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

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

instability of the cauchy kovalevskaya solution for a class of nonlinear systems we prove that in any c infty neighborhood of an analytic cauchy datum there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first order nonlinear systems we use a method initiated by g m etivier for elliptic systems based on the representation of solutions and on the fbi transform in our case the system can be hyperbolic at initial time but the characteristic roots leave the real line at positive times

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

a connection between quantum dynamics and approximation of markov diffusions it is observed that the existence of an attracting set in the class of solutions to the stochastic lagrangian variational principle leads to a natural problem of convergence of diffusions in the carlen class it is then shown how dynamical properties enable one to prove some convergence results in the two dimensional gaussian case

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

higher order resonances and instability of high frequency wkb solutions abstract this paper focuses on the destabilizing role of resonances in high frequency wkb solutions specifically we study higher order resonances associated with higher order harmonics generated by nonlinearities we give examples of systems and solutions for which such resonances generate instantaneous instabilities even though the equations linearized around the leading wkb terms are initially stable meaning in particular that the key destabilizing terms are not present in the data

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

lagrangian variational principle in stochastic mechanics gauge structure and stability the lagrangian variational principle with the classical action leads in stochastic mechanics to madelung s fluid equations if only irrotational velocity fields are allowed while new dynamical equations arise if rotational velocity fields are also taken into account the new equations are shown to be equivalent to the gauge invariant system of a schrodinger equation involving a four vector potential a and the coupled evolution equation of magnetohydrodynamical type for the vector field a a general energy theorem can be proved and the stability properties of irrotational and rotational solutions investigated

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

signal detection in fractional gaussian noise several results related to the reproducing kernel hilbert space of fractional brownian motion are presented to facilitate the study of signal detection in additive fractional gaussian noise this hilbert space is completely characterized and an alternative characterization for the restriction of this class of functions to a compact interval 0 t is given infinite and finite interval whitening filters for fractional brownian motion are also developed the application of these results to the signal detection problem yields necessary and sufficient conditions for a deterministic or stochastic signal to produce a nonsingular shift when embedded in additive fractional gaussian noise a formula for the likelihood ratio corresponding to any deterministic nonsingular shift is developed

global solutions for the gravity water waves equation in dimension 3 we show existence of global solutions for the gravity water waves equation in dimension 3 in the case of small data the proof combines energy estimates which yield control of l 2 related norms with dispersive estimates which give decay in l 8 to obtain these dispersive estimates we use an analysis in fourier space the study of space and time resonances is then the crucial point

a limit theorem for financial markets with inert investors we study the effect of investor inertia on stock price fluctuations with a market microstructure model comprising many small investors who are inactive most of the time it turns out that semi markov processes are tailor made for modelling inert investors with a suitable scaling we show that when the price is driven by the market imbalance the log price process is approximated by a process with long range dependence and non gaussian returns distributions driven by a fractional brownian motion consequently investor inertia may lead to arbitrage opportunities for sophisticated market participants the mathematical contributions are a functional central limit theorem for stationary semi markov processes and approximation results for stochastic integrals of continuous semimartingales with respect to fractional brownian motion

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

layer oscillations in reaction diffusion systems this paper considers a two component system of reaction diffusion equations involving two parameters varepsilon and tau varepsilon tau u 1 varepsilon 2 u xx f u v qquad v

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

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

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

convergence of nelson diffusions with time dependent electromagnetic potentials some recent results on the convergence of nelson diffusions are extended to the case of schrodinger operators with time dependent electromagnetic potentials it is proven that the sequence p n n 1 of measures on the canonical space of physical trajectories associated to the solutions of schrodinger equations in nelson s scheme corresponding to the sequence vn an n 1 c1 r r l2 r3 converges in the total variation norm under the assumptions that for every fixed t the scalar potentials vn t converge in r the space of rollnik class potentials and the vector potentials an t converge in lloc r l2 r3 in order to prove these results conditions are given under which solutions of schrodinger equations are continuous in the time dependent electromagnetic potentials in the norm of the sobolev space h1 r3

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

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

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

on hyperbolicity and gevrey well posedness part two scalar or degenerate transitions abstract for first order quasi linear systems of partial differential equations we formulate an assumption of a transition from initial hyperbolicity to ellipticity this assumption bears on the principal symbol of the first order operator under such an assumption we prove a strong hadamard instability for the associated cauchy problem namely an instantaneous defect of holder continuity of the flow from g to l 2 with 0 0 the limiting gevrey index 0 depending on the nature of the transition we restrict here to scalar transitions and non scalar transitions in which the boundary of the hyperbolic zone satisfies a flatness condition as in our previous work for initially elliptic cauchy problems b morisse on hyperbolicity and gevrey well posedness part one the elliptic case arxiv 1611 07225 the instability follows from a long time cauchy kovalevskaya construction for highly oscillating solutions this extends recent work of n lerner t nguyen and b texier the onset of instability in first order systems to appear in j eur math soc

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

derivation of the zakharov equations this article continues the study initiated in 27 7 of the validity of the zakharov model which describes langmuir turbulence we give an existence theorem for a class of singular quasilinear equations this theorem is valid for prepared initial data we apply this result to the euler maxwell equations which describes laser plasma interactions to obtain in a high frequency limit an asymptotic estimate that describes solutions of the euler maxwell equations in terms of wkb approximate solutions the leading terms of which are solutions of the zakharov equations due to the transparency properties of the euler maxwell equations evidenced in 27 this study is carried out in a supercritical highly nonlinear regime in such a regime resonances between plasma waves electromagnetric waves and acoustic waves could create instabilities in small time the key of this work is the control of these resonances the proof involves the techniques of geometric optics of joly metivier and rauch 12 13 recent results by lannes on norms of pseudodifferential operators 14 and a semiclassical paradifferential calculus

convergence of nelson fiffusions let t t n n 1 be solutions of schrodinger equations with potentials form bounded by 1 2 and initial data inh1 d letp p n n 1 be the probability measures on the path space c d given by the corresponding nelson diffusions we show that if t n n 1 converges to t inh1 d uniformly int over compact intervals then open image in new window converges to open image in new window in total variation t 0 moreover if the potentials are in the kato classk d we show that the above result follows fromh1 convergence of initial data andk d convergence of potentials

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