reaction kinetics of a b 0 a method to find the exact reaction rate density function of an a b 0 reaction on a finite 1d bar is derived we consider the case where the a and b species diffuse at the same rate have equal initial numbers with the initial location of either species on the bar being arbitrary the extension of the method to study a b 0 on fractals is also shown additionally we introduce the notion of a continual in time reaction front and a ceasing in time reaction front and provide a physical criterion for when either occurs

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

solving langevin equation with the bicolour rooted tree method national basic research program of china 973 program 2007cb814800 national natural science foundation of china 10475067

universal property of the housekeeping entropy production the entropy production of a nonequilibrium system with broken detailed balance is a random variable whose mean value is nonnegative the housekeeping entropy production which is a part of total entropy production is associated with the heat dissipation in maintaining a nonequilibrium steady state we derive a langevin type stochastic differential equation for the housekeeping entropy production the equation allows us to define a housekeeping entropic time ensuremath tau remarkably it turns out that the probability distribution of the housekeeping entropy production at a fixed value of ensuremath tau is given by the gaussian distribution regardless of system details the gaussian distribution is universal for any systems whether in the steady state or in the transient state and whether they are driven by time independent or time dependent driving forces we demonstrate the universal distribution numerically for model systems

synchrony and variability induced by spatially correlated additive and multiplicative noise in the coupled langevin model the synchrony and variability of the coupled langevin model subjected to spatially correlated additive and multiplicative noise are discussed we have employed numerical simulations and the analytical augmented moment method which is the second order moment method for local and global variables h hasegawa phys rev e 67 041903 2003 it has been shown that the synchrony of an ensemble is increased decreased by a positive negative spatial correlation in both additive and multiplicative noise although the variability for local fluctuations is almost insensitive to spatial correlations that for global fluctuations is increased decreased by positive negative correlations when a pulse input is applied the synchrony is increased for the correlated multiplicative noise whereas it may be decreased for correlated additive noise coexisting with uncorrelated multiplicative noise an application of our study to neuron ensembles has demonstrated the possibility that information is conveyed by the variance and synchrony in input signals which accounts for some neuronal experiments

reaction diffusion branching models of stock price fluctuations abstract several models of stock trading bak et al physica a 246 1997 430 are analyzed in analogy with one dimensional two species reaction diffusion branching processes using heuristic and scaling arguments we show that the short time market price variation is subdiffusive with a hurst exponent h 1 4 biased diffusion towards the market price and blind eyed copying lead to crossovers to the empirically observed random walk behavior h 1 2 at long times the calculated crossover forms and diffusion constants are shown to agree well with simulation data

population rate codes carried by mean fluctuation and synchrony of neuronal firings abstract a population of firing neurons is expected to carry information not only by the mean firing rate but also by fluctuation and synchrony among neurons in order to examine this possibility we have studied responses of neuronal ensembles to three kinds of inputs mean fluctuation and synchrony driven inputs the generalized rate code model including additive and multiplicative noise h hasegawa phys rev e 75 2007 051904 has been studied by direct simulations dss and the augmented moment method amm in which equations of motion for mean firing rate fluctuation and synchrony are derived results calculated by the amm are in good agreement with those by dss the independent component analysis ica of our results has shown that mean firing rate fluctuation or variability and synchrony may carry independent information in the population rate code model the input output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise as recently observed in prefrontal cortex a comparison is made between results obtained by the integrate and fire if model and our rate code model

strong convergence of the stopped euler maruyama method for nonlinear stochastic differential equations in this paper numerical methods for the nonlinear stochastic differential equations sdes with non global lipschitz drift coefficient are discussed the existing known results have only so far shown that the classical explicit euler maruyama em approximate solutions converge to the true solution in probability 22 23 more recently the authors in 16 proved that the classical em method will diverge in l 2 sense for the underlying sdes in this paper and those sdes with superlinearly growing coefficients these strongly indicate that the classical em method is not good enough for the highly nonlinear sdes however in this paper we introduce a modified em method using stopping time and show successfully that the discrete version of the modified em approximate solution converges to the true solution in the strong sense namely in l 2 with a order arbitrarily close to a half

exotic reaction fronts in the steady state abstract simulations of the steady state reaction diffusion front formed by opposing currents of species a and b are carried out for reaction schemes of the form ma nb 0 and for symmetrized reaction schemes of the form a nb 0 na b 0 the reaction rate profile is described by a single length scale the scaling properties including upper critical dimension and scaling exponents of this length are in good agreement with theoretical predictions

