the limits of stochastic integrals of differential forms this paper is concerned with the integration of 1 forms against the markov stochastic process associated with a second order elliptic differential operator in divergence form it focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity this extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure applications are given for example it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved cartan hadamard manifold has an asymptotic direction recovering and strengthening the previous arguments of pratt sullivan and others the approach can also be used to construct a levy area for such processes to study the thinness of sets for the elliptic operator and probably has wider applications

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

algorithms for phase field computation of the dendritic operating state at large supercoolings this paper deals with numerical solutions of the phase field model of solidification in two dimensions in the context of dendritic growth finite difference methods associated with vectorized algorithms are employed to solve the phase field equations the temperature equation is solved by an alternating direction implicit scheme and the equation for the phase field by an explicit euler scheme we identify a region of parameter space namely large supercoolings where we can compute dendritic morphologies that display steady state dendrite tip operating conditions that are independent of computational parameters

weak convergence of symmetric diffusion processes in this paper we show the convergence of forms in the sense of mosco associated with the part form on relatively compact open set of dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular dirichlet space setting we also get the same assertion under dirichlet space in infinite dimensional setting as a result of this we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by modified pseudo metric used in k th sturm

optimal perturbation for enhanced chaotic transport the issue of determining the best perturbation which results in optimal chaotic flux across a separatrix is addressed using the melnikov function and lobe dynamics this theoretical analysis is motivated mainly through micro fluidic devices for which this problem has become important recently both two and three dimensional flows are analysed utilising a fourier transform representation the nature of the perturbation which maximises this flux for each frequency value is obtained the resulting optimally attainable flux is computed a concise bound on this flux is presented in terms of the supremum norm of the normal component of the perturbing velocity and the size of the heteroclinic manifold in this instance where the spatial part of the perturbation is permitted to be chosen based on the frequency it is shown that greater flux is achievable for smaller frequencies the theory is illustrated through two examples

phase field computations of single needle crystals crystal growth and motion by mean curvature the phase field model for free boundaries consists of a system of parabolic differential equations in which the variables represent a phase or order parameter and temperature respectively the parameters in the equations are related directly to the physical observables including the interfacial width epsilon which we can regard as a free parameter in computation the phase field equations can be used to compute a wide range of sharp interface problems including the classical stefan model its modification to incorporate surface tension and or surface kinetic terms the cahn allen motion by mean curvature the hele shaw model etc also included is anisotropy in the equilibrium and dynamical forms generally considered by materials scientists by adjusting the parameters the computations can be varied continuously from single needle dendritic to faceted crystals the computational method consists of smoothing a sharp interface problem within the scaling of distinguished limits of the phase field e

structure of motion near saddle points and chaotic transport in hamiltonian systems generic symmetry and transport properties of near separatrix motion in 11 2 degree of freedom hamiltonian systems are studied first the rescaling invariance of motion near saddle points with respect to the transformation epsilon lambdaepsilon chi chi pi of the amplitude epsilon and phase chi of the time periodic perturbation is recalled the rescaling parameter lambda depends only on the frequency of the perturbation and the behavior of an unperturbed hamiltonian near a saddle point additional rescaling symmetry of the motion with respect to transformation epsilon lambda 1 2 epsilon chi chi pi 2 is found for some hamiltonian systems possessing symmetry in the phase space it is shown that these rescaling invariance properties of motion lead to strong periodic or quasiperiodic dependencies of all statistical characteristics of the chaotic motion near the separatrix on log 10 epsilon with the period log 10 lambda these properties are examined for different models of chaotic motion by direct numerical integrations of equations of motion as by well as using a computationally efficient method of the separatrix mapping

a simple level set method for solving stefan problems a simple level set method for solving stefan problems is presented this method can be applied to problems involving dendritic solidification our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance our method is accurate with respect to some exact solutions of the stefan problem results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification

harmonicity and invariance on slices of the boolean cube in a recent work with kindler and wimmer we proved an invariance principle for the slice for low influence low degree harmonic multilinear polynomials a polynomial in x 1 ldots x n is harmonic if it is annihilated by sum i 1 n frac partial partial x i here we provide an alternative proof for general low degree harmonic multilinear polynomials with no constraints on the influences we show that any real valued harmonic multilinear polynomial on the slice whose degree is o sqrt n has approximately the same distribution under the slice and cube measures our proof is based on ideas and results from the representation theory of s n along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales while such decompositions have been used in the past for stationary reversible markov chains our decomposition is applied in a non stationary non reversible setup we also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof finally we provide independent simple proofs for the known facts that 1 one cannot distinguish between the slice and the cube based on functions of o n coordinates and 2 boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o sqrt n

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

canonical maps near separatrix in hamiltonian systems a systematic and rigorous method to construct symplectic maps near separatrix of generic hamiltonian systems subjected to time periodic perturbations is developed it is based on the method of canonical transformation of variables to construct hamiltonian maps j phys a 35 2811 2002 using canonical transformation of variables and the first order approximation for the generating function the general form of mapping in terms of time and energy variables is obtained different limiting cases of the mapping are considered the method is illustrated for simple hamiltonian systems with one and a large number of saddle points it is also applied to derive mappings for the periodic driven morse oscillator describing the process of stochastic excitation and dissociation of diatomic molecules the so called canonical kepler map is derived for the one dimensional hydrogen atom in a microwave field

weak convergence of diffusion processes on wiener space let be a gaussian measure on a suslin space x h be the corresponding cameron martin space and e i h be an orthonormal basis of h suppose that n n is a sequence of probability measures which converges weakly to a probability measure consider a sequence of dirichlet forms varepsilon n where varepsilon n f int x nabla h f 2 h rho n rm d gamma and sqrt rho n in w 2 1 gamma we prove some sufficient conditions for mosco convergence varepsilon n to varepsilon where varepsilon f int x nabla h f 2 h rho rm d gamma in particular if x is a hilbert space sup n sqrt rho n w 2 1 gamma infty and partial e i rho n rho n can be uniformly approximated by finite dimensional conditional expectations rm ie mathcal f n mu n big partial e i rho n rho n big for every fixed e i then under broad assumptions varepsilon n to varepsilon mosco and the distributions of the associated stochastic processes converge weakly

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

non symmetric perturbations of symmetric dirichlet forms abstract we provide a path space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local dirichlet form the representation is a combination of feynman kac and girsanov formulas and extends previously known results in the framework of symmetric diffusion processes through the use of the hardy class of smooth measures which contains the kato class of smooth measures

n v limit for stochastic dynamics in continuous particle systems we provide an n v limit for the infinite particle infinite volume stochastic dynamics associated with gibbs states in continuous particle systems on d d 1 starting point is an n particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset d with finite volume lebesgue measure v the aim is to approximate the infinite particle infinite volume stochastic dynamic by the above n particle dynamic in as n and v such that n v where is the particle density first we derive an improved ruelle bound for the canonical correlation functions under an appropriate relation between n and v then tightness is shown by using the lyons zheng decomposition the equilibrium measures of the accumulation points are identified as infinite volume canonical gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle infinite volume stochastic dynamics via the associated martingale problem assuming a property closely related to markov uniqueness and weaker than essential self adjointness via mosco convergence techniques we can identify the accumulation points as markov processes and show uniqueness i e all accumulation corresponding to one invariant canonical gibbs measure coincide the proofs work for general repulsive interaction potentials of ruelle type and all temperatures densities and dimensions d 1 respectively may have a nontrivial negative part and infinite range as e g the lennard jones potential additionally our result provides as a by product an approximation of grand canonical gibbs measures by finite volume canonical gibbs measures with empty boundary condition

ito formula for c 1 functions of semimartingales we establish an ito formula forc1 functions of processes whose time reversal are semimartingales and forc1 functions whose first derivatives are holder continuous of any parameter and the process comes out from a stochastic flow of homeomorphism

