a uniformly convergent difference scheme for a semilinear singular perturbation problem we present a difference scheme for solving a semilinear singular perturbation problem with any number of turning points of arbitrary orders it is shown that a solution of the scheme converges uniformly in a perturbation parameter to that of the continuous problem

forward backward and symmetric stochastic integration we define three types of non causal stochastic integrals forward backward and symmetric our approach consists in approximating the integrator two optics are considered the first one is based on traditional usual stochastic calculus and the second one on wiener distributions

singularly perturbed finite element methods this paper introduces a new piecewise linear finite element which is designed to handle singularly perturbed ordinary differential equations both pointwise and global estimates which are independent of the perturbation parameter are obtained

consistency and validity of perturbative formulations for scattering from rough multilayers two different perturbative approaches have been recently presented for the evaluation of scattering from layered media characterized by rough interfaces the boundary perturbation theory bpt and the volumetric perturbative reciprocal theory vprt these two approaches are conceptually different they employ different strategies to provide closed form solutions to the scattering problem and at first sight they rely on different basic assumptions so that different regimes of validity seem to be expected in spite of that both approaches lead to the same expression of the first order scattered field to solve this seeming incongruence in this paper a careful theoretical analysis of basic assumptions of vprt is carried out this requires also considering the procedure to obtain the vprt solution that in this paper is reformulated relying on a more physically sound way the performed analysis demonstrates that actually basic assumptions of vprt are equivalent to those of bpt so that their ranges of validity are consistently also the same thus the seeming incongruence is solved the two theoretical constructs can be now regarded in a conceptually coherent frame

accurate discretization for singular perturbations the one dimensional case this paper develops a discretization method for one dimensional singular perturbation problems based on a petrov galerkin finite element or an equivalent finite volume scheme the method is unique in that its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter it satisfies a local discrete conservation law and it exhibits a discrete maximum principle numerical results are included for comparison with the theory

a uniform scheme for the singularly perturbed riccati equation we consider the singularly perturbed riccati equation varepsilon u x a x u 2 x b x u x c x 0 u 0 a and present a new finite difference scheme which is exact when the coeffcientsa b andc are constant we show that under certain restrictions the solution u i h of this difference scheme converges uniformly in e tou xi the solution of the differential equation on the uniform meshxi ih numerical results are presented

integration questions related to fractional brownian motion let b h u u be a fractional brownian motion fbm with index h 0 1 and b h be the closure in l 2 of the span sp b h of the increments of fbm b h it is well known that when b h b 1 2 is the usual brownian motion bm an element x b 1 2 can be characterized by a unique function f x l 2 in which case one writes x in an integral form as x f x u db 1 2 u from a different though equivalent perspective the space l 2 forms a class of integrands for the integral on the real line with respect to bm b 1 2 in this work we explore whether a similar characterization of elements of b h can be obtained when h 0 1 2 or h 1 2 1 since it is natural to define the integral of an elementary function f k 1 n f k 1 uk uk 1 by k 1 n f k b h u k 1 b h u k we want the spaces of integrands to contain elementary functions these classes of integrands are inner product spaces if the space of integrands is not complete then it characterizes only a strict subset of b h when 0 h 1 2 by using the moving average representation of fbm b h we construct a complete space of integrands when 1 2 h 1 however an analogous construction leads to a space of integrands which is not complete when 0 h 1 2 or 1 2 h 1 we also consider a number of other spaces of integrands while smaller and henceincomplete they form a natural choice and are convenient to workwith we compare these spaces of integrands to the reproducing kernel hilbert space of fbm

on richardson extrapolation for fitted operator finite difference methods abstract recently there has been a great interest towards the higher order methods for singularly perturbed problems as compared to their lower order counterparts they provide better accuracy with fewer mesh points construction and or implementation of direct higher order methods is usually very complicated thus a natural choice is to use some convergence acceleration techniques e g richardson extrapolation however as we see in this article such techniques do not perform equally well on all type of methods to investigate this we consider two fitted operator finite difference methods fofdms developed by patidar k c patidar high order fitted operator numerical method for self adjoint singular perturbation problems appl math comput 171 1 2005 547 566 and lubuma and patidar j lubuma k c patidar uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems j comput appl math 191 2006 228 238 referred to as fofdm i and fofdm ii respectively the fofdm i is fourth and second order accurate for moderate and smaller values of e respectively unfortunately richardson extrapolation does not improve the order of this method the fofdm ii is second order uniformly convergent and we show that its order can be improved up to four by using richardson extrapolation both the methods are analyzed for convergence and comparative numerical results supporting theoretical estimates are provided