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

particle smoothing in continuous time a fast approach via density estimation we consider the particle smoothing problem for state space models where the transition density is not available in closed form in particular for continuous time nonlinear models expressed via stochastic differential equations sdes conventional forward backward and two filter smoothers for the particle filter require a closed form transition density with the linear gaussian euler maruyama discretization usually applied to the sdes to achieve this we develop a pair of variants using kernel density approximations to relieve the dependence and in doing so enable use of faster and more accurate discretization schemes such as runge kutta in addition the new methods admit arbitrary proposal distributions providing an avenue to deal with degeneracy issues experimental results on a functional magnetic resonance imaging fmri deconvolution task demonstrate comparable accuracy and significantly improved runtime over conventional techniques

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

a random map implementation of implicit filters implicit particle filters for data assimilation generate high probability samples by representing each particle location as a separate function of a common reference variable this representation requires that a certain underdetermined equation be solved for each particle and at each time an observation becomes available we present a new implementation of implicit filters in which we find the solution of the equation via a random map as examples we assimilate data for a stochastically driven lorenz system with sparse observations and for a stochastic kuramoto sivashinsky equation with observations that are sparse in both space and time

strong predictor corrector euler maruyama methods for stochastic differential equations with markovian switching in this paper numerical methods for solving stochastic differential equations with markovian switching sdewmss are developed by pathwise approximation the proposed family of strong predictor corrector euler maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path this paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions a natural analogue of the p stability criterion is studied numerical examples are given to illustrate the computational efficiency of the new predictor corrector euler maruyama approximation

weak second order splitting schemes for lagrangian monte carlo particle methods for the composition pdf fdf transport equations we study a class of methods for the numerical solution of the system of stochastic differential equations sdes that arises in the modeling of turbulent combustion specifically in the monte carlo particle method for the solution of the model equations for the composition probability density function pdf and the filtered density function fdf this system consists of an sde for particle position and a random differential equation for particle composition the numerical methods considered advance the solution in time with weak second order accuracy with respect to the time step size the four primary contributions of the paper are i establishing that the coefficients in the particle equations can be frozen at the mid time while preserving second order accuracy ii examining the performance of three existing schemes for integrating the sdes iii developing and evaluating different splitting schemes which treat particle motion reaction and mixing on different sub steps and iv developing the method of manufactured solutions mms to assess the convergence of monte carlo particle methods tests using mms confirm the second order accuracy of the schemes in general the use of frozen coefficients reduces the numerical errors otherwise no significant differences are observed in the performance of the different sde schemes and splitting schemes

stochastic resonance in bistable systems with nonlinear dissipation and multiplicative noise a microscopic approach abstract we have studied the stochastic resonance sr of bistable systems coupled to a bath with a nonlinear system bath interaction by using the microscopic generalized caldeira leggett cl model the adopted cl model yields the non markovian langevin equation with nonlinear dissipation and state dependent multiplicative diffusion which preserve the fluctuation dissipation relation fdr results of our simulations are given as follows 1 the spectral power amplification spa exhibits sr not only for a and b but also for while the stationary probability distribution function is independent of them where a and b denote magnitudes of multiplicative and additive noises respectively and expresses the relaxation time of ornstein uhlenbeck ou colored noise 2 the spa for coexisting additive and multiplicative noises has a single peak but two peak structure as functions of a b and or results 1 and 2 are qualitatively different from previous ones obtained by phenomenological langevin models where the fdr is not held or indefinite these show an importance of the fdr in a study on sr of open bistable systems

runge kutta methods for numerical solution of stochastic differential equations the way to obtain deterministic runge kutta methods from taylor approximations is generalized for stochastic differential equations now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an ito process a class of explicit runge kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed also two runge kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients numerical examples that compare the proposed schemes to standard ones are presented

