a variation embedding theorem and applications fractional sobolev spaces also known as besov or slobodetski spaces arise in many areas of analysis stochastic analysis in particular we prove an embedding into certain q variation spaces applications include a new route to a regularity result by kusuoka for stochastic differential equations integration against besov paths a regularity criterion for rough paths and a new regularity result for cameron martin paths associated to fractional brownian motion

simple diagonal locally finite lie algebras the ground field f is an algebraically closed field of zero characteristic and bbb n is the set of natural numbers let l be a locally finite dimensional or em locally finite for brevity lie algebra assume for simplicity that l has countable dimension then one can choose finite dimensional subalgebras l i i in bbb n of l in such a way that l i subset l i 1 for all i and l bigcup i in bbb n l i the set l i mid i in bbb n is called a em local system of l assume that l is simple then all l i can be chosen to be perfect that is l i l i l i we shall call such local systems em perfect let s i s i 1 oplus ldots oplus s i n i be a levi subalgebra of l i where s i 1 ldots s i n i are simple components of s i and let v i k be the standard s i k module since l i is perfect for every k there exists a unique irreducible l i module cal v i k such that the restriction cal v i k downarrow s i k is isomorphic to v i k an embedding l i to l i 1 is called em diagonal if for each l any non trivial composition factor of the restriction cal v i 1 l downarrow l i is isomorphic to cal v i k or its dual cal v i k for some k we prove that a simple locally finite lie algebra l can be embedded into a locally finite associative algebra if and only if there exists a perfect local system l i i in bbb n of l such that all embeddings l i to l i 1 are diagonal note that this result can be considered as a version of ado s theorem for simple locally finite lie algebras let v be a vector space an element x in frak gl v is called em finitary if dim xv infty the finitary transformations of v form an ideal frak fgl v of frak gl v and any subalgebra of frak fgl v is called a em finitary lie algebra we classify all finitary simple lie algebras of countable dimension it turns out that there are only three frak sl infty f frak so infty f and frak sp infty f the author has classified all finitary simple lie algebras and will describe this in a subsequent paper the analogous problem in group theory has been recently solved by j i hall he classified simple locally finite groups of finitary linear transformations 1991 mathematics subject classification 17b35 17b65

duration of transient fronts in a bistable reaction diffusion equation in a one dimensional bounded domain the duration of transient fronts in a bistable reaction diffusion equation in a bounded domain is considered the speed of the front decreases exponentially with the length of the domain and the duration increases exponentially with the domain length the duration of the fronts generated from random initial conditions is distributed in a power law form up to a cutoff time the cutoff time then increases exponentially with the domain length so that the power law distribution dominates for large domains further external noise of intermediate strength increases the mean duration of the fronts the increases in the duration with the domain length become almost linear however in the presence of asymmetry in the cubic nonlinearity

functional differential equations driven by a fractional brownian motion in this paper we prove a global existence and uniqueness result for the solution of a stochastic functional differential equation driven by a fractional brownian motion with hurst parameter h 1 2 we also study the dependence of the solution on the initial condition

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

stochastic differential games in a non markovian setting stochastic differential games are considered in a non markovian setting typically in stochastic differential games the modulating process of the diffusion equation describing the state flow is taken to be markovian then nash equilibria or other types of solutions such as pareto equilibria are constructed using hamilton jacobi bellman hjb equations but in a non markovian setting the hjb method is not applicable to examine the non markovian case this paper considers the situation in which the modulating process is a fractional brownian motion fractional noise calculus is used for such models to find the nash equilibria explicitly although fractional brownian motion is taken as the modulating process because of its versatility in modeling in the fields of finance and networks the approach in this paper has the merit of being applicable to more general gaussian stochastic differential games with only slight conceptual modifications this work has applications in finance to stock price modeling which incorporates the effect of institutional investors and to stochastic differential portfolio games in markets in which the stock prices follow diffusions modulated with fractional brownian motion