well posedness of stochastic kdv bo equation driven by fractional brownian motion the cauchy problem for the korteweg de vries benjamin ono equation driven by cylindrical fractional brownian motion is discussed in this paper fractional brownian motion is a family of processes b h it is known that the smaller the value of hurst parameter h is the worse of the regularity of fbm is using bourgain restriction method we obtain the lower bound of the hurst parameter h for the driving processes b h with h 38 we prove local existence results with initial value in classical sobolev spaces of negative indices i e h s with s 18

wave propagation in parallel plate metallic waveguide with finite conductivity and three dimensional roughness we study the effects of random roughness on wave propagation in a parallel plate metallic waveguide with finite conductivity the rough surface is three dimensional 3d with roughness heights varying in both horizontal directions integral equations are obtained from the extinction theorem formulated with layered medium green s function the second order small perturbation method is then applied to solve the integral equations a closed form expression for the coherent wave is derived which is expressed in terms of a three fold sommerfeld type integral due to the waveguide structure approximate methods are applied to calculate the sommerfeld integral the results based on the approximations are shown to agree with that of direct integration the coherent wave enhancement factors of absorption are computed results of the present 3d roughness case are compared with the previous results of two dimensional 2d roughness the results for waveguides are also compared with the results obtained for a plane wave incident on a metal surface with 3d roughness results are illustrated for 3d roughness with a variety of power spectra it is shown that enhancement factors of 3d roughness are larger than for 2d roughness we also show that enhancement factors for a waveguide are larger than that of the plane wave case comparisons are also made between the theoretical results and measurements

spatial asymptotics for the parabolic anderson models with generalized time space gaussian noise partially motivated by the recent papers of conus joseph and khoshnevisan ann probab 41 2013 2225 2260 and conus et al probab theory related fields 156 2013 483 533 this work is concerned with the precise spatial asymptotic behavior for the parabolic anderson equation u t t x 12 u t x v t x u t x u 0 x u0 x where the homogeneous generalized gaussian noise v t x is among other forms white or fractional white in time and space associated with the cole hopf solution to the kpz equation in particular the precise asymptotic form limr logr 2 3logmax x ru t x 342t3 3a s is obtained for the parabolic anderson model tu 12 2xxu w u with the 1 1 white noise w t x in addition some links between time and space asymptotics for the parabolic anderson equation are also pursued

the p version of the finite element method for a singularly perturbed boundary value problem the p version of the finite element method is applied to solve the singularly perturbed two point boundary value problem with or without turning point with the special choice of mesh points global error estimates are derived in some cases the exponential rate of convergence is obtained some numerical results are given to show the performance of the proposed method

fernique type inequalities and moduli of continuity for anisotropic gaussian random fields this paper is concerned with sample path properties of anisotropic gaussian random fields we establish fernique type inequalities and utilize them to study the global and local moduli of continuity for anisotropic gaussian random fields applications to fractional brownian sheets and to the solutions of stochastic partial differential equations are investigated

linear stochastic differential equations and wick products we establish the existence and uniqueness of the solution to a multidimensional linear skorohod stochastic differential equation with deterministic diffusion matrix using the notions of wick product andstransform if the diffusion matrix is constant and has real eigenvalues the solution is a stochastic process with moments of all orders provided that the initial condition is differentiable up to a suitable order the case of a diffusion matrix in the first wiener chaos is discussed in the last section

an error analysis for a difference scheme of exponential type applied to a nonlinear singular perturbation problem without turning points abstract a difference scheme of exponential type given in 5 is analyzed again for solving a nonlinear singular perturbation problem without turning points it is shown that a solution of the scheme converges uniformly in a perturbation parameter to an exact solution of the continuous problem with order one in l1 sense

evolution equations driven by a fractional brownian motion abstract in this paper we study nonlinear stochastic evolution equations in a hilbert space driven by a cylindrical fractional brownian motion with hurst parameter h 1 2 and nuclear covariance operator we establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter h this result is applied to stochastic parabolic equation perturbed by a fractional white noise in this case if the coefficients are lipschitz continuous and bounded the existence and uniqueness of a solution holds if h d 4 the proofs of our results combine techniques of fractional calculus with semigroup estimates

