the inertial term and consider a noise with finite correlation time our analysis is non perturbative and based on a recursive adiabatic elimination scheme a reduced effective langevin dynamics for the slow action variable is obtained after averaging out the fast angular variable in the conservative case we show that the physical observables grow algebraically with time and calculate the associated anomalous scaling exponents and generalized diffusion constants in the case of small dissipation we derive an analytic expression of the stationary probability distribution function pdf which differs from the canonical boltzmann gibbs distribution our results are in excellent agreement with numerical simulations

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

harmonic oscillator with multiplicative noise nonmonotonic dependence on the strength and the rate of dichotomous noise the output signal of a undamped linear oscillator with a random frequency subject to a periodic force shows nonmonotonic dependence on the strength and the rate of color noise stochastic resonance the effect is absent for white noise

influence of the noise spectrum on the anomalous diffusion in a stochastic system we use an effective markovian description to study the long time behavior of a nonlinear second order langevin equation with a gaussian noise when dissipation is neglected the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high frequency distribution of the noise the asymptotic expression of the probability distribution function in phase space is calculated analytically the results are extended to the case where small dissipative effects are taken into account

remarks on the local existence of solutions to the debye system abstract in this paper we study the cauchy problem of the debye system for initial data in the lorentz space l n 1 r n and the besov space b l n 1 r n for 0 1 some local existence theorems are proved

finite size scaling for near critical continuum fluids at constant pressure we consider the application of finite size scaling methods to isothermal isobaric constantnpt simulations of pure continuum fluids a finite size scaling ansatz is made for the dependence of the relevant scaling operators on the particle number to test the proposed scaling form constant pressure simulations of the lennard jones fluid at its liquid vapour critical point are performed the critical scaling operator distributions are obtained and their scaling with particle number is found to be consistent with the proposed behaviour the forms of these sealing distributions are shown to be identical to their ising model counterparts the relative merits of employing the eonstant npt and grand canonical constant vt ensembles for simulations of fluid criticality are also discussed

on diffusion equations for dynamical systems driven by noise a unified formalism is presented to study hamiltonian linear systems driven by noise with this formalism the phase averaging approximation valid at weak noise is easily performed already known results are straightforwardly recovered and new ones are obtained after introducing this formalism on the exactly solvable one degree of freedom problem with uncorrelated noise one studies the corresponding exponentially correlated case the validity of the approximate results thus obtained is considered by investigating the systematic weak disorder expansion beyond the quasilinear approximation in particular it is argued that this expansion behaves uniformly for weak and large correlation time the two degrees of freedom problem is completely solved at the low disorder approximation and this result is applied to the two channel anderson localization problem the invariant measure and the two positive lyapunov exponents are computed at all coupling between the channels for systems withn degree of freedom the phase averaging leads to a fokker planck equation for the measure in action space describing the system however it is argued that it is not solvable except in a special case which is explicitly displayed and solved nevertheless in the large n limit it is possible to compute the largest lyapunov exponent moreover generalized lyapunov exponents are calculated in this limit and they do not exhibit a dispersion in particular log log 1 where is the energy of the system and where the brackets denote averaging over the noise on the other hand it is possible to compute at weak noise the sum of all the positive lyapunov exponents taking into account all these results allows more insight on the whole spectrum of lyapunov exponents

large time behavior for convection diffusion equations in rn abstract we describe the large time behavior of solutions of the convection diffusion equation u t u a u q 1 u in 0 r n with a r n and q 1 1 n n 1 when q 1 1 n we prove that the large time behavior of solutions with initial data in l 1 r n is given by a uniparametric family of self similar solutions the relevant parameter is the mass of the solution that is conserved for all t our result extends to dimensions n 1 well known results on the large time behavior of solutions for viscous burgers equations in one space dimension the proof is based on la salle s invariance principle applied to the equation written in its self similarity variables when q 1 1 n the convection term is too weak and the large time behavior is given by the heat kernel in this case the result is easily proved applying standard estimates of the heat kernel on the integral equation related to the problem

morrey spaces and classification of global solutions for a supercritical semilinear heat equation in rn abstract we prove the boundedness of global classical solutions for the semilinear heat equation u t u u p 1 u in the whole space r n with n 3 and supercritical power p n 2 n 2 this is proved without any radial symmetry or sign assumptions unlike in all the previously known results for the cauchy problem and under spatial decay assumptions on the initial data that are essentially optimal in view of the known counter examples moreover we show that any global classical solution has to decay in time faster than t 1 p 1 which is also optimal and in contrast with the subcritical case the proof relies on nontrivial modifications of techniques developed by chou du and zheng 8 and by blatt and struwe 3 5 for the case of convex bounded domains they are based on weighted energy estimates of giga kohn type combined with an analysis of the equation in a suitable morrey space this method actually works for any convex bounded or unbounded smooth domain but at the same time captures some of the specific behaviors associated with the case of the whole space r n as a consequence we also prove that the set of initial data producing global solutions is open in suitable topologies and we show that the so called borderline global weak solutions blow up in finite time and then become classical again and decay as t all these results confirm the key role played by the morrey space m 2 4 p 1 in the understanding of the structure of the set of global solutions for p p s