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

finitary lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple and finitary irreducible lie algebras over an algebraically closed field of characteristic 2 3

random attractor for the 3d viscous primitive equations driven by fractional noises abstract we develop a new and general method to prove the existence of the random attractor strong attractor for the primitive equations pes of large scale ocean and atmosphere dynamics under non periodic boundary conditions and driven by infinite dimensional additive fractional wiener processes in contrast to our new method the common method compact sobolev embedding theorem is to obtain the time uniform a priori estimates in some sobolev space whose regularity is higher than the solution space but this method can not be applied to the 3d stochastic pes with the non periodic boundary conditions therefore the existence of universal attractor weak attractor was established in previous works see 15 16 the main idea of our method is that we first derive that p almost surely the solution operator of stochastic pes is compact then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set we should point out that our method has some advantages over the common method of using compact sobolev embedding theorem i e using our method we only need to obtain time uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system while the common method need to establish time uniform a priori estimates in a more regular functional space than the solution space take the stochastic pes for example as the unique strong solution to the stochastic pes belongs to c 0 t h 1 3 in view of our method we only need to obtain the time uniform a priori estimates in the solution space h 1 3 to prove the existence of random attractor for this stochastic system while the common method need to establish time uniform a priori estimates for the solution in the functional space h 2 3 however time uniform a priori estimates in h 2 3 for the solution to stochastic pes are too difficult to be established the present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by wiener noises fractional noises and even jump noises in a forth coming paper using this new method we 46 prove the existence of random attractor for the stochastic nematic liquid crystals equations this is the first result about the long time behavior of stochastic nematic liquid crystals equations

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

duration of transient oscillations in ring networks of unidirectionally coupled neurons abstract properties of the duration of long lasting transient oscillations in ring networks of unidirectionally coupled sigmoidal neurons are derived with a kinematical model of traveling waves in the network the duration of the transient oscillations occurring from random initial conditions increases exponentially as the number of neurons the distribution of the duration is approximated by a power law function when the number of neurons is large further transient oscillations which oscillate about one thousand cycles before ceasing are observed in a network of forty neurons in circuit experiments though the duration decreases owing to random biases

semimartingale approximation of fractional brownian motion and its applications the aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration this result leads us to approximate the fractional black scholes model by a model driven by semimartingales and a european option pricing formula is found

asymptotic behavior of stochastic lattice systems with a caputo fractional time derivative abstract the asymptotic behavior of a stochastic lattice system with a caputo fractional time derivative is investigated in particular the existence of a global forward attracting set in the weak mean square topology is established a general theorem on the existence of solutions for a fractional sde in a hilbert space under the assumption that the nonlinear term is weakly continuous in a given sense is established and applied to the lattice system the existence and uniqueness of solutions for a more general fractional sdes is also obtained under a lipschitz condition

bifurcation and stabilization of oscillations in ring neural networks with inertia effects of inertia on oscillations in ring networks of unidirectionally coupled sigmoidal neurons are studied it is known that ring neural networks without inertia are bistable and the duration of transient oscillations increases exponentially with the number of neurons in this paper a kinematical description of traveling waves in the oscillations in the networks is extended to networks with inertia when the inertia is below a critical value and the state of each neuron is over damped properties of the networks are qualitatively the same as those without inertia the duration of the transient oscillations then increases with inertia and the increasing rate of the logarithm of the duration becomes more than double when the inertia exceeds a critical value and the state of each neuron becomes under damped properties of the networks qualitatively change the periodic solution is then stabilized through the pitchfork bifurcation as inertia increases more bifurcations occur so that various periodic solutions are generated and the stability of the periodic solutions changes alternately further stable oscillations generated with inertia are observed in an experiment on an analog circuit

on some fractional stochastic delay differential equations abstract we consider the cauchy problem for an abstract stochastic delay differential equation driven by fractional brownian motion with the hurst parameter h 1 2 we prove the existence and uniqueness for this problem when the coefficients have enough regularity the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of any order we prove the theorem by using the convergence of the picard lindel o f iterations in l 2 to a solution of this problem which admits a smooth density with respect to lebesgue s measure on r

metastable bubble solutions for the allen cahn equation with mass conservation in a multidimensional domain the slow motion behavior of internal layer solutions with spherical interfaces referred to as bubble solutions is analyzed for the nonlocal allen cahn equation with mass conservation this problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint the bubble is shown to drift exponentially slowly across the domain without change of shape toward the closest point on the boundary of the domain an explicit ordinary differential equation for the motion of the center of the bubble is derived by extending to a multidimensional setting the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension an asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point an analogy between slow bubble motion and the classical exit prob

rough path analysis via fractional calculus using fractional calculus we define integrals of the form b a f x t dy t where x and y are vector valued holder continuous functions of order p 1 3 1 2 and f is a continuously differentiable function such that f is a holder continuous for some a 1 2 under some further smooth conditions on f the integral is a continuous functional of x y and the tensor product x y with respect to the holder norms we derive some estimates for these integrals and we solve differential equations driven by the function y we discuss some applications to stochastic integrals and stochastic differential equations

