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

geometrical treatment of systems driven by coloured noise the authors use geometrical methods and functional derivative techniques to analyse the stochastic behaviour of multidegree of freedom nonlinear systems driven by gaussian noises with arbitrary correlation functions the method has the advantage of being also applicable to systems with an infinite number of degrees of freedom partial differential equations with some kind of hamiltonian structure and gives some insight into the nature of the approximations that lead to some of the main effective fokker planck equations used in the literature

the projection approach to the fokker planck equation i colored gaussian noise it is shown that the fokker planck operator can be derived via a projection perturbation approach using the repartition of a more detailed operator into a perturbation 1 and an unperturbed part 0 the standard fokker planck structure is recovered at the second order in 1 whereas the perturbation terms of higher order are shown to provoke the breakdown of this structure to get rid of these higher order terms a key approximation local linearization ll is made in general to evaluate at the second order in 1 the exact expression of the diffusion coefficient which simulates the influence of a gaussian noise with a finite correlation time a resummation up to infinite order in must be carried out leading to what other authors call the best fokker planck approximation bfpa it is shown that due to the role of terms of higher order in 1 the bfpa leads to predictions on the equilibrium distributions that are reliable only up to the first order in t the ll on the contrary in addition to making the influence of terms of higher order in 1 vanish results in a simple analytical expression for the term of second order that is formally coincident with the complete resummation over all the orders in t provided by the fox theory the corresponding diffusion coefficient in turn is shown to lead in the limiting case to exact results for the steady state distributions therefore over the whole range 0 the ll turns out to be an approximation much more accurate than the global linearization proposed by other authors for the same purpose of making the terms of higher order in 1 vanish in the short region the ll leads to results virtually coincident with those of the bfpa in the large region the ll is a more accurate approximation than the bfpa itself these theoretical arguments are supported by the results of both analog and digital simulation

external non white noise theory and experiment abstract we consider a system under the influence of an external gaussian noise with a flat spectrum and finite cut off frequency the stationery properties of the system are studied as a function of the correlation time of the noise theoretical predictions based on an expansion in powers of the correlation time are corroborated by experimental measurements on an electric circuit with a digital noise generator the two main qualitative properties experimentally observed which can not be explained in a white noise limit theory are i appearance of relative extrema of the stationery distribution for high noise intensity when the correlation time is not too small ii dependence on the noise characteristics of the position of the maximum of the stationary distribution for low noise intensity

a note on the regularity criterion for the 3d navier stokes equations via the gradient of one velocity component abstract in this paper we consider the regularity criterion for weak solutions to the 3d navier stokes equations we show that if u 3 l 32 7 0 t l 2 r 3 then the solution is smooth on 0 t which improves the regularity condition u 3 l 24 5 0 t l 2 r 3 in 42

a remark on the regularity criterion for the 3d navier stokes equations involving the gradient of one velocity component abstract in this paper we consider the regularity criterion for weak solutions to the 3d navier stokes equations we show that if u 3 belongs to some multiplier spaces then the solution actually is smooth on 0 t in particular we have the regularity condition in the bmo spaces u 3 l 4 3 0 t bmo which improves previous results

on regularity criteria for the n dimensional navier stokes equations in terms of the pressure abstract we study the cauchy problem for the n dimensional navier stokes equations n 3 and prove some regularity criteria involving the integrability of the pressure or the pressure gradient for weak solutions in the morrey besov and multiplier spaces

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

brownian dynamics time averaging and colored noise we propose a method to obtain the equilibrium distribution for positions and velocities of a one dimensional particle via time averaging and laplace transformations we apply it to the case of a damped harmonic oscillator in contact with a thermal bath the present method allows us to treat among other cases a gaussian noise function exponentially correlated in time e g gaussian colored noise we obtain the exact equilibrium solution and study some of its properties

linear quadratic fractional gaussian control in this paper a control problem for a linear stochastic system driven by a noise process that is an arbitrary zero mean square integrable stochastic process with continuous sample paths and a cost functional that is quadratic in the system state and the control is solved an optimal control is given explicitly as the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and the prediction of the response of a system to the future noise process the optimal cost is also given the special case of a noise process that is an arbitrary standard fractional brownian motion is noted explicitly with an explicit expression for the prediction of the future response of a system to the noise process that is used the optimal control

