mesoscopic simulation of ostwald ripening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening this phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange ising model this model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations the typical cluster size is observed to grow like t1 3 over range of times with faster growth at later times the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level

higher order interpolated lattice schemes for multidimensional option pricing problems this article presents a new class of higher order space time discretization schemes for multidimensional diffusions via lattice systems which involve space interpolation techniques through the explanation of the methodology the author shows that the algorithm can be simply represented by multiplications of sparse matrices the key idea is to combine the weak approximation approach for stochastic differential equations and some techniques on high dimensional spaces to break the curse of dimensionality cubature formulas and sparse grid interpolation techniques the first objective is to investigate the error estimates derived from short time asymptotics of certain semigroup operators together with the discussion of numerical stability as the second objective several computational experiments for some derivative pricing models are presented in one and three dimensional settings

exact simulation of stochastic volatility and other affine jump diffusion processes the stochastic differential equations for affine jump diffusion models do not yield exact solutions that can be directly simulated discretization methods can be used for simulating security prices under these models however discretization introduces bias into the simulation results and a large number of time steps may be needed to reduce the discretization bias to an acceptable level this paper suggests a method for the exact simulation of the stock price and variance under hestons stochastic volatility model and other affine jump diffusion processes the sample stock price and variance from the exact distribution can then be used to generate an unbiased estimator of the price of a derivative security we compare our method with the more conventional euler discretization method and demonstrate the faster convergence rate of the error in our method specifically our method achieves an o s 1 2 convergence rate where s is the total computational budget the convergence rate for the euler discretization method is o s 1 3 or slower depending on the model coefficients and option payoff function

nonuniversality in low volume fraction ostwald ripening we study ostwald ripening in the regime of small volume fraction and consider spatially periodic systems whose size is smaller than the screening length within the snapshot perspective we obtain an explicit characterization of the leading order deviation to the classical mean field theory by lifshitz slyozov and wagner lsw using this representation we show that the corrections are not universal in the sense that the mean value has a strong dependence on geometry and arbitrarily large fluctuations can happen with finite probability

sound penetration through an arbitrarily shaped aperture in a soft screen analytical approach van bladel has reduced the problem of low frequency scattering through an aperture in a soft screen to a sequence of static integral equations analytical solutions are known at the moment for a circular and an elliptic aperture only a new analytical method is proposed here which is valid for the apertures of arbitrary shape the method is based on an integral representation for the reciprocal distance between two points obtained by the author earlier specific approximate formulae are derived for evaluating the average value of the zeroth order term in the low frequency expansion for an aperture in the shape of a polygon a triangle a rectangle a rhombus a circular sector and a circular segment all the formulae are checked against the numerical solutions known in the literature and a good agreement is confirmed

exact sampling of jump diffusions this paper develops a method for the exact simulation of a skeleton a hitting time and other functionals of a one dimensional jump diffusion with state dependent drift volatility jump intensity and jump size the method requires the drift function to be c1 the volatility function to be c2 and the jump intensity function to be locally bounded no further structure is imposed on these functions the method leads to unbiased simulation estimators of security prices transition densities hitting probabilities and other quantities numerical results illustrate its features

an euler type method for the strong approximation of the cox ingersoll ross process we analyse the strong approximation of the cox ingersoll ross cir process in the regime where the process does not hit zero by a positivity preserving drift implicit euler type method as an error criterion we use the p th mean of the maximum distance between the cir process and its approximation on a finite time interval we show that under mild assumptions on the parameters of the cir process the proposed method attains up to a logarithmic term the convergence of order 1 2 this agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with lipschitz coefficients despite the fact that the cir process has a non lipschitz diffusion coefficient

localization and exact simulation of brownian motion driven stochastic differential equations generating sample paths of stochastic differential equations sde using the monte carlo method finds wide applications in financial engineering discretization is a popular approximate approach to generating those paths it is easy to implement but prone to simulation bias this paper presents a new simulation scheme to exactly generate samples for sdes the key observation is that the law of a general sde can be decomposed into a product of the law of standard brownian motion and the law of a doubly stochastic poisson process an acceptance rejection algorithm is devised based on the combination of this decomposition and a localization technique the numerical results corroborates that the mean square error of the proposed method is in the order of ot 1 2 which is superior to the conventional discretization schemes furthermore the proposed method also can generate exact samples for sde with boundaries which the discretization schemes usually find difficulty in dealing with