well posedness of the cauchy problem for the fractional power dissipative equations abstract this paper studies the cauchy problem for the nonlinear fractional power dissipative equation u t u f u for initial data in the lebesgue space l r r n with either r r d n b 2 d or the homogeneous besov space b p r n with 2 d b n p and 1 p where 0 f u f u or q d f u with q d being a homogeneous pseudo differential operator of order d 0 2 and f u is a function of u which behaves like u b u with b 0

anomalous diffusion in nonlinear oscillators with multiplicative noise the time asymptotic behavior of undamped nonlinear oscillators with a random frequency is investigated analytically and numerically we find that averaged quantities of physical interest such as the oscillator s mechanical energy root mean square position and velocity grow algebraically with time the scaling exponents and associated generalized diffusion constants are calculated when the oscillator s potential energy grows as a power of its position u x approximately x 2n for x infinity correlated noise yields anomalous diffusion exponents equal to half the value found for white noise

random oscillator with general gaussian noise we study the long time behaviour of a nonlinear oscillator subject to a random multiplicative noise with a spectral density or power spectrum that decays as a power law at high frequencies when the dissipation is negligible physical observables such as the amplitude the velocity and the energy of the oscillator grow as power laws with time we calculate the associated scaling exponents and we show that their values depend on the asymptotic behaviour of the external potential and on the high frequencies of the noise our results are generalized to include dissipative effects and additive noise

adiabatic invariants and asymptotic behavior of lyapunov exponents of the schrodinger equation we give an upper bound for the high energy behavior of the lyapunov exponent of the one dimensional schrodinger equation we relate this behavior to the diffrentiability properties of the potential as an application this result provides an upper bound for the asymptotic length of the gaps of the schrodinger equation

diffusion over a fluctuating barrier in underdamped dynamics we apply a langevin model by imposing additive and multiplicative noises to study thermally activated diffusion over a fluctuating barrier in underdamped dynamics the barrier fluctuation is characterized by gaussian colored noise with exponential correlation we present the exact solutions for the first and second moments furthermore we use direct simulations to calculate the asymptotic probability for a brownian particle passing over the fluctuating barrier the results indicate that the correlation of the fluctuating barrier is crucial for barrier crossing dynamics

global radial solutions in classical keller segel model of chemotaxis abstract we consider the simplest parabolic elliptic model of chemotaxis in the whole space in several dimensions criteria for the existence of radially symmetric global in time solutions in terms of suitable morrey norms are derived

on the validity of the picard algorithm for nonlinear parabolic equations we discuss some ill posedness results for solutions arising from the picard iterations algorithm i e the banach fixed point theorem in the case of the nonlinear heat equation the viscous hamilton jacobi equation the convection diffusion equation and the incompressible navier stokes system

scaling in nonlinear parabolic equations the cauchy problem for parabolic equations with quadratic nonlinearity is studied we investigate the existence of global in time solutions and their large time behavior assuming some scaling property of the equation as well as of the norm of the banach space in which the solutions are constructed some applications of these abstract results are indicated

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

smooth or singular solutions to the navier stokes system abstract the existence of singular solutions of the incompressible navier stokes system with singular external forces the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions including stationary ones and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type

critical point and coexistence curve properties of the lennard jones fluid a finite size scaling study monte carlo simulations within the grand canonical ensemble are used to explore the liquid vapor coexistence curve and critical point properties of the lennard jones fluid attention is focused on the joint distribution of density and energy fluctuations at coexistence in the vicinity of the critical point this distribution is analyzed using mixed field finite size scaling techniques aided by histogram reweighting methods the analysis yields highly accurate estimates of the critical point parameters as well as exposing the size and character of corrections to scaling in the subcritical coexistence region the density distribution is obtained by combining multicanonical simulations with histogram reweighting techniques it is demonstrated that this procedure permits an efficient and accurate mapping of the coexistence curve even deep within the two phase region

large global in time solutions to a nonlocal model of chemotaxis abstract we consider the parabolic elliptic model for the chemotaxis with fractional anomalous diffusion global in time solutions are constructed under nearly optimal assumptions on the size of radial initial data moreover criteria for blowup of radial solutions in terms of suitable morrey spaces norms are derived