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

a study on the action of non gaussian noise on a brownian particle abstract we analytically address the non equilibrium problem of a brownian particle in contact with a thermal reservoir by means of a non gaussian langevin noise term t the presence of noise kurtosis is akin to a second temperature reservoir acting on the system and we exploit its consequences by means of studying a converging exact form for the stationary probability distribution

discrete time mean field stochastic linear quadratic optimal control problems ii infinite horizon case this paper first presents results on the equivalence of several notions of l2l2 stability for linear mean field stochastic difference equations with random initial value then it is shown that the optimal control of a mean field linear quadratic optimal control with an infinite time horizon uniquely exists and the optimal control can be expressed as a linear state feedback involving the state and its mean via the minimal nonnegative definite solution of two coupled algebraic riccati equations as a byproduct the open loop l2l2 stabilizability is proved to be equivalent to the closed loop l2l2 stabilizability moreover the minimal nonnegative definite solution the maximal solution the stabilizing solution of the algebraic riccati equations and their relations are carefully investigated specifically it is shown that the maximal solution is employed to construct the optimal control and value function to another infinite time horizon mean field linear quadratic optimal control in addition the maximal solution being the stabilizing solution is completely characterized by properties of the coefficients of the controlled system this enriches the existing theory about stochastic algebraic riccati equations finally the notion of exact detectability is introduced with its equivalent characterization of stochastic versions of the popov belevitch hautus criteria it is then shown that the minimal nonnegative definite solution is the stabilizing solution if and only if the uncontrolled system is exactly detectable

extension of statistical replacement to systems with time correlated fluctuations nonlinear systems with correlated stochastic parameters are approximated by simpler systems this method is an extension of an earlier version of statistical replacement and statistical linearization the extended method is applicable to systems with correlated fluctuations we show how this general method reduces to the earlier methods in special cases

global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor in this paper we provide a sufficient condition in terms of only one of the nine entries of the gradient tensor that is the jacobian matrix of the velocity vector field for the global regularity of strong solutions to the three dimensional navier stokes equations in the whole space as well as for the case of periodic boundary conditions

solvable stochastic differential games in rank one compact symmetric spaces some nonlinear stochastic differential games are formulated in the family of complex and quaternion projective spaces that are among the rank one compact symmetric spaces that consist of the spheres the projective spaces over r c and h and one arising from an exceptional lie algebra called the cayley plane the payoff functionals for the differential games are obtained from some eigenfunctions of the radial part of the laplacians for these riemannian manifolds and these payoffs induce symmetries for the game problems that reduce the required analysis to a radial direction in these manifolds these projective spaces are given a natural riemannian metric from the killing forms of the compact lie groups for these spaces that are the special unitary groups su n and the symplectic groups sp n explicit optimal control strategies are obtained for these differential games and the explicit payoffs are given a countable family of distinct solvable stochastic differential games can be obtained for

linear quadratic optimal control problems for mean field stochastic differential equations linear quadratic optimal control problems are considered for mean field stochastic differential equations with deterministic coefficients by a variational method the optimality system is derived which is a linear mean field forward backward stochastic differential equation using a decoupling technique two riccati differential equations are obtained which are uniquely solvable under certain conditions then a feedback representation is obtained for the optimal control

regularity criteria for the 3d mhd equations involving one current density and the gradient of one velocity component abstract in this paper we consider the cauchy problem for the 3 d mhd equations and provide a regularity criterion involving u 3 and j 3 where u 3 and j 3 are the third component of the velocity and the current density respectively as by products we show that assumptions on two vorticity and one current density component are enough to ensure the smoothness of the solution and improve some previous results skalak 2014 zhou and pokorn 2009 2010 for the navier stokes equations

linear exponential quadratic gaussian control in this technical note an optimal control problem for a linear stochastic system with brownian motion and a cost that is an exponential of a quadratic functional of the state and the control is solved by obtaining explicitly an optimal control and the optimal cost while this solution has been previously obtained the approach given here is direct and elementary and does not use the well known solution methods of the hamilton jacobi bellman equation or the stochastic maximum principle the approach given here presents a basic insight in the solution by providing a simple explanation for the additional term in the riccati equation for the optimal control as compared to the riccati equation for the linear quadratic gaussian control problem

a regularity criterion for the navier stokes equations based on the gradient of one velocity component abstract we show that if u is a leray solution to the navier stokes equations in the full three dimensional space with an initial condition from w 0 1 2 t 0 and u l t 0 t l s where 2 t 3 s 59 30 for s 2 30 13 and 2 t 3 s 7 4 1 2 s for s 30 13 3 then u is regular on 0 t we prove our result as a special case of a more general method which might possibly bring a further improvement