statistical mechanical theory of coarsening of spherical droplets abstract a new statistical mechanical theory of diffusion controlled droplet coarsening is presented with the aid of a scaling expansion method a spatial graining is carried out in a manner consistent with an expansion in droplet volume fraction to obtain kinetic equations for a single distribution function of droplets it is shown that there two characteristic stages of coarsening depending on their space time scales an intermediate stage and a late stage in both stages new kinetic equations are systematically derived to order these equations have two terms at order a collisionless drift term and a collision term the collision term is shown to be different from the conventional encounter integral discussed by lifshitz and slyozov since the former is of order and describes distant soft collisions while the latter is of order and describes close hard collisions it is shown in both stages that the mean droplet radius increases as the cube root of the time t 1 3 a scaling behavior of the distribution function is also found in both stages in particular in the late stage this scaling behavior is shown to coincide with that obtained by lifshitz and slysov in the limit o it is also pointed out that a naive expansion in powers of breaks down due to the long range nature of interactions among droplets through diffusions fluctuations around the kinetic equations are also explicitly explored they are shown to be nonthermal fluctuations generated by the soft collision process and to be small especially in the late stage they are shown to obey a linear gaussian markov process satisfying a fluctuation dissipation relation of the second kind

elementary derivation of kinetic equations for ostwald ripening a direct and elementary derivation is presented for the kinetic equation of the single droplet distribution function and the variance equation that describe ostwald ripening the derivation is basically algebraical and the screening effects of diffusional interactions among droplets is taken into account automatically by a projection technique the results correct up to the square root of the volume fraction confirm the kinetic equations derived previously by tokuyama and kawasaki except for a correction to the variance equation

averaging for diffusions with fine grained boundaries this article investigates the limiting behavior of a diffusion in a half space with a complicated boundary condition the boundary condition implements a reflection condition everywhere except a number of small sets or holes that meet dirichlet or mixed boundary conditions probabilistic methods associated with the feynman kac formula are used to find the limiting behavior of the diffusion equations as the number of holes gets large and the size of each hole is reduced with particular scaling homogenization occurs and we see that the complicated boundary condition is replaced by a simple mixed boundary condition depending on the capacitance and distribution of the holes

fluctuation results for the wiener sausage on demontre des resultats de fluctuation pour le volume de la saucisse de wiener associe au mouvement brownien a d dimensions et a un ensemble compact de capacite positive

ostwald ripening for dilute systems under quasistationary dynamics we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying size evolving according to quasistationary dynamics under the scaling hypotheses that 1 size distance 2 deviation from sphericity size are both initially small we show robustness of the almost spherical shape by associating centers and radii to each particle and then derive rigorously a set of o d e s for the radii which we relate to the lifschitz slyozov wagner theory of coarsening and also establish an estimate for the speed of the centers

dynamics of fluctuations in ostwald ripening a new equation of motion for the structure function abstract a new method of finding an equation of motion for the structure function s k t from the kinetic equations recently derived for the late stage of phase separation is presented s k t is shown to be scaled by two kinds of characteristics lengths the average droplet radius r t and the screening length l t as s k t r t 3 s x q q 1 2 with x k r t and q kl t where q is the volume fraction of the minority phase and r t and l t satisfy the power laws t q 1 2 r t and r t t 1 3 the scaled structure function s x q q 1 2 is found to obey a first order differential equation with two source terms which originate from two kinds of fluctuations initial thermal fluctuations and nonthermal fluctuations generated by soft collisions among droplets the soft collision process is shown to produce a new type of narrowing of s x q q 1 2 which is different from the usual one that occurs as q increases the results are compared with other theories

derivation of the lsw theory for ostwald ripening by homogenization methods the coarsening of a solid phase in an undercooled liquid is described by a stefan problem with surface tension the solid phase is characterized by a large number of balls with small volume fraction and small capacity we solve the quasi static equation as well as the parabolic problem and construct approximate solutions by means of comparison principles the framework of young measures is used to pass to the limit of infinitely many particles we obtain that the capacity is the crucial quantity to get the conservation law for the particle size distribution which is studied in the classical theory for ostwald ripening by lifshitz slyozov wagner lsw