gaussian type lower bounds for the density of solutions of sdes driven by fractional brownian motions in this paper we obtain gaussian type lower bounds for the density of solutions to stochastic differential equations sdes driven by a fractional brownian motion with hurst parameter h in the one dimensional case with additive noise our study encompasses all parameters h 0 1 while the multidimensional case is restricted to the case h 1 2 we rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools

the gibbs ensemble of a vortex filament we introduce a statistical ensemble for a single vortex filament of a three dimensional incompressible fluid the core of the vortex is modeled by a quite generic stochastic process we prove the existence of the partition function for both positive and a limited range of negative temperatures

variational solutions for partial differential equations driven by a fractional noise abstract in this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset d r d and driven by an infinite dimensional multiplicative fractional noise we introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an l 2 d valued fractional wiener process w h with hurst parameter h 1 1 1 whose covariance operator satisfies appropriate integrability conditions and where 0 1 denotes the holder exponent of the derivative of the nonlinearity in the stochastic term of the equations we also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of faedo galerkin approximations while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to w h

mixed stochastic differential equations with long range dependence existence uniqueness and convergence of solutions for a mixed stochastic differential equation involving standard brownian motion and an almost surely holder continuous process z with holder exponent c 1 2 we establish a new result on its unique solvability we also establish an estimate for difference of solutions to such equations with different processes z and deduce a corresponding limit theorem as a by product we obtain a result on existence of moments of a solution to a mixed equation under an assumption that z has certain exponential moments

stability and chaos in an inertial two neuron system abstract inertia is added to a continuous time hopfield 1984 effective neuron system we explore the effects on the stability of the fixed points of the system a two neuron system with one or two inertial terms added is shown to exhibit chaos the chaos is confirmed by lyapunov exponents power spectra and phase space plots

finitary simple lie algebras abstract an algebra is called finitary if it consists of finite rank transformations of a vector space we classify finitary simple lie algebras over a field of characteristic 0 we also describe finitary irreducible lie algebras

convergence of solutions of mixed stochastic delay differential equations with applications the paper is concerned with a mixed stochastic delay differential equation involving both a wiener process and a holder continuous process with 1 2 e g a fractional brownian motion with hurst parameter greater than 1 2 it is shown that its solution depends continuously on the coefficients and the initial data two applications of this result are given the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of euler approximations for mixed stochastic differential equations as a side result of independent interest the integrability of solution to mixed stochastic delay differential equations is established

exponential transients in continuous time liapunov systems we consider the convergence behavior of a class of continuous time dynamical systems corresponding to so called symmetric hopfield nets studied in neural networks theory we prove that such systems may have transient times that are exponential in the system dimension i e number of neurons despite the fact that their dynamics are controlled by liapunov functions this result stands in contrast to many proposed uses of such systems in e g combinatorial optimization applications in which it is often implicitly assumed that their convergence is rapid an additional interesting observation is that our example of an exponential transient continuous time system a simulated binary counter in fact converges more slowly than any discrete time hopfield system of the same representation size this suggests that continuous time systems may be worth investigating for gains in descriptional efficiency as compared to their discrete time counterparts

floquet multipliers of a metastable rotating wave in a chua yang ring network abstract a ring of n 2 m identical neuron cells with piecewise linear and saturated bidirectional coupling nonlinearities is considered the rotating wave under investigation exists for most values of the coupling parameters 0 and the dominant floquet multiplier is unstable and converges exponentially to 1 in the number of cells the remaining 2 m 2 nontrivial floquet multipliers converge exponentially to 0 a heteroclinic bifurcation curve and also the heteroclinic orbit connections are described by explicit formulas the entire work was motivated by electrical circuit experiments

relaxation closing probabilities and transition from oscillatory to chaotic attractors in asymmetric neural networks attractors in asymmetric neural networks with deterministic parallel dynamics present a chaotic regime at symmetry where the average length of the cycles increases exponentially with system size and an oscillatory regime at high symmetry where the average length of the cycles is 2 we show both with analytic arguments and numerically that there is a sharp transition at a critical symmetry between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size and the weights of their attraction basins are distributed as in a random map with reversal symmetry the timescale after which cycles are reached grows exponentially with system size n and the exponent vanishes in the symmetric limit where the transition can be related to the dynamics of the infinite system where cycles are never reached using the closing probabilities as a tool we also study the relaxation of the function where is the local field experienced by the neuron i in the symmetric system it plays the role of a ljapunov function which drives the system towards its minima through steepest descent this interpretation survives even if only on the average also for small asymmetry this acts like an effective temperature the larger the asymmetry the faster the relaxation and the higher the asymptotic value reached e reaches very deep minima in the fixed points of the dynamics which are reached with vanishing probability and attains a larger value on the typical attractors which are cycles of length 2