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

reduction of overdamped stochastic processes in the coherent state representation the adiabatic elimination of fast variables from overdamped stochastic processes by functional integration is demonstrated adiabatic elimination entails the evaluation of a reduced evolution operator from the full evolution equation for fokker planck processes the reduced evolution operator may be expressed as a ground state expectation and it is shown how this is represented as a coherent state path integral the elimination is then achieved by functionally integrating out all reference to the fast variables the end result is a decoupling of the full evolution equation into separate equations for the fast and slow variables the method is demonstrated for brownian motion and for a system with multiplicative colored noise

the generalized wiener process ii finite systems a class of langevin like equations non markovian processes are studied in the presence of non natural boundary conditions exact results for all cumulants and the corresponding kolmogorov hierarchy of distributions are given in terms of our functional approach we previously reported 1997 j phys a math gen 30 8427 the generalized wiener processes on finite domains are completely characterized for reflecting and periodic boundary conditions some examples are given to show the behaviour of the moments and the probability distributions for different noises the interplay between the boundary conditions and the structure of the noises is also pointed out

on a regularity criterion for the navier stokes equations involving gradient of one velocity component we improve the regularity criterion for the incompressible navier stokes equations in the full three dimensional space involving the gradient of one velocity component the method is based on recent results of cao and titi see regularity criteria for the three dimensional navier stokes equations indiana univ math j 57 2643 2008 and kukavica and ziane see navier stokes equations with regularity in one direction j math phys 48 065203 2007 in particular for s 2 3 we get that the solution is regular if u3 lt 0 t ls r3 2 t 3 s 2312

theory of nonlinear langevin equation with quadratic noise a generalized langevin equation with a multiplicative linear and quadratic ornstein uhlenbeck stochastic noise is investigated an exact generalized fokker planck equation for this langevin equation is obtained this generalized fokker planck equation has the form of an integro differential equation an exact form of the integral kernel is given in terms of an operator valued matrix continued fraction in the limit of small fluctuations and short coherence time of the noise a standard fokker planck equation with drift and diffusion is derived in a systematic way

computing a non maxwellian velocity distribution from first principles we investigate a family of single particle anomalous velocity distribution by solving a particular class of stochastic liouville equations the stationary state is obtained analytically and the maxwell boltzmann distribution is reobtained in a particular limit we discuss the comparison with other different methods to obtain the stationary state extensions when the models cannot be solved in an exact way are also pointed out in connection with the one ficton approximation

what should a statistical mechanics satisfy to reflect nature there is no compelling reason imposing that the methods of statistical mechanics should be restricted to the dynamical systems which follow the usual boltzmann gibbs prescriptions more specifically ubiquitous natural and artificial systems exhibit generic stationary states which are not ergodic nor close to it in any geometrically simple subset of the a priori allowed phase space in any even extended trivial sense a vast class of such systems appears nevertheless to be tractable within thermostatistical methods completely analogous to the usual ones the question posed in the title arises then naturally some answer to this complex question is advanced in the present review of nonextensive statistical mechanics and its recent connections

discrete time mean field stochastic linear quadratic optimal control problems this paper firstly presents necessary and sufficient conditions for the solvability of discrete time mean field stochastic linear quadratic optimal control problems secondly the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method thirdly by introducing several sequences of bounded linear operators the problem is formulated as an operator stochastic linear quadratic optimal control problem by the kernel range decomposition representation of the expectation operator and its pseudo inverse the optimal control is derived using solutions to two algebraic riccati difference equations finally by completing the square the two riccati equations and the optimal control are also obtained

