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

linear quadratic control for stochastic equations in a hilbert space with fractional brownian motions a linear quadratic control problem with a finite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional gaussian noise is formulated and solved the feedback form of the optimal control and the optimal cost are given explicitly the optimal control is the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise the covariance of the noise as well as the control operator in the system equation can in general be unbounded so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain some examples of controlled stochastic partial differential equations are given

nonlinear filtering and measure valued processes we construct a sequence of branching particle systems with time and space dependent branching mechanisms whose expectation converges to the solution of the zakai equation this gives an alternative numerical method to solve the filtering problem

non linear rough heat equations this article is devoted to define and solve an evolution equation of the form dyt yt dt dx t yt wherestands for the laplace operator on a space of the form l p r n and x is a finite dimensional noisy nonlinearity whose typical form is given by xt n i 1 x i fi where each x x 1 x n is a holder function generating a rough path and each fi is a smooth enough function defined on l p r n the generalization of the usual rough path theory allowing to cope with such

approximate controllability of fractional stochastic differential equations driven by mixed fractional brownian motion via resolvent operators abstractin this manuscript we investigate the approximate controllability of a class of fractional stochastic differential equations driven by mixed fractional brownian motion in hilbert space with hurst parameter h 12 1 the main results are obtained by employing fractional calculus analytic resolvent operators and with the help of schaefer s fixed point theorem under the assumption that the corresponding linear system is approximately controllable finally in order to validate the obtained theoretical results an illustrative example is provided

generalized covariations local time and stratonovich ito s formula for fractional brownian motion with hurst index h 1 4 given a locally bounded real function g we examine the existence of a 4 covariation g b h b h b h b h where b h is a fractional brownian motion with a hurst index h ge tfrac 1 4 we provide two essential applications first we relate the 4 covariation to one expression involving the derivative of local time in the case h tfrac 1 4 generalizing an identity of bouleau yor type well known for the classical brownian motion a second application is an ito formula of stratonovich type for f b h the main difficulty comes from the fact b h has only a finite 4 variation

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

vector valued stochastic delay equations a weak solution and its markovian representation abstract a class of stochastic delay equations in a type 2 umd banach space e driven by the cylindrical wiener process is studied we investigate two concepts of solutions weak and generalised strong and give conditions under which they are equivalent we present an evolution equation approach in a banach space e p e l p 1 0 e proving that the solutions can be reformulated as e p valued markov processes based on the markovian representation we prove the existence and continuity of the solutions the results are applied to stochastic reaction diffusion equations with delays

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

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

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

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

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

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

superprocesses in a brownian environment superprocesses are idealized models for the evolution of a population with geographical structure motion and unisex reproductive behaviour in this paper the reproductive behaviour in particular the offspring distribution is related to an exterior process the environment we characterize these superprocesses as the unique solution of certain evolution equations we also prove that they can be obtained as limits of systems of branching particles with minimal variance branchings and faster non classical branching rates

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

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

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

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

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

logarithmic corrections for the percolative properties of the four state potts model clusters in the potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state droplets can be obtained by adding bonds with probability p 1 exp k between the sites in a cluster where k is the potts model s inverse temperature when q to 4 renormalization group rg fixed points describing clusters and droplets will coalesce leading to logarithmic corrections we calculate the precise form of these corrections used in a differential rg method our predictions are then tested using extensive monte carlo calculations theory and the simulations are found to be in excellent agreement

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

retarded evolution systems driven by fractional brownian motion with hurst parameter h 1 2 abstract this paper is devoted to the study of pathwise solutions of delayed evolution systems driven by a hilbert valued holder continuous function with holder index greater than 1 2 in the first part we shall obtain the existence and uniqueness of a mild solution for such a system which turns out to define a non autonomous dynamical system in the second part we consider retarded stochastic systems driven by a hilbert valued fractional brownian motion with hurst parameter bigger than 1 2 and show that the corresponding mild solutions generate a random dynamical system provided that the canonical version of the fractional brownian motion is chosen as the metric dynamical system

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

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

viability for stochastic functional differential equations in hilbert spaces driven by fractional brownian motion abstract in this paper we consider a class of stochastic functional differential equations in hilbert spaces driven by a fractional brownian motion with hurst parameter 1 2