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

on the associated graded rings of parameter ideals in buchsbaum rings is an invariant 1 a of a not depending on the particular choice of a parameter ideal q of a where i a q and e a denote the length of the amodule a q and the multiplicity of a relative to q respectively this is equivalent to saying that every system a a ad of parameters for a is a weak a sequence i e the equality a ai a i a ai m holds for all 0 i d 1 21 thus every cohen macaulay local ring a is clearly buchsbaum with i a 0 and vice versa in this sense the notion of buchsbaum rings is an extension of that of cohen macaulay rings and is originated in an answer of vogel 29 to a problem of buchsbaum 11 notice that the concept of buchsbaum rings may be canonically generalized to that of buchsbaum modules cf e g 11 221 now let us fix a parameter ideal q of a and let g denote the associated graded ring g a ona q qn of q recall that the ring g is cohenmacaulay if a is so actually g is a polynomial ring over a q in this case then what happens when a is buchsbaum this is a motive problem in the present research and our conclusion is briefly stated

numerical procedures for sample structures on stochastic differential equations procedures are discussed for the numerical construction of sample processes on stochastic differential equations structures of a relevant class of gaussian white noise functionals are analyzed to some closed forms and new numerical algorithms for single noise problems are presented up to the accuracy of o h3 or within errors of o h7 2 in the sense of the mean square norm

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

asymptotic expansions and lie algebras for some nonlinear filtering problems state estimation problems for systems involving small parameters are treated by both analytical and lie algebraic approximation techniques an asymptotic expansion for the unnormalized conditional density corresponding to the case of observations of a gauss markov process through a weak polynomial nonlinearity is computed and a convergence result is derived the expansion is related to certain approximations of the associated estimation lie algebra the convergence result is based on arguments used recently to prove existence and uniqueness and to estimate the tail behavior of solutions to nonlinear filtering problems with unbounded coefficients these arguments are in turn adapted from the analytical theory of parabolic pde s simulation results are presented to further assess the performance of the resulting approximate filters

buchsbaumness in rees modules associated to ideals of minimal multiplicity in the equi i invariant case abstract the present article gives certain conditions for rees modules to obtain buchsbaumness suppose a given m primary ideal is of minimal multiplicity in the equi i invariant case then it is shown that the positively graded submodule of the rees module must be buchsbaum and moreover that the rees module itself is also buchsbaum if the dimension of a given module is greater than one

lagrangian quadrature schemes for computing weak solutions of quantum stochastic differential equations lagrangian quadrature schemes for computing weak solutions of lipschitzian quantum stochastic differential equations are introduced and studied this is accomplished within the framework of the hudson parthasarathy formulation of quantum stochastic calculus and subject to matrix elements of solution being sufficiently differentiable results concerning convergence of these schemes in the topology of the locally convex space of solution are presented numerical examples are given

buchsbaumness of certain generalization of the associated graded modules in the equi i invariant case the present article introduces a generalization of the associated graded modules writing g a e suppose that the equi i invariant case occurs then it is shown that our graded module g a e is buchsbaum over r a for any 0 if the given a module e is so over a moreover it is also shown that e is buchsbaum over a iff g m e is so over r m for all 0 where m is the maximal ideal of a

stochastic calculus application to dynamic bifurcations and threshold crossings for the dynamic pitchfork bifurcation in the presence of white noise the statistics of the last time at zero are calculated as a function of the noise level and the rate of change of the parameter the threshold crossing problem used for example to model the firing of a single cortical neuron is considered concentrating on quantities that may be experimentally measurable but have so far received little attention expressions for the statistics of pre threshold excursions occupation density and last crossing time of zero are compared with results from numerical generation of paths

a haar like construction for the ornstein uhlenbeck process the classical haar construction of brownian motion uses a binary tree of triangular wedge shaped functions this basis has compactness properties which make it especially suited for certain classes of numerical algorithms we present a similar basis for the ornstein uhlenbeck process in which the basis elements approach asymptotically the haar functions as the index increases and preserve the following properties of the haar basis all basis elements have compact support on an open interval with dyadic rational endpoints these intervals are nested and become smaller for larger indices of the basis element and for any dyadic rational only a finite number of basis elements is nonzero at that number thus the expansion in our basis when evaluated at a dyadic rational terminates in a finite number of steps we prove the covariance formulae for our expansion and discuss its statistical interpretation