viability for differential equations driven by fractional brownian motion abstract in this paper we prove a viability result for multidimensional time dependent stochastic differential equations driven by fractional brownian motion with hurst parameter 1 2 h 1 using pathwise approach the sufficient condition is also an alternative global existence result for the fractional differential equations with restrictions on the state

unstable invariant manifolds for stochastic pdes driven by a fractional brownian motion abstract in this paper we consider a class of stochastic partial differential equations spdes driven by a fractional brownian motion fbm with the hurst parameter bigger than 1 2 the existence of local random unstable manifolds is shown if the linear parts of these spdes are hyperbolic for this purpose we introduce a modified lyapunov perron transform which contains stochastic integrals by the singularities inside these integrals we obtain a special lyapunov perron s approach by treating a segment of the solution over time interval 0 1 as a starting point and setting up an infinite series equation involving these segments as time evolves using this approach we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium

existence of solutions for nonlinear fractional stochastic differential equations abstract the fractional stochastic differential equations have wide applications in various fields of science and engineering this paper addresses the issue of existence of mild solutions for a class of fractional stochastic differential equations with impulses in hilbert spaces using fractional calculations fixed point technique stochastic analysis theory and methods adopted directly from deterministic fractional equations new set of sufficient conditions are formulated and proved for the existence of mild solutions for the fractional impulsive stochastic differential equation with infinite delay further we study the existence of solutions for fractional stochastic semilinear differential equations with nonlocal conditions examples are provided to illustrate the obtained theory

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

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

dynamics of simple electronic neural networks abstract we examine systems of one and two nonlinear threshold switching elements neurons of the kind used in electronic neural networks characteristics of these systems which deviate from standard ideal models are found to induce complex dynamics when the neurons possess a finite frequency response or a transfer characteristic with a time delay underdamped transients and instability leading to oscillation can occur inertia in the neuron connections is found to cause ringing about fixed points convoluted basin boundaries instability and spontaneous oscillation and chaotic behavior when driven furthermore the collective behavior of a network of multiple neurons can be underdamped even when the individual connections are overdamped these results imply that care should be exercised in implementing networks with electronic devices or when adding inertia to enhance the performance of optimizing networks

local mild solutions for rough stochastic partial differential equations abstract we investigate mild solutions for stochastic evolution equations driven by a fractional brownian motion fbm with hurst parameter h 1 3 1 2 in infinite dimensional banach spaces using elements from rough paths theory we introduce an appropriate integral with respect to the fbm this allows us to solve pathwise our stochastic evolution equation in a suitable function space

existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations abstract fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering this paper is concerned with the square mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a hilbert space the main objective of this paper is to establish the existence and uniqueness of square mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations a new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy finally an example is provided to illustrate the obtained theory

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

the riemann approach to stochastic integration using non uniform meshes in this note we shall prove that the stochastic integral with respect to a semimartingale can be defined by riemann s approach however in this approach we use non uniform meshes instead of the usual uniform meshes

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

metastable patterns in solutions of ut ϵ2uxx f u we consider the equation in question on the interval 0 x 1 having neumann boundary conditions with f u f u where f is a double well energy density with equal minima at u 1 the only stable states of the system are patternless constant solutions but given two phase initial data a pattern of interfacial layers typically forms far out of equilibrium the ensuing nonlinear relaxation process is extremely slow patterns persist for exponentially long times proportional to exp a l where a f 1 and l is the minimum distance between layers physically a tiny potential jump across a layer drives its motion we prove the existence and persistence of these metastable patterns and characterise accurately the equations governing their motion the point of view is reminiscent of center manifold theory a manifold parametrising slowly evolving states is introduced a neighbourhood is shown to be normally attracting and the parallel flow is characterised to high relative accuracy proofs involve a detailed study of the dirichlet problem spectral gap analysis and energy estimates