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

impulsive neutral stochastic functional integro differential equations with infinite delay driven by fbm in this paper we study a class of impulsive neutral stochastic functional integro differential equations with infinite delay driven by a standard cylindrical wiener process and an independent cylindrical fractional brownian motion fbm with hurst parameter h 1 2 1 in the hilbert space we prove the existence and uniqueness of the mild solution for this kind of equations with the coefficients satisfying some non lipschitz conditions which include the classical lipschitz conditions as special case an example is provided to illustrate the theory some well known results are generalized and extended

viability for delay evolution equations with nonlocal initial conditions abstract we prove a sufficient condition for a time dependent closed set to be viable with respect to a delay evolution inclusion subjected to an implicit nonlocal initial condition an application to a comparison problem is also included

conformal invariance finite size scaling and surface magnetic exponent of the potts model in two dimensions the analysis of renormalisation group equations for the q state potts model in two dimensions is generalised to include the presence of a surface field hs besides the bulk one hb combined with conformal invariance results this allows prediction of a surface susceptibility behaviour delta 2 f delta hs delta hb approximately l 1 8 1 c lnl 15 16 for a q 4 system of size l at the critical temperature this prediction is checked by a monte carlo based finite size scaling analysis which also nicely reproduces the exact and conjectured magnetic surface exponents for q 2 and 3 respectively for the baxter wu model similar methods give results consistent with the q 4 behaviour without logarithmic corrections c 0

mutually catalytic branching in the plane finite measure states we study a pair of populations in r 2 which undergo diffusion and branching the system is interactive in that the branching rate of each type is proportional to the local density of the other type for a diffusion rate sufficiently large compared with the branching rate the model is constructed as the unique pair of finite measure valued processes which satisfy a martingale problem involving the collision local time of the solutions the processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations in the long term limit global extinction of one type is shown the process constructed is a rescaled limit of the corresponding z 2 lattice model studied by d a dawson and e a perkins ann probab 26 1998 1088 1138 and resolves the large scale mass time space behavior of that model

viability for nonautonomous semilinear differential equations abstract let x be a reflexive banach space a d a x x the infinitesimal generator of a compact c0 semigroup s t x x t 0 d a locally closed subset in x and t x f t x a function on a b d which is locally integrably bounded measurable with respect to t and continuous with respect to x the main result of the paper is t heorem under the general assumptions above a necessary and sufficient condition in order that for each a b dthere exists at least one mild solutionu t dof du dt t au t f t u t satisfyingu is the tangency condition below there is a negligible subset z of a b such that for eacht a b zand eachx d lim inf h 0 1 h d s h x hf t x d 0

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

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

logarithmic corrections to finite size scaling in strips abstract the corrections to the finite size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width which occur when the hamiltonian contains a marginal operator are computed using conformal invariance they show a calculable universal logarthmic character for the four state potts model they agree with numerical data

measurable viability theorems and the hamilton jacobi bellman equation abstract we prove viability and invariance theorems for systems with dynamics depending on time in a measurable way and having time dependent state constraints x t f t x t x t p t in the above t p t is an absolutely continuous set valued map and t x f t x is a set valued map which is measurable with respect to t and upper semicontinuous or continuous or locally lipschitz with respect to x for this aim we investigate infinitesimal generators of reachable maps and the lebesgue points of set valued maps the results are applied to define and to study lower semicontinuous solutions of the hamilton jacobi bellman equation u t h t x u x 0 with the hamiltonian h measurable with respect to time locally lipschitz with respect to x and convex in the last variable

exact rates of convergeance for a branching particle approximation to the solution of the zakai equation in crisan gaines and lyons siam j appl probab 58 1998 313 342 we describe a branching particle algorithm that produces a particle approximation to the solution of the zakai equation and find an upper bound for the rate of convergence of the mean square error in this paper the exact rate of convergence of the mean square error is deduced also several variations of the branching algorithm with better rates of convergence are introduced

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

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

generalized integration and stochastic odes stochastic forward integrals for processes more general than semimartingales are shown to exist generalized forms of ito wentzell formula and covariation formula are proved and one dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved this generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift

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

ergodicity of stochastic differential equations driven by fractional brownian motion we study the ergodic properties of finite dimensional syste ms of sdes driven by non degenerate additive fractional brownian motion with arbitrary hurst parameter h 2 0 1 a general framework is constructed to make precise the notions of invariant measure and stationary state for such a system we then prove under rather weak dissipativity conditions that such an sde possesses a unique stationary solution and that the convergence rate of an arbitrary solut ion towards the stationary one is at least algebraic a lower bound on the exponent is also given

stochastic heat equation driven by fractional noise and local time the aim of this paper is to study the d dimensional stochastic heat equation with a multiplicative gaussian noise which is white in space and has the covariance of a fractional brownian motion with hurst parameter h 0 1 in time two types of equations are considered first we consider the equation in the ito skorohod sense and later in the stratonovich sense an explicit chaos expansion for the solution is obtained on the other hand the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the brownian motion

nonlinear fractional stochastic pdes and bdsdes with hurst parameter in 1 2 1 abstract we study a class of semilinear stochastic partial differential equations driven by a fractional brownian motion with hurst parameter h 1 2 1 for this end we use the doubly stochastic interpretation through a backward doubly stochastic differential equations driven by both a standard and an independent fractional brownian motion the doss sussmann transformation is employed to establish the link between the backward doubly stochastic differential equation and a backward stochastic differential equation driven only by the standard brownian motion through which the stochastic viscosity solution of the stochastic partial differential equation is studied

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

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

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

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

the existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion abstract in this paper we investigate the existence uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional brownian motion b q h t d x t a x t f t x t d t g t d b q h t with hurst parameter h 1 2 1 we also consider the existence of weak solutions

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

convergence of a branching particle method to the solution of the zakai equation we construct a sequence of branching particle systems un convergent in distribution to the solution of the zakai equation the algorithm based on this result can be used to solve numerically the filtering problem the result is an improvement of the one presented in a recent paper crisan and t lyons prob theory related fields 109 1997 pp 217 244 because it eliminates the extra degree of randomness introduced there

sample paths estimates for stochastic fast slow systems driven by fractional brownian motion we analyze the effect of additive fractional noise with hurst parameter h 1 2 on fast slow systems our strategy is based on sample paths estimates similar to the approach by berglund and gentz in the brownian motion case yet the setting of fractional brownian motion does not allow us to use the martingale methods from fast slow systems with brownian motion we thoroughly investigate the case where the deterministic system permits a uniformly hyperbolic stable slow manifold in this setting we provide a neighborhood tailored to the fast slow structure of the system that contains the process with high probability we prove this assertion by providing exponential error estimates on the probability that the system leaves this neighborhood we also illustrate our results in an example arising in climate modeling where time correlated noise processes have become of greater relevance recently

superprocesses in random environments we study the limiting behavior of large branching particle systems undergoing random motion whose branching mechanism is affected by a random environment the weak convergence result is established for a sequence of such particle systems and the limiting process is characterized as the unique solution of a martingale problem the proof of uniqueness of the solution for the martingale problem requires an extension of standard duality techniques

a class of second order mckean vlasov stochastic evolution equations driven by fractional brownian motion and poisson jumps abstract this paper focuses on a class of second order mckean vlasov stochastic evolution equations driven by a fractional brownian motion and poisson jumps specifically we allow nonlinearities and the jump term to depend not only of the state of the solution but also on the corresponding probability law of the state the global existence and uniqueness of mild solutions is established under various growth conditions and a related stability result is discussed an example is presented to illustrate the applicability of the theory

sharp gaussian regularity on the circle and applications to the fractional stochastic heat equation abstract a sharp regularity theory is established for homogeneous gaussian fields on the unit circle two types of characterizations for such a field to have a given almost sure uniform modulus of continuity are established in a general setting the first characterization relates the modulus to the field s canonical metric the full force of fernique s zero one laws and talagrand s theory of majorizing measures is required the second characterization ties the modulus to the field s random fourier series representation as an application it is shown that the fractional stochastic heat equation has up to a non random constant a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation s additive noise a random fourier series characterization is also given

viability theorems for a class of differential operator inclusions abstract this paper discusses the following viability problem of a differential inclusion x t ax t f x t x 0 x0 k t 0 x t k where k is a compact subset of a banach space x a is the infinitesimal generator of a compact differentiable semigroup of bounded linear operators on x and f k x is upper semicontinuous with compact convex values we show that a natural tanegential condition is necessary and sufficient for the existence of a global solution to this problem