accurate monte carlo tests of the stochastic ginzburg landau model with multiplicative colored noise an accurate and fast monte carlo algorithm is proposed for solving the ginzburg landau equation with multiplicative colored noise the stable cases of solution for choosing time steps and trajectory numbers are discussed

exponential timestepping with boundary test for stochastic differential equations we present new numerical methods for scalar stochastic differential equations successive time increments are independent random variables with an exponential distribution we perform numerical experiments using a double well potential exponential timestepping algorithms are efficient for escape time problems because a simple boundary test can be performed at the end of each step

on the buchsbaum property of associated graded rings abstract let i be an m primary ideal in a buchsbaum local ring a m in this paper we investigate the buchsbaum property of the associated graded ring of i when the equality i 2 q i holds for some minimal reduction q of i however the buchsbaum property does not always follow even if i 2 q i so we give certain conditions for associated graded rings to be buchsbaum in this situation

balanced implicit methods for stiff stochastic systems this paper introduces some implicitness in stochastic terms of numerical methods for solving stiff stochastic differential equations and especially a class of fully implicit methods the balanced methods their order of strong convergence is proved numerical experiments compare the stability properties of these schemes with explicit ones

stability of weak numerical schemes for stochastic differential equations abstract this paper considers numerical stability and convergence of weak schemes solving stochastic differential equations a relatively strong notion of stability for a special type of test equations is proposed these are stochastic differential equations with multiplicative noise for different explicit and implicit schemes the regions of stability are also examined

statistical mechanics of kinks for quasi exactly solvable potentials we study the thermodynamics of kinks in the presence of quasi exactly solvable qes potentials using a langevin code implemented on a massively parallel computer specifically we study the potential v 2 1 2 for which certain exact solutions of the schrodinger equation are known provided the parameters and satisfy certain constraints for 1 this potential has two degenerate minima otherwise it has only one minimum thus a system with this potential is capable of describing a second order phase transition a la the 4 model however no exact solutions of the schrodinger equation for this potential exist we have obtained an exact solution for the kink in this model in addition we have calculated in a 1d model such quantities as the probability density function pdf field configuration and field field correlation functions both above and below the transition temperature tc for several temperatures these quantities help us understand the contribution to the specific heat from coherent structures such as domain walls kinks as opposed to the contribution from lattice vibrations we have calibrated our results against known exact solutions for limiting cases with very high accuracy

theoretical analysis for critical fluctuations of relaxation trajectory near a saddle node bifurcation a langevin equation whose deterministic part undergoes a saddle node bifurcation is investigated theoretically it is found that statistical properties of relaxation trajectories in this system exhibit divergent behaviors near a saddle node bifurcation point in the weak noise limit while the final value of the deterministic solution changes discontinuously at the point a systematic formulation for analyzing a path probability measure is constructed on the basis of a singular perturbation method in this formulation the critical nature turns out to originate from the neutrality of exiting time from a saddle point the theoretical calculation explains results of numerical simulations

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

noetherian local rings with buchsbaum associated graded rings is an invariant of a not depending on the choice of a parameter ideal q of a where f a q and e q denote respectively the length of the a module a q and the multiplicity of a relative to q see section 2 for detail here recall that if a is buchsbaum the local ring a is cohen macaulay and dim a d dim a p for any prime ideal p p m of a moreover the local cohomology modules a i d of a are vector spaces that is m h a 0 and one has the equality

diffusion limited annihilation with initially separated reactants a diffusion limited annihilation process a b ensuremath rightarrow o with species initially separated in space is investigated a heuristic argument suggests the form of the reaction rate in dimensions less or equal to the upper critical dimension mathit d mathit c 2 using this reaction rate we find that the width of the reaction front grows as mathit t 1 4 in one dimension and as mathit t 1 6 lnt 1 3 in two dimensions

computer simulations of multiplicative stochastic differential equations a class of robust algorithms for the computer simulation of stochastic differential equations with multiplicative noise is investigated excellent agreement is obtained with the known analytic behaviour of the kubo oscillator in the white noise limit the algorithms include a known first order one dimensional explicit method as well as implicit methods of increased stability a distinction is drawn between classes of stochastic differential equations depending on the type of spatial variation or curvature defined by the diffusion tensor this allows greatly simplified numerical implementations of the new algorithms in certain cases the results of different techniques are compared for the case of the kubo oscillator where a semi implicit technique gives the greatest accuracy