on weak approximation of stochastic differential equations through hard bounds by mathematical programming upper and lower hard bounds of the expected value on stochastic differential equations can be obtained with the help of mathematical programming and the dynkin formula without recourse to monte carlo sample paths simulation in this paper we show that feasible solutions of those optimization approaches further provide useful additional information namely feasible solutions provide upper and lower bounds for arbitrary intermediate times and or different initial states we also show that the optimization approach can be applied to stochastic differential equations with a random initial state our theoretical analyses are illustrated by numerical results on the survival probability for a square root diffusion on the moment estimation of a doleans dade stochastic exponential with jumps and on pricing of a barrier basket option under the multiasset black scholes model

on the macroscopic description of slow viscous flow past a random array of spheres we give rigorous derivation of brinkman s equation as the effective equation for slow viscous flow in porous media with high porosity the medium is composed of spherical obstacles distributed randomly and the microscopic flow is described by the stokes equation along the road we givew21 convergence estimates for the point sources approximation

a law of large numbers and a central limit theorem for the schrodinger operator with zero range potentials we consider the schrodinger operator with zero range potentials onn points of three dimensional space independently chosen according to a common distributionv x under some assumptions we prove that whenn goes to infinity the sequence converges to a schrodinger operator with an effective potential the fluctuations around the limit operator are explicitly characterized

a local time approach to the self intersections of brownian paths in space we study the brownian functional alpha x b int limits b int delta x w t w s dsdt wherewt is a brownian path in two or three dimensions forb off the diagonal we identify x b with a local time and establish the holder continuity of x b in bothx andb

on an elaboration of m kac s theorem concerning eigenvalues of the laplacian in a region with randomly distributed small obstacles we removem balls of centersw 1 w m with the same radius m from a bounded domain inr 3 with smooth boundary let k m w m denote thek th eigenvalue of the laplacian in m balls under the dirichlet condition we consider k m w m as a random variable on a probability space w 1 w m and we examine a precise behaviour of k m w m asm we give an elaboration of m kac s theorem

about non differentiable functions we develop the notion of local fractional derivative introduced by kolvankar and gangal it allows a fine study of the local structure of irregular fractal functions using this tool we extend classical theorems of analysis extrema rolle to non differentiable functions in particular we prove a generalized taylor expansion theorem we introduce a new derivative of real order and discuss its properties

hydrodynamic friction and the capacitance of arbitrarily shaped objects the translational friction coefficient and the capacitance of a variety of objects are calculated with a probabilistic method involving hitting the probed objects with random walks launched from an enclosing spherical surface this method is applied to exactly solvable examples to test the program accuracy and to physically important and analytically intractable examples cube chain of spheres at the vertices of self avoiding and random walks etc large fluctuations in the friction of polymer chains with a random coil structure are found to give large deviations from the mean field kirkwood riseman theory and hydrodynamic fluctuation effects are found to diminish with the chain swelling accompanying excluded volume interaction capacity applications are reviewed and our probabilistic estimates of polymer friction are compared with previous calculations using alternative methods transients to the capacity and related properties are expressed in terms of fluctuations in the wiener sausage volume volume swept out by a brownian particle where a repeated visit to a spatial region does not contribute to the volume increase in time

etude asymptotique par des mesures de 135 1135 1135 1de saucisses de wiener localisees let b t t 0 be a brownian motion in 3 we denote its wiener sausage byw t 1 n w t 1 n left m in mathbb r 3 exists s leqq t parallel b s m parallel leqq frac 1 n right spitzer has proved in 64 that lim n vol w t 1 n 2 t

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

coarsening kinetics in finite clusters we address the problem of diffusional interactions in a finite sized cluster of spherical particles for volume fractions v v in the range 0 0 01 we determined the quasistatic monopole diffusion solution for n particles distributed at random in a continuous matrix a global mass conservation condition is employed obviating the need for any external boundary condition the numerical results provide the instantaneous snapshot growth or shrinkage rate of each particle precluding the need for extensive time dependent computations the close connection between these snapshot results and the coarse grained kinetic constants are discussed a square root dependence of the deviations of the rate constants from their zero volume fraction value is found for the higher v v investigated the behavior is consistent with predictions from the diffusion debye h uckel screening theory by contrast a cube root dependence reported in earlier numerical studies is found for the lower v v investigated the roll over region of the volume fraction where the two asymptotics merge depends on the number of particles n alone a theoretical estimate for the roll over point predicts that the corresponding v v varies as n ensuremath 2 in good agreement with the numerical results