remarks on the regularity criterion to the navier stokes equations via the gradient of one velocity component abstract in this paper we consider the classical three dimensional 3d incompressible navier stokes equations by making use of a new anisotropic sobolev inequality we improve a regularity criterion for the weak solution to the 3d navier stokes equations involving the gradient of one velocity component this result improves the previous work of zhou and pokorn 2010

on the regularity of the solutions to the navier stokes equations via the gradient of one velocity component abstract we improve a regularity criterion for the solutions to the navier stokes equations in the full three dimensional space involving the gradient of one velocity component revising the method used in pokorn and zhou 2009 2010 we show that a weak solution u is regular on 0 t provided that u 3 l t 0 t l s where 2 t 3 s 19 10 for s 30 19 10 3 and 2 t 3 s 7 4 1 2 s for s 10 3 it improves the known results for s 30 19 150 77 and s 10 3

statistical replacement for systems with delta correlated fluctuations nonlinear systems with stochastic parameters are approximated by simpler systems using a method that we call statistical replacement this method is an extension of the previously developed agree which was restricted to systems with additive fluctuations statistical replacement incorporates the distinctions between globally stable thermodynamically closed systems and thermodynamically open systems that can be unstable

functional characterization of generalized langevin equations we present an exact functional formalism to deal with linear langevin equations with arbitrary memory kernels and driven by an arbitrary noise structure characterized through its characteristic functional no other hypothesis is assumed over the noise neither do we use the fluctuation dissipation theorem we find that the characteristic functional of the linear process can be expressed in terms of noise functional and the green function of the deterministic memory like dissipative dynamics this yields a procedure for calculating the full kolmogorov hierarchy of the non markov process as examples we have characterized through the 1 time probability a noise induced interplay between the dissipative dynamics and the structure of different noises conditions that lead to non gaussian statistics and distributions with long tails are analysed the introduction of arbitrary fluctuations in fractional langevin equations has also been pointed out

ordered cumulant technique and differential equations for probability density abstract this paper is concerned with stochastic differential equations of langevin type in which the stochastic forces are gaussian and not of white noise type the problem of finding explicitly the differential equation for the probability density of the process is solved by means of an expansion in the small correlation time of the noise for this purpose we will use the cumulant technique different situations are studied such as markovian or fokker planck equation first correction to the markovian approximation but with the fokker planck form and the corrections of non fokker planck form

a note on the regularity of the solutions to the navier stokes equations via the gradient of one velocity component we present a regularity criterion for the solutions to the navier stokes equations based on the gradient of one velocity component starting with the method developed by cao and titi global regularity criterion for the 3d navier stokes equations involving one entry of the velocity gradient tensor arch ration mech anal 202 919 932 2011 for the case of one entry of the velocity gradient and using further some inequalities concerning the anisotropic sobolev spaces we show as a main result that a weak solution u is regular on 0 t t 0 provided that u3 lt 0 t ls where 2 t 3 s 3 2 3 4s and s 3 2 2 it improves the known results for s 3 2 15 8

a regularity criterion for the tridimensional navier stokes equations in term of one velocity component in this paper we investigate the regularity criterion of the tridimensional navier stokes equations via one velocity component our strategy is to establish the following version of regularity criterions of leray hopf weak solutions in the framework of anisotropic lebesgue space 0t ui lip lj k d 2 2 1p 34 12 p 1 or 0t ui ljp li k d 2 2 1p 34 14 14p p 2 fori j this allows us to obtain regularity criterion of leray hopf weak solutions via only one element i uj with 0 1 and i j 1 2 3 that is 0t i uj l d here 1 2 and are the sets of indexes which appear in our results and the fractional operator i i2 this extends and improves some known regularity criterions of leray hopf weak solutions in term of one velocity component including the notable works of c cao and e s titi 4 more importantly by making full use of the bony paraproduct decomposition we show that leray hopf weak solutions are smooth on 0 t if 0t uj bmo83d j 1 2 3 or 0t i uj b 2083 2 d 0 1 fori j 1 2 3 which fill the gap of endpoint

geometrical derivation of the intrinsic fokker planck equation and its stationary distribution starting from an intrinsic langevin equation we give a geometrical derivation of the fokker planck equation we also present a method for obtaining a stationary distribution and for deriving potential conditions when the diffusion matrix is singular