locally simple subalgebras of diagonal lie algebras abstract we describe up to isomorphism all locally simple subalgebras of any diagonal locally simple lie algebra

effects of noise and variations on the duration of transient oscillations in unidirectionally coupled bistable ring networks we study effects of spatiotemporal noise and spatial variations on long lasting transient oscillations in ring networks of unidirectionally coupled bistable elements neurons the duration of which increases exponentially with the number of neurons on the one hand spatiotemporal noise tends to sustain the transient oscillations the duration of the oscillations occurring from fixed initial conditions changes nonmonotonically with noise strength and takes the maximum value at intermediate noise strength further the duration of the oscillations is distributed in the form of the second power law and the mean duration increases with the number of neurons in the presence of an optimal noise on the other hand spatial variations degrade the exponential increases in the duration of the oscillations with the number of neurons in the presence of fixed biases in the steady states of the neurons there is a flat region in the distribution of the duration of the oscillations occurring under random initial conditions and an increase in the mean duration is almost linear with the number of neurons further the duration of the oscillations in an ensemble of the networks with random biases drawing from an identical distribution is distributed in the form of the second power law and the ensemble mean increases in proportion to the five halves power of the number of neurons

equation of motion for interacting pulses we develop a systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension the theory is applicable to both dissipative and dispersive systems in the case of the time dependent ginzburg landau equation which is a typical example of a dissipative system the front equation obtained is the same as has been obtained previously the pulse interaction is also derived for the korteweg char21 de vries equation emphasizing the difference between the cases with and without dissipative terms

the rate of convergence for the smoluchowski kramers approximation for stochastic differential equations with fbm in this paper we study the rate in the smoluchowski kramers approximation for the solution of the equation x t x b t h int 0 t b x s ds where b t h t in 0 t is a fractional brownian motion with hurst parameter h in big frac 1 2 1 big based on the techniques of malliavin calculus we provide an explicit bound on total variation distance for the rate of convergence

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

evolution equations driven by a fractional brownian motion abstract in this paper we study nonlinear stochastic evolution equations in a hilbert space driven by a cylindrical fractional brownian motion with hurst parameter h 1 2 and nuclear covariance operator we establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter h this result is applied to stochastic parabolic equation perturbed by a fractional white noise in this case if the coefficients are lipschitz continuous and bounded the existence and uniqueness of a solution holds if h d 4 the proofs of our results combine techniques of fractional calculus with semigroup estimates

locally semisimple and maximal subalgebras of the finitary lie algebras gl sl so and sp abstract we describe all locally semisimple subalgebras and all maximal subalgebras of the finitary lie algebras gl sl so and sp for simple finite dimensional lie algebras these classes of subalgebras have been described in the classical works of a malcev and e dynkin

complicated dynamics of a ring neural network with time delays this paper studies the dynamical behavior of a ring neural network with time delays on the basis of lyapunov s method the asymptotic stability of the equilibrium is first investigated and the delay dependent criteria ensuring global stability for the ring neural network are obtained moreover based on the global hopf bifurcation theorem for fde the conditions that guarantee the global existence of periodic solutions are determined it shows that periodic solutions bifurcating from the trivial equilibrium can continue when the time delay is far away from the critical value some examples are induced to illustrate our results in addition complicated dynamics of the model are investigated with the help of numerical simulation the study results show that the model exhibits period doubling bifurcations which lead eventually to chaos and the chaos can also directly occur via the bifurcations from the quasi periodic solutions

how delays affect neural dynamics and learning we investigate the effects of delays on the dynamics and in particular on the oscillatory properties of simple neural network models we extend previously known results regarding the effects of delays on stability and convergence properties we treat in detail the case of ring networks for which we derive simple conditions for oscillating behavior and several formulas to predict the regions of bifurcation the periods of the limit cycles and the phases of the different neurons these results in turn can readily be applied to more complex and more biologically motivated architectures such as layered networks in general the main result is that delays tend to increase the period of oscillations and broaden the spectrum of possible frequencies in a quantifiable way simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior adaptable delays are then proposed as one additional mechanism through which neural systems could tailor their own dynamics accordingly we derive recurrent backpropagation learning formulas for the adjustment of delays and other parameters in networks with delayed interactions and discuss some possible applications