global well posedness of a dissipative system arising in electrohydrodynamics in negative order besov spaces in this paper we study the navier stokes nernst planck poisson system arising from electrohydrodynamics global well posedness of this system for small initial data is proven in negative order besov spaces as a corollary to this result we obtain the existence of self similar solutions to this system asymptotic stability of self similar solutions as time goes to infinity is also proven

a nonlinear oscillator with parametric coloured noise some analytical results the asymptotic behaviour of a nonlinear oscillator subject to a multiplicative ornstein uhlenbeck noise is investigated when the dynamics is expressed in terms of energy angle coordinates it is observed that the angle is a fast variable as compared to the energy thus an effective stochastic dynamics for the energy can be derived if the angular variable is averaged out however the standard elimination procedure performed earlier for a gaussian white noise fails when the noise is coloured because of correlations between the noise and the fast angular variable we develop here a specific averaging scheme that retains these correlations this allows us to calculate the probability distribution function pdf of the system and to derive the behaviour of physical observables in the long time limit

finite correlation time effects in nonequilibrium phase transitions i dynamic equation and steady state properties abstract we determine an approximate renormalized equation of evolution for an arbitrary nonlinear single degree of freedom system externally driven by gaussian parametric fluctuations of finite correlation time the renormalization scheme used here gives a second order equation with a time and state dependent diffusion coefficient we are able to calculate the diffusion coefficient in closed form the steady state distribution can easily be obtained from the evolution equation we are thus able to determine the parameter dependence of the steady state distribution and in particular the influence of a correlation time of the fluctuations which does not vanish on the steady state distribution

asymptotic localization of stationary states in the nonlinear schrodinger equation the mapping of the nonlinear schrodinger equation with a random potential on the fokker planck equation is used to calculate the localization length of its stationary states the asymptotic growth rates of the moments of the wave function and its derivative for the linear schrodinger equation in a random potential are computed analytically and resummation is used to obtain the corresponding growth rate for the nonlinear schrodinger equation and the localization length of the stationary states

delocalization phenomena in one dimensional models with long range correlated disorder a perturbative approach we study the nature of electronic states in one dimensional continuous models with weak correlated disorder using a perturbative approach we compute the inverse localization length lyapunov exponent up to terms proportional to the fourth power of the potential this makes it possible to analyse the delocalization transition which takes place when the disorder exhibits specific long range correlations we find that the transition consists of a change of the lyapunov exponent which switches from a quadratic to a quartic depending on the strength of the disorder within the framework of the fourth order approximation we also discuss the different localization properties which distinguish gaussian from non gaussian random potentials

well posedness of the cauchy problem for the fractional power dissipative equation in critical besov spaces abstract in this paper we study the cauchy problem for the semilinear fractional power dissipative equation u t u f u for the initial data u 0 in critical besov spaces b 2 r with n 2 2 d b where 0 f u p d u b 1 with p d being a homogeneous pseudo differential operator of order d 0 2 and b 0 being an integer making use of some estimates of the corresponding linear equation in the frame of mixed time space spaces the so called mono norm method which is different from the kato s double norm method fourier localization technique and littlewood paley theory we get the well posedness result in the case n 2

besov morrey spaces function space theory and applications to non linear pde this paper is devoted to the analysis of function spaces modeled on besov spaces and their applications to non linear partial differential equations with emphasis on the incompressible isotropic navier stokes system and semilinear heat equations specifically we consider the class introduced by hideo kozono and masao yamazaki of besov spaces based on morrey spaces which we call besov morrey or bm spaces we obtain equivalent representations in terms of the weierstrass semigroup and wavelets and various embeddings in classical spaces we then establish pseudo differential and para differential estimates our results cover non regular and exotic symbols although the heat semigroup is not strongly continuous on morrey spaces we show that its action defines an equivalent norm in particular homogeneous bm spaces belong to a larger class constructed by grzegorz karch to analyze scaling in parabolic equations we compare karch s results with those of kozono and yamazaki and generalize them by obtaining short time existence and uniqueness of solutions for arbitrary data with subcritical regularity we exploit pseudo differential calculus to extend the analysis to compact smooth boundaryless riemannian manifolds bm spaces are defined by means of partitions of unity and coordinate patches and intrinsically in terms of functions of the laplace operator

effects of parametric noise on a nonlinear oscillator we study a model of a nonlinear oscillator with a random frequency and derive the asymptotic behavior of the probability distribution function when the noise is white in the small damping limit we show that the physical observables grow algebraically with time before the dissipative time scale is reached and calculate the associated anomalous diffusion exponents in the case of colored noise with a non zero but arbitrarily small correlation time the characteristic exponents are modified we determine their values thanks to a self consistent ansatz