perturbations of eigenvalues embedded at threshold i one and three dimensional solvable models we examine perturbations of eigenvalues and resonances for a class of multi channel quantum mechanical model hamiltonians describing a particle interacting with a localized spin in dimension d 1 3 we consider unperturbed hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various types of perturbations on the spectral singularities we provide algorithms to obtain convergent series expansions for the coordinates of the singularities

theory of phase separation dynamics in quenched binary mixtures a systematic theory of the dynamics of phase separation of quenched binary systems into metastable states is presented not only the kinetic equation for the single droplet size distribution function f r t but also the linear equation for the structure function s k t which are valid over the entire time region after the nucleation stage are derived from a unified point of view the functions f r t and s k t are shown to satisfy the dynamical scaling relations f r t n t r t f r r t and s k t mathit k mathit m t mathrm ensuremath mathit d ensuremath phi t mathrm ensuremath delta ensuremath psi k mathit k mathit m t with mathit k mathit m mathrm ensuremath 1 t ensuremath propto r mathrm ensuremath phi 1 mathit d where n is the number density of the minority phase r the average droplet radius ensuremath phi the volume fraction of the minority phase mathit k mathit m the peak position of s k t and d 3 here three characteristic stages are then shown to exist after the nucleation stage the growth stage with ensuremath delta 2 where the power laws are given by r ensuremath propto mathit t 1 2 n ensuremath propto mathit t 0 mathit k mathit m ensuremath propto mathit t 0 and ensuremath phi ensuremath propto mathit t 3 2 the intermediate stage with ensuremath delta 1 d where r ensuremath propto mathit t 1 4 n ensuremath propto mathit t mathrm ensuremath 2 3 mathit k mathit m ensuremath propto mathit t mathrm ensuremath 2 9 and ensuremath phi ensuremath propto mathit t 1 12 and the coarsening stage with ensuremath delta 1 d where r ensuremath propto mathit t 1 3 n ensuremath propto mathit t mathrm ensuremath 1 mathit k mathit m ensuremath propto mathit t mathrm ensuremath 1 3 and ensuremath phi ensuremath propto mathit t 0 two crossovers are then observed in the time exponents the time evolution of the scaling functions f ensuremath rho t and ensuremath psi x t are thus studied explicitly including their asymptotic behavior and their dependence on ensuremath phi

non self similar behavior in the lsw theory of ostwald ripening the classical lifshitz slyozov wagner theory of domain coarsening predicts asymptotically self similar behavior for the size distribution of a dilute system of particles that evolve by diffusional mass transfer with a common mean field here we consider the long time behavior of measure valued solutions for systems in which particle size is uniformly bounded i e for initial measures of compact support we prove that the long time behavior of the size distribution depends sensitively on the initial distribution of the largest particles in the system convergence to the classically predicted smooth similarity solution is impossible if the initial distribution function is comparable to any finite power of distance to the end of the support we give a necessary criterion for convergence to other self similar solutions and conditional stability theorems for some such solutions for a dense set of initial data convergence to any self similar solution is impossible

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

the laplacian in regions with many small obstacles fluctuations around the limit operator we consider the laplacian m in 3 or in a bounded region of 3 with dirichlet boundary conditions on the surfaces of some identical small neighborhoods ofm randomly distributed points in the limit whenm goes to infinity and their linear size decreases as 1 m we give here a stronger form of the result showing the convergence of the above operator to c x wherec x is the limit density of electrostatic capacity of the obstacles in particular results on the rate of convergence and on the fluctuations of m around the limit operator are given

high order discretization schemes for the cir process application to affine term structure and heston models this paper presents weak second and third order schemes for the cox ingersoll ross cir process without any restriction on its parameters at the same time it gives a general recursive construction method to get weak second order schemes that extends the one introduced by ninomiya and victoir cite nv combining these both results this allows to propose a second order scheme for more general affine diffusions simulation examples are given to illustrate the convergence of these schemes on cir and heston models algorithms are stated in a pseudocode language