stochastic shell models driven by a multiplicative fractional brownian motion abstract we prove existence and uniqueness of the solution of a stochastic shell model the equation is driven by an infinite dimensional fractional brownian motion with hurst parameter h 1 2 1 and contains a non trivial coefficient in front of the noise which satisfies special regularity conditions the appearing stochastic integrals are defined in a fractional sense first we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise for which we are able to derive important uniform estimates in some functional spaces then thanks to a compactness argument and these estimates we prove that these variational solutions converge to a limit solution which turns out to be the unique pathwise mild solution associated to the shell model with fractional noise as driving process

a note on stochastic schrodinger equations with fractional multiplicative noise abstract this work is devoted to non linear stochastic schrodinger equations with multiplicative fractional noise where the stochastic integral is defined following the riemann stieljes approach of zahle under the assumptions that the initial condition is in the sobolev space h q r n for a dimension n less than three and q an integer greater than or equal to zero that the noise is a q fractional brownian motion with hurst index h 1 2 1 and spatial regularity h q 4 r n as well as appropriate hypotheses on the non linearity we obtain the local existence of a unique pathwise solution in c 0 0 t h q r n c 0 0 t h q 2 r n for any 0 h contrary to the parabolic case standard fixed point techniques based on the mild formulation of the spde cannot be directly used because of the weak smoothing in time properties of the schrodinger semigroup we follow here a different route and our proof relies on a change of phase that removes the noise and leads to a schrodinger equation with a magnetic potential that is not differentiable in time

viability for stochastic functional differential equations with infinite memory driven by a fractional brownian motion abstract in this paper we prove a viability result for stochastic functional differential equation with infinite memory driven by a fractional brownian motion with hurst parameter 1 2 h 1 using pathwise approach

transient oscillations in continuous time excitatory ring neural networks with delay a ring neural network is a closed chain in which each unit is connected unidirectionally to the next one numerical investigations indicate that continuous time excitatory ring networks composed of graded response units can generate oscillations when interunit transmission is delayed these oscillations appear for a wide range of initial conditions the mechanisms underlying the generation of such patterns of activity are studied the analysis of the asymptotic behavior of the system shows that i trajectories of most initial conditions tend to stable equilibria ii undamped oscillations are unstable and can only exist in a narrow region forming the boundary between the basins of attraction of the stable equilibria therefore the analysis of the asymptotic behavior of the system is not sufficient to explain the oscillations observed numerically when interunit transmission is delayed this analysis corroborates the hypothesis that the oscillations are transient in fact it is shown that the transient behavior of the system with delay follows that of the corresponding discrete time excitatory ring network the latter displays infinitely many nonconstant periodic oscillations that transiently attract the trajectories of the network with delay leading to long lasting transient oscillations the duration of these oscillations increases exponentially with the inverse of the characteristic charge discharge time of the neurons indicating that they can outlast observation windows in numerical investigations therefore for practical applications these transients cannot be distinguished from stationary oscillations it is argued that understanding the transient behavior of neural network models is an important complement to the analysis of their asymptotic behavior since both living nervous systems and artificial neural networks may operate in changing environments where long lasting transients are functionally indistinguishable from asymptotic regimes

stochastic derivatives for fractional diffusions in this paper we introduce some fundamental notions related to the so called stochastic derivatives with respect to a given fieldq in our framework we recall well known results about markov wiener diffusions we then focus mainly on the case where x is a fractional diffusion and where q is the past the future or the present of x we treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of x when x solves a stochastic differential equation driven by a fractional brownian motion with hurst index h 1 2 we give explicit formulas 1 introduction there exist various ways to generalize the notion of differentiation on deterministic functions we may think of fractional derivatives or differentiation in the sense of the theory of distributions in both cases we lose a dynamical or geometric interpretation of tangent vectors velocities e g in the present work we seek to construct derivatives on stochastic processes which conserve a dynamical meaning our goal is motivated by the stochastic embedding of dynamical systems introduced in 2 this procedure aims at comprehending the following question how can we write an equation which contains the dynamical meaning of an initial ordinary differential equation and which extends this dynamical meaning to stochastic processes we refer to 3 for more details unfortunately for most of the stochastic processes used in physical models the limit zt h zt h

gradient type noises ii systems of stochastic partial differential equations abstract the present paper is the second and main part of a study of partial differential equations under the influence of noisy perturbations existence and uniqueness of function solutions in the mild sense are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems if the noise data are random the results may be seen as a pathwise approach to spde s for typical examples such as spatially one dimensional stochastic heat equations with additive or multiplicative perturbations of fractional brownian type we recover and extend known results in addition we propose to consider partial noises of low order