distribution of characteristic exponents in the thermodynamic limit the existence of the thermodynamic limit for the spectrum of the lyapunov characteristic exponents is numerically investigated for the fermi pasta ulam p model we show that the shape of the spectrum for energy density well above the equipartition threshold e allows the kolmogorov sinai entropy to be expressed simply in terms of the maximum exponent i the presence of a power law behaviour e is investigated the analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some universal properties

phase diagram of the random frequency oscillator the case of ornstein uhlenbeck noise abstract we study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an ornstein uhlenbeck noise this system undergoes a noise induced bifurcation when the amplitude of the noise grows the critical curve that separates the absorbing phase from an extended non equilibrium steady state corresponds to the vanishing of the lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy we derive various expressions for this lyapunov exponent by using different approximation schemes this allows us to study quantitatively the phase diagram of the random parametric oscillator

classical harmonic oscillator with multiplicative noise abstract the first two moments of an underdamped harmonic oscillator were found for additive and multiplicative noises a detailed analysis is performed of the case of a random damping which supplements the case of a random frequency which has received the most study in both cases the second moments are unstable for the sufficient large noise strength the difference appears in the stability conditions of the first moments which can diverge already for white noise for the random damping and only for color noise for the random frequency the difference between these two types of multiplicative noises shows up also in the response to an external periodic force namely white noise influences the first moment only in the case of a random damping however the second moments in both cases show the non monotonic dependence on the noise strength and correlation rate stochastic resonance in linear systems

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

a characteristic of the lyapunov spectrum for the multichannel anderson localisation in the thermodynamical limit the distribution of lyapunov exponents of the n channel anderson problem is investigated the kolmogorov entropy and the maximum lyapunov exponent are calculated in the thermodynamical limit at weak disorder the ratio i of these two quantities which characterises the convexity of the distribution is shown to depend on the disorder the energy and the energy spectrum of the noiseless system moreover i is found to be in general greater than two in agreement with the convex spectrum obtained previously by some authors through numerical experiments it is also shown that in contrast to the one dimensional case the generalised lyapunov exponent of order two and the largest lyapunov exponent are equal in the limit n to infinity

stationary energy probability density of oscillators driven by a random external force we derive rigorous analytical results for the stationary energy probability density function of linear and nonlinear oscillators driven by additive gaussian noise our study focuses on two cases i a harmonic oscillator subjected to gaussian colored noise with an arbitrary correlation function and ii nonlinear oscillators with a general potential driven by gaussian white noise we also derive analytical expressions for the stationary moments of the energy and investigate the partition of the mean energy between kinetic and potential energy to illustrate our general results we consider specifically the case of exponentially correlated noise for i and power law and bistable potentials for ii our theoretical results are substantiated by langevin simulations between the kinetic energy and the potential energy of the oscillator our studies focus on two cases first we consider a harmonically bound particle driven by general additive gaussian colored noise second we consider a particle in a general potential driven by gaussian white noise in both cases we develop a general theory and study specific illustrative examples the paper is organized as follows in sec ii we formulate the langevin equations for a particle with inertia in a general potential driven by an additive random force we also derive the general relation between the stationary energy pdf and the stationary pdf for the particle s position and velocity section iii is devoted to the motion of a particle in a harmonic potential driven by colored gaussian noise with an arbitrary correlation function the stationary energy pdf as well as the first and second moments of the energy is derived analytically we apply these results to the case of exponentially correlated gaussian noise and to gaussian white noise for these specific cases we also determine the partition of the energy between kinetic and potential energy we analyze the effect of the potential shape on the stationary energy pdf for oscillators driven by an additive gaussian white noise in sec iv w e apply our analytical results to two specific cases a general power law potential and a bistable potential the first type of oscillator without friction and with additive noise 36 and multiplicative noise 37 has recently been studied in the context of anomalous diffusion the second type the duffing oscillator has been investigated as an alternative to the linear oscillator for harvesting energy from external random sources 28 29 35 38 39

gevrey regularity of mild solutions to the parabolic elliptic system of drift diffusion type in critical besov spaces abstract this paper is devoted to studying analyticity of mild solutions to the parabolic elliptic system of drift diffusion type with small initial data in critical besov spaces more precisely by using multilinear singular integrals and fourier localization argument we show that global in time mild solutions are gevrey regular i e e t 1 v e t 1 w l t b p r 2 n p r n l t 1 b p r n p r n for all 1 p 2 n and 1 r where 1 is the fourier multiplier whose symbol is given by 1 i 1 n i as a corollary we obtain decay estimates of mild solutions in critical besov spaces without using cumbersome recursive estimation of higher order derivatives