markov chains in smooth banach spaces and gromov hyperbolic metric spaces a metric space x has markov type 2 if for any reversible flnite state markov chain fztg with z0 chosen according to the stationary distribution and any map f from the state space to x the distance dt from f z0 to f zt satisfles e d 2 k 2 te d 2 for some k k x 2 has markov type 2 this proves a conjecture of ball we also show that trees hyperbolic groups and simply connected riemannian manifolds of pinched negative curvature have markov type 2 our results are applied to settle several conjectures on lipschitz extensions and embeddings in particular we answer a question posed by johnson and lindenstrauss in 1982 by showing that for 1 q 2 p 1 any lipschitz mapping from a subset of lp to lq has a lipschitz extension deflned on all of lp

tail estimates for markovian rough paths we work in the context of markovian rough paths associated to a class of uniformly subelliptic dirichlet forms 26 and prove a better than exponential tail estimate for the accumulated local p variation functional which has been introduced and studied in 17 we comment on the significance of these estimates to a range of currently studied problems including the recent results of ni hao 32 and chevyrev and lyons 18

a two sided stochastic integral and its calculus let x be a forward diffusion and y a backward diffusion both defined on 0 1 x t and y t being respectively adapted to the past of a wiener process w and to its future increments we construct a two sided stochastic integral of the form mathop smallint limits 0 t phi u x u y u dw u which generalizes the backward and forward ito integrals simultaneously our construction is quite intuitive and leads to a generalized stochastic calculus it is also shown that for each fixed t our integral coincides with that defined by skorohod in 18

diffusions on path and loop spaces existence finite dimensional approximation and holder continuity we construct ornstein uhlenbeck processes and more general diffusion processes on path and loop spaces of riemannian manifolds by finite dimensional approximation we also show holder continuity of the sample paths w r t the supremum norm the proofs are based on the lyons zheng decomposition

scaling limit of stochastic dynamics in classical continuous systems we investigate a scaling limit of gradient stochastic dynamics associated with gibbs states in classical continuous systems on rd d 1 the aim is to derive macroscopic quantities from a given microscopic or mesoscopic system the scaling we consider has been investigated by brox in 1980 rost in 1981 spohn in 1986 and guo and papanicolaou in 1985 under the assumption that the underlying potential is in c30 and positive we prove that the dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the dirichlet form of a generalized ornstein uhlenbeck process the proof is based on the analysis and geometry on the configuration space which was developed by albeverio kondratiev and rockner in 1998 and works for general gibbs measures of ruelle type hence the underlying potential may have a singularity at the origin only has to be bounded from below and may not be compactly supported therefore singular interactions of physical interest are covered as for example the one given by the lennard jones potential which is studied in the theory of fluids furthermore using the lyons zheng decomposition we give a simple proof for the tightness of the scaled processes we also prove that the corresponding generators however do not converge in the l2 sense this settles a conjecture formulated by brox by rost and by spohn

phase transitions with semi diffuse interfaces in this paper we examine new phase field models with semi diffuse interfaces these models have the property that the 1 1 planar phase transitions take place over a finite interval the models also support multiple interface solutions with interfaces centered at arbitrary points l sub 1 l sub 2 l sub n these solutions correspond to local minima of an entropy functional rather than saddle points and are dynamically stable the classical models have no such exact solutions but they do support solutions with n equally spaced transition points where the order parameter transitions between valves p sub min n and p sub max n satisfying 1 p sub min n 0 p sub max n 1 these solutions of the classical model are saddle points of the entropy functional associated with those models and are not dynamically stable

formes de dirichlet generales et densite des variables aleatoires reelles sur l espace de wiener abstract it is shown that the main properties of dirichlet spaces can be proved without the classical local compacity hypothesis to this purely measurable framework we extend the functional calculus and the energy image density property these results are then applied to the ornstein uhlenbeck semi group on wiener space where it is obtained that the real random variables which are solutions of lipschitzian stochastic differential equations always have probability densities after perhaps an initial deterministic phase