random media and eigenvalues of the laplacian let be a fixed number 1 we remove m balls of centersw1 w m beta with the same radius m from a bounded domain inr3 we consider the asymptotic behaviour of thekth eigenvalue of the laplacian in omega backslash overline m beta balls under the dirichlet condition as a random variable on a probability space w 1 w m beta in omega m beta whenm

a limit evolution problem for time dependent point interactions abstract we study the diffusion in r 3 of a particle interacting with n fixed points through point interactions whose strength varies in time under mild assumptions on the time dependence of the strengths we prove existence for all times and uniqueness of the solution for which we provide a rather explicit expression we also prove that under a suitable rescaling of the interaction strengths the solution converges when n to the solution of a diffusion equation with a regular killing term potential we use properties of the local self adjoint extensions of the laplacian and results from the theory of fractional integrals and derivatives

high order approximation of pearson diffusion processes this paper focuses on pearson diffusions and the spectral high order approximation of their related fokker planck equations the pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient they are widely used in the physical and chemical sciences engineering rheology environmental sciences and financial mathematics in recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations analytical solutions have been derived for some of the pearson diffusions including the ornstein uhlenbeck cox ingersoll ross and jacobi processes however analytical investigations and computations for diffusions with so called heavy tailed ergodic distributions are more difficult to perform the novelty of this research is the development of an accurate and efficient numerical method to solve the fokker planck equations associated with pearson diffusions with different boundary conditions comparisons between the numerical predictions and available time dependent and equilibrium analytical solutions are made the solution of the fokker planck equation is approximated using a reduced basis spectral method the advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving fokker planck type equations

the effect of the geometry of the particle distribution in ostwald ripening based on 1 we derive equations for the radii and the centers that we relate to the lifshitz slyozov wagner theory

on the effect of stochastic fluctuations in the dynamics of the lifshitz slyozov wagner model in this paper the dynamics of a system of spherical particles that fill a small volume fraction of the space and that evolves in a concentration field is discussed corrections to the lifshitz slyozov wagner lsw model that take into account the stochastic character of the problem are computed it is proved under suitable smallness assumptions for the volume fraction filled by the particles that the effect of these corrections does not modify much the dynamics of the self similar solutions of the lsw system of equations

on calculus of local fractional derivatives abstract local fractional derivative lfd operators have been introduced in the recent literature chaos 6 1996 505 513 being local in nature these derivatives have proven useful in studying fractional differentiability properties of highly irregular and nowhere differentiable functions in the present paper we prove leibniz rule chain rule for lfd operators generalization of directional lfd and multivariable fractional taylor series to higher orders have been presented

on first order corrections to the lsw theory ii finite systems we consider first order corrections to the classical theory by lifshitz slyozov and wagner lsw for systems with a finite number of particles numerical simulations in v e fradkov etal phys rev e 53 3925 3932 1996 show a cross over in the scaling of the correction term from 1 3 to 1 2 is the volume fraction of particles when the system size becomes larger than the screening length we rigorously derive this cross over for the rate of change of the energy starting from the monopole approximation the proof exploits the fact that the rate of change of energy has a variational characterization

on the local fractional derivative abstract we present the necessary conditions for the existence of the kolwankar gangal local fractional derivatives kg lfd and introduce more general but weaker notions of lfds by using limits of certain integral averages of the difference quotient by applying classical results due to stein and zygmund 1965 16 we show that the kg lfd is almost everywhere zero in any given intervals we generalize some of our results to higher dimensional cases and use integral approximation formulas obtained to design numerical schemes for detecting fractional dimensional edges in signal processing

the equations of ostwald ripening for dilute systems we consider a dilute mixture in 3d of a finite number of particles initially close to spherical but of varying sizes and representing one of the phases evolving according to the quasistatic dynamics under the scaling hypotheses that 1 typical size typical distance and 2 deviation from sphericity typical size are small we associate centers and radii to each particle for the whole evolution and derive rigorously a set of odes fo the radii which we relate to the lifschitz slyosov wagner theory of coarsening

screening in interacting particle systems we consider the green s function of the laplace operator in domains with spherical holes particles under natural assumptions on the distribution of particles we show that the green s function decays exponentially over distances larger than the screening length this result is fundamental for example when deriving effective equations for coarsening systems in unbounded domains