some serrin type regularity criteria for weak solutions to the navier stokes equations we consider the regularity criteria for a weak solution u u1 u2 u3 to the navier stokes equations in r3 denoting by 1 2 3 curlu the vorticity we then prove that u is smooth provided u3 3 or 3u3 3 are in some serrin type integrability classes

harmonic potential driven by long range correlated noise the probability distribution of a particle in a quadratic potential driven by gaussian long range correlated noise has been obtained the long time asymptotic relaxation of the stochastic process has been characterized in terms of the long range correlated noise appearing in the corresponding stochastic differential equation defining the process the particular case when the particle is driven by a gaussian color noise has also been revisited in this way giving its exact probability distribution for all time by using a characteristic functional technique reported previously results for a non gaussian long range correlated noise are also shown emphasis has been placed on solving the plane rotator in the presence of arbitrary random torques having long range correlations in order to show strong and weak non markovian effects coming from different sources of noise the undamped free particle under the influence of arbitrary accelerations has been analyzed we also analyze in detail the structure of the trajectories of the overdamped and undamped free particle

a new approximate stationary probability distribution for processes driven by ornstein uhlenbeck noise a novel approximate probability distribution for a class of nonlinear processes driven by multiplicative linear ornstein uhlenbeck noise is presented a comparison is made with other approximate distributions reported in the literature

the generalized ornstein uhlenbeck process langevin like equations have been studied in the presence of arbitrary noise the characteristic functional of the generalized langevin process has been built up exact results for all cumulants are given particular stress has been put on the campbell dichotomous and radioactive decay noises transient relaxation susceptibility and diffusion constants for different noisy media have been sketched in order to exemplify the theory the generalized ornstein uhlenbeck and wiener processes have been completely characterized the generalized kubo oscillator has been worked out and all its 1 time moments have been calculated for different noise structures

a mean field game of evacuation in multilevel building this paper puts forward a simple mean field game that captures some of the key dynamic features of crowd and pedestrian flows in multilevel building evacuations it considers both microscopic and macroscopic route choice by strategic agents to achieve this we use mean field differential game with local congestion measure based on the location of the agent in the building including the local mean field term and its evolution along the path causes a sort of dispersion of the flow the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing costs at the stairs and exits each agent state is represented by a center of a box that follows a simple first order dynamical system in an euclidean space each agent will move to one of the closest exits that is safer and with less congested path first we formulate the problem and derive optimality equations using maximum principle and dynamic programming with boundary conditions second well posedness and existence results are provided numerics and simulations are carried out to illustrate mean field equilibria of a safer evacuation process finally the methodology is shown to be flexible enough to include movement noises and stochastic structural component of the building

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

levy noise levy flights levy fluctuations this paper is concerned with a levy noise characterized for t 0 by the functional with 0 2 then levy flights can be defined through a stochastic differential equations rather than the usual chapmann kolmogorov equation we have used this functional approach to solve a plane rotor in the presence of levy noise the linear damped stochastic process driven by levy noise is revisited and its non autonomous and non markovian generalizations have been solved in the context of our functional analysis

langevin like equation with colored noise consider a stochastic differential equation of the form of a langevin equation but in which the noise source is not white if it is nearly white i e its autocorrelation time is short a systematic approximation method is known it leads to a fokker planck equation with successive higher order corrections to obtain the coefficients more explicitly a secondary expansion may be employed the validity of the resulting double series approximation is discussed and confronted with the various results given in the literature in addition an alternative approximation method is obtained using the technique for eliminating fast variables it produces the same terms in a different sequence

forward and backward equations for a nonmarkovian process a cumulant approach abstract this paper deals with nonmarkovian processes induced by colored noise the forward equations of motion for both the single and joint probability distribution functions are derived at the same time the backward equation for the probability density of a nonmarkovian process is for the first time deduced these results are obtained by means of the systematic use of the ordered cumulant technique

numerical method for solving stochastic differential equations with dichotomous noise we propose a numerical method for solving stochastic differential equations with dichotomous markov noise the numerical scheme is formulated such that i the stochastic formula used follows the stratonovichtaylor form over the entire range of noise correlation times including the gaussian white noise limit and ii the method is readily applicable to dynamical systems driven by arbitrary types of noise provided there exists a way to describe the random increment of the stochastic process expressed in the stratonovich taylor form we further propose a simplified taylor scheme that significantly reduces the computation time while still satisfying the moment properties up to the required order the accuracies and efficiencies of the proposed algorithms are validated by applying the schemes to two prototypical model systems that possess analytical solutions