stochastic calculus over symmetric markov processes without time reversal we refine stochastic calculus for symmetric markov processes without using time reverse operators under some conditions on the jump functions of locally square integrable martingale additive functionals we extend nakao s divergence like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi everywhere starting points which are refinements of the previous results under the law for almost everywhere starting points this refinement of stochastic calculus enables us to establish a generalized fukushima decomposition for a certain class of functions locally in the domain of dirichlet form and a generalized ito formula

convergence of dirichlet forms and associated schrodinger operators abstract we study the convergence of local dirichlet forms and associated schrodinger operators conditions are given for n in order that the schrodinger operators n n associated with dirichlet forms u x 2 n 2 x dx generate contraction semigroups which converge strongly in l2 r d moreover conditions are given which permit one to identify the limit contraction semigroups and schrodinger operators

change of variable formulas for non anticipative functionals on path space we derive a change of variable formula for non anticipative functionals defined on the space of rd valued right continuous paths with left limits the functionals are only required to possess certain directional derivatives which may be computed pathwise our results lead to functional extensions of the ito formula for a large class of stochastic processes including semimartingales and dirichlet processes in particular we show the stability of the class of semimartingales under certain functional transformations

anisotropy propagation failure and wave speedup in traveling waves of discretizations of a nagumo pde this article is concerned with effect of spatial and temporal discretizations on traveling wave solutions to parabolic pdes nagumo type possessing piecewise linear bistable nonlinearities solution behavior is compared in terms of waveforms and in terms of the so called a c relationship where a is a parameter controlling the bistable nonlinearity by varying the potential energy difference of the two phases and c is the wave speed of the traveling wave uniform spatial discretizations and a stable linear multistep methods in time are considered results obtained show that although the traveling wave solutions to parabolic pdes are stationary for only one value of the parameter a a0 spatial discretization of these pdes produce traveling waves which are stationary for a nontrivial interval of a values which include a0 i e failure of the solution to propagate in the presence of a driving force this is true no matter how wide the interface is with respect to the discretization for temporal discretizations at large wave speeds the set of parameter a values for which there are traveling wave solutions is constrained an analysis of a complete discretization points out the potential for nonuniqueness in the a c relationship

adaptive mesh refinement computation of solidification microstructures using dynamic data structures we study the evolution of solidification microstructures using a phase field model computed on an adaptive finite element grid we discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the phase field model at low undercooling in particular we show that the computational complexity of solving any phase boundary problem scales with the interface arclength when using an adapting mesh moreover the use of dynamic data structures allows us to simulate system sizes corresponding to experimental conditions which would otherwise require lattices greater than 217 217elements we examine the convergence properties of our algorithm we also present two dimensional time dependent calculations of dendritic evolution with and without surface tension anisotropy we benchmark our results for dendritic growth with microscopic solvability theory finding them to be in good agreement with theory for high undercoolings at low undercooling however we obtain higher values of velocity than solvability theory at low undercooling where transients dominate in accord with a heuristic criterion which we derive

on the small time behavior of ornstein uhlenbeck processes with unbounded linear drifts in this paper we study the small time behavior of ornstein ulenbeck processes with unbounded linear drifts on hilbert spaces the large deviations estimates are obtained the general theory of dirichlet forms lyons zheng s decompositions and the convolution representation of the processes play an important role

on the small time asymptotics of diffusion processes on hilbert spaces in this paper we establish a small time large deviation principle and obtain the following small time asymptotics lim t 0 2t log p x 0 b x t c d 2 b c for diffusion processes on hilbert spaces where d b c is the intrinsic metric between two subsets b and c associated with the diffusions the case of perturbed ornstein uhlenbeck processes is treated separately at the end of the paper

integrability of functionals of dirichlet processes probabilistic representations of semigroups and estimates of heat kernels abstract this paper consists of three parts in part i we obtain results on the integrability of functional of exponential type of dirichlet processes in part ii we give a striking probabilistic representation of semigroup probably non markovian associated with a non divergence operator part iii is devoted to perturbation bounds on the operator norm of semigroups and a new short proof of the off diagonal estimates of the heat kernel associated with a divergence operator the theory of dirichlet forms and forward backward martingales decompositions play a central role in the whole paper