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

statistical mechanics of double sinh gordon kinks abstract we study the classical thermodynamics of the double sinh gordon dshg theory in 1 1 dimensions this model theory has a double well potential v cosh2 n 2 when n thus allowing for the existence of kinks and antikinks though it is non integrable the dshg model is remarkably amenable to analysis below we obtain exact single kink and kink lattice solutions as well as the asymptotic kink antikink interaction in the continuum limit finding the classical partition function is equivalent to solving for the ground state of a schrodinger like equation obtained via the transfer integral method for the dshg model this equation turns out to be quasi exactly solvable we exploit this property to obtain exact energy eigenvalues and wavefunctions for several temperatures both above and below the symmetry breaking transition temperature provided n 1 2 6 the availability of exact results provides an excellent testing ground for large scale langevin simulations the probability distribution function pdf calculated from langevin dynamics is found to be in striking agreement with the exact pdf obtained from the ground state wavefunction this validation points to the utility of a pdf based computation of thermodynamics utilizing langevin methods in addition to the pdf field field and field fluctuation correlation functions were computed and also found to be in excellent agreement with the exact results

stationary solutions of linear stochastic delay differential equations applications to biological systems recently k uchler and mensch stochastics stochastics rep 40 23 1992 derived exact stationary probability densities for linear stochastic delay differential equations this paper presents an alternative derivation of these solutions by means of the fokker planck approach introduced by guillouzic phys rev e 59 3970 1999 61 4906 2000 applications of this approach which is argued to have greater generality are discussed in the context of stochastic models for population growth and tracking movements

molecular modeling of proteins and mathematical prediction of protein structure this paper discusses the mathematical formulation of and solution attempts for the so called protein folding problem the static aspect is concerned with how to predict the folded native tertiary structure of a protein given its sequence of amino acids the dynamic aspect asks about the possible pathways to folding and unfolding including the stability of the folded protein from a mathematical point of view there are several main sides to the static problem the selection of an appropriate potential energy function the parameter identification by fitting to experimental data and the global optimization of the potential the dynamic problem entails in addition the solution of because of multiple time scales very stiff ordinary or stochastic differential equations molecular dynamics simulation or in case of constrained molecular dynamics of differential algebraic equations a theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs the present paper gives a self contained introduction to the necessary background from physics and chemistry and surveys some of the literature it also discusses the various mathematical problems arising some deficiencies of the current models and algorithms and possible past and future attacks to arrive at solutions to the protein folding problem

weak order stochastic runge kutta methods for commutative stochastic differential equations a new explicit stochastic runge kutta scheme of weak order 2 is proposed under a commutativity condition which is derivative free and which attains order 4 for ordinary differential equations the weak order conditions are derived by utilizing multi colored rooted tree analysis and a solution is found in a transparent way the scheme is compared with other derivative free and weak second order schemes in numerical experiments

numerical solution of the kardar parisi zhang equation in one two and three dimensions abstract on a course grained level a family of microscopic growth processes may be described by a stochastic differential equation which is solved numerically for surface dimensions d 1 2 and 3 dimensional analysis shows that the spatial discretization parameter has the meaning of an effective coupling constant the numerical stability of the euler integration scheme is discussed for the strong coupling exponents defined by surface width time the following effective values were obtained d 1 0 330 0 004 and d 2 0 24 0 005 considering the width and its ensemble fluctuations at constant dimensionless time the transition between strong and weak coupling phases is located in d 3 for the largest coupling for which reliable data are available we obtain an effective exponent close to the best estimates on discrete models d 3 0 17