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

evolution equations on locally closed graphs and applications abstract let x be a banach space let a d a x x be the infinitesimal generator of a c 0 semigroup let i be a nonempty bounded interval and let k i x be a given multi valued function by using the concept of a quasi tangent set introduced by c rj necula vrabie o c rj m necula i i vrabie necessary and sufficient conditions for viability for a semilinear differential inclusions trans amer math soc 361 2009 343 390 o c rj m necula i i vrabie viability invariance and applications north holland a mathematics studies vol 207 elsevier 2007 and using a tangency condition expressed in the terms of this concept we establish a necessary and sufficient condition for mild viability referring to evolution inclusions of the form u t a u t f t u t where f is a multi function defined on the graph of k as applications we deduce a sufficient condition for null controllability and a comparison result for semilinear evolution inclusions

bulk surface and hull fractal dimension of critical ising clusters in d 2 the authors present accurate numerical calculations of the fractal dimension d and surface dimension ds of the critical ising cluster in d 2 the results clearly support the values d 187 96 ds 5 6 which are consistent with ising clusters being described by tricritical q 1 potts model exponents from this the hull dimension dh of critical ising clusters is found to be dh 11 8 consistent with numerical work of other authors

stochastic averaging for a class of two time scale systems of stochastic partial differential equations abstract this paper focuses on systems of stochastic partial differential equations that have a slow component driven by a fractional brownian motion and a fast component driven by a fast varying diffusion we establish an averaging principle in which the fast varying diffusion process acts as a noise and is averaged out in the limit the slow process is shown to have a limit in the l 2 sense which is characterized by the solution of a stochastic partial differential equation driven by a fractional brownian motion whose coefficients are averages of that of the original slow process with respect to the stationary measure of the fast varying diffusion this averaging principle paves a way for reduction of computational complexity the implication is that one can ignore the complex original systems and concentrate on the average systems instead

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

fractional stochastic differential equations satisfying fluctuation dissipation theorem we propose in this work a fractional stochastic differential equation fsde model consistent with the over damped limit of the generalized langevin equation model as a result of the fluctuation dissipation theorem the differential equations driven by fractional brownian noise to model memory effects should be paired with caputo derivatives and this fsde model should be understood in an integral form we establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to gibbs measure in the linear forcing regime we show rigorously the algebraic convergence to gibbs measure when the fluctuation dissipation theorem is satisfied and this verifies that satisfying fluctuation dissipation theorem indeed leads to the correct physical behavior we further discuss possible approaches to analyze the ergodicity and convergence to gibbs measure in the nonlinear forcing regime while leave the rigorous analysis for future works the fsde model proposed is suitable for systems in contact with heat bath with power law kernel and subdiffusion behaviors

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

flow invariance for semilinear evolution equations under generalized dissipativity conditions abstract let x be a real banach space let a d a x x be a linear operator which is the infinitesimal generator of a c 0 semigroup and let b d x x be a nonlinear perturbation which is continuous on level sets of d with respect to a lower semicontinuous l s c functional we discuss the existence of a nonlinear semigroup s providing mild solutions to the semilinear abstract cauchy problem sp x u t a b u t t 0 u 0 x d and satisfying a certain lipschitz like estimation and an exponential growth condition using the discrete schemes approximation it is proved that the combination of a subtangential condition and a semilinear stability condition in terms of a metric like functional is necessary and sufficient for the generation of such a semigroup s

fractal dimensions of potts clusters potts clusters are connected sets of nearest neighbour sites for which the potts variable is in the same state at criticality there exists a fractal cluster using arguments from the renormalization group conformal invariance and numerical simulations the author determines the bulk surface hull and red bond dimension of these clusters as a function of the number of potts states

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

clusters and droplets in the q state potts model a potts correlated polychromatic percolation is studied the clusters are made of sites corresponding to a given value of the q state potts variables connected by bonds being active with probability pb to treat this problem an s state potts hamiltonian diluted with q state potts variables instead of lattice gas variables is introduced to which the the migdal kadanoff renormalisation group is applied it is found for a particular choice of pb 1 e k where k is the potts coupling constant divided by the boltzmann factor that these clusters called droplets diverge at the potts critical point with potts exponents

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