linear exponential quadratic stochastic differential games in this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved the optimal strategies form a nash equilibrium the value of the game is also determined explicitly the solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem

discrete time linear quadratic mean field type repeated games perfect incomplete and imperfect information in this paper we study linear quadratic mean field type repeated games with arbitrary number of interacting decision makers different information structures are examined i perfect knowledge of the model ii imperfect knowledge of the model under perfect state observation we provide semi explicit solutions for both non cooperative and cooperative cases the state dynamics is given by a stochastic difference equation that includes mean field terms i e the expected value for both the state and control inputs in addition the cost function incorporates not only the expected value of the states but also the variance of both system state and control inputs the effect of the information is quantified using price of anarchy and price of imperfect information finally some numerical examples are presented

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

regularity criterion via two components of vorticity on weak solutions to the navier stokes equations in r 3 we consider the regularity of weak solutions to the navier stokes equations in r3 let u be a weak solution in r3 0 t w curlu and w w1 w2 0 it is proved that u becomes a classical solution if w lq 0 t b r 0 for 2 q 3 r 2 32 r and 2r 3 this is an improvement of the result given by kozono yatsu extension criterion via two components of vorticity on strong solution to the 3d navier stokes equations math z 246 2003 55 68

weak noise expansions through functional integrals for colored noise we use path integral methods to obtain expansions for the correlation functions of the non markovian stochastic processes generated by stochastic differential equations with colored noise

dynamical properties of non markovian stochastic differential equations we study nonstationary non markovian processes defined by langevin type stochastic differential equations with an ornstein uhlenbeck driving force we concentrate on the long time limit of the dynamical evolution we derive an approximate equation for the correlation function of a nonlinear nonstationary non markovian process and we discuss its consequences non markovicity can introduce a dependence on noise parameters in the dynamics of the correlation function in cases in which it becomes independent of these parameters in the markovian limit several examples are discussed in which the relaxation time increases with respect to the markovian limit for a brownian harmonic oscillator with fluctuating frequency the non markovicity of the process decreases the domain of stability of the system and it can change an infradamped evolution into an overdamped one

a computable evolution equation for the joint response excitation probability density function of stochastic dynamical systems by using functional integral methods we determine a computable evolution equation for the joint response excitation probability density function of a stochastic dynamical system driven by coloured noise this equation can be represented in terms of a superimposition of differential constraints i e partial differential equations involving unusual limit partial derivatives the first one of which was originally proposed by sapsis athanassoulis a connection with the classical response approach is established in the general case of random noise with arbitrary correlation time yielding a fully consistent new theory for non markovian systems we also address the question of computability of the joint response excitation probability density function as a solution to a boundary value problem involving only one differential constraint by means of a simple analytical example it is shown that in general such a problem is undetermined in the sense that it admits an infinite number of solutions this issue can be overcome by completing the system with additional relations yielding a closure problem which is similar to the one arising in the standard response theory numerical verification of the equations for the joint response excitation density is obtained for a tumour cell growth model under immune response

navier stokes equations with regularity in one direction we consider sufficient conditions for the regularity of leray hopf solutions of the navier stokes equations we prove that if the third derivative of the velocity u x3 belongs to the space lts0lxr0 where 2 s0 3 r0 2 and 9 4 r0 3 then the solution is regular this extends a result of beirao da veiga chin ann math ser b 16 407 412 1995 c r acad sci ser i math 321 405 408 1995 by making a requirement only on one direction of the velocity instead of on the full gradient the derivative u x3 can be substituted with any directional derivative of u

an approximate master equation for systems driven by linear ornstein uhlenbeck noise abstract by use of an operator method we construct a novel approximate evolution equation for a one dimensional probability distribution of a single degree of freedom system driven by ornstein uhlenbeck noise this equation is an integro differential equation of the time convolutionless type and its steady state solution is presented a class of non linear equations with multiplicative noise is considered