stochastic heat equation with rough dependence in space this paper studies the nonlinear one dimensional stochastic heat equation driven by a gaussian noise which is white in time and which has the covariance of a fractional brownian motion with hurst parameter 1 4 h 1 2 in the space variable the existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a lipschitz derivative and vanishes at 0 in the case of a multiplicative noise that is the linear equation we derive the wiener chaos expansion of the solution and a feynman kac formula for the moments of the solution these results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution

on a jump type stochastic fractional partial differential equation with fractional noises abstract we study a class of stochastic fractional partial differential equations of order 1 driven by a pure jump levy space time white noise and a fractional noise we prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions

renormalized self intersection local time for fractional brownian motion let b h t be a d dimensional fractional brownian motion with hurst parameter h 0 1 assume d 2 we prove that the renormalized self intersection local time l t 0 t 0 b h t b h s ds dt e t 0 t 0 b h t b h s ds dt exists in l 2 if and only if h h 3 2d r e l e converges in distribution to a normal law n 0 t 2 as e tends to zero where l e is an approximation of l defined through 2 and r e loge 1 if h 3 2d and r e e d 3 2h if 3 2d h

intermittency for the stochastic heat equation driven by a rough time fractional gaussian noise this paper studies the stochastic heat equation driven by time fractional gaussian noise with hurst parameter h in 0 1 2 we establish the feynman kac representation of the solution and use this representation to obtain matching lower and upper bounds for the l p omega moments of the solution

asymptotic and numerical analysis of singular perturbation problems a survey this paper is intended to be a brief survey of the asymptotic and numerical analysis of singular perturbation problems the purpose is to find out what problems are treated and what numerical asymptotic methods are employed with an eye toward the goal of developing general methods to solve such problems a summary of the results of some recent methods is presented and this leads to conclusions and recommendations about what methods to use on singular perturbation problems finally some areas of current research are indicated a bibliography of about 130 items is provided

scattering from two rough surfaces with inhomogeneous dielectric profiles a rigorous analytical technique to derive the scattering coefficients of three dimensional 3 d two rough surfaces with arbitrary dielectric profiles is presented the proposed method applies a perturbation approach to a surface integral equation ie formulation that is derived from the extended boundary conditions ebc first using ebc or the surface equivalence principle sep and the dyadic green s functions of the resulting simpler geometries a system of ies for the surface fields of the two rough interfaces is established then the fourier domain surface fields are derived using a perturbation method the scattered fields are subsequently determined by a second application of sep in general the approach could be used to present the general formulation for multiple rough layers and also higher order solutions but the focus of this paper is on deriving compact closed form solutions for inhomogeneous dielectric profiles and thus only two rough interfaces are considered accordingly the first order closed form scattered fields are represented in a remarkably compact form that is suitable for physical interpretation and also extension to higher orders finally the derived expressions are compared to known solutions of special cases available in the literature the results are shown to be exactly consistent with existing ones and hence validated analytically and numerically

a conservative uniformly accurate difference method for a singular perturbation problem in conservation form a conservative three point finite difference method is presented for the numerical solution of the singular perturbation problem varepsilon u xx b x u x f x 0 x 1 u 0 and u 1 given b x ne 0 certain a priori estimates are established for this problem and are then used to obtain uniform error estimates for the difference scheme some illustrative numerical results are also given

generalized brownian functionals and the solution to a stochastic partial differential equation abstract a class of generalized wiener functionals related to those of hida and watanabe is introduced and the malliavin calculs associated with these functionals is developed these notions are applied to the derivation of a solution to the stochastic partial differential equation y t ly y where l is a second order partial differential operator in the n space variables denotes a white noise in the n 1 space time variables y denotes a skorohod type stochastic integral and is a non random function of the n 1 independent variables

feynman kac formula for heat equation driven by fractional white noise we establish a version of the feynman kac formula for the multidimensional stochastic heat equation with a multiplicative fractional brownian sheet we use the techniques of malliavin calculus to prove that the process defined by the feynman kac formula is a weak solution of the stochastic heat equation from the feynman kac formula we establish the smoothness of the density of the solution and the holder regularity in the space and time variables we also derive a feynman kac formula for the stochastic heat equation in the skorokhod sense and we obtain the wiener chaos expansion of the solution

on the stochastic heat equation with spatially colored random forcing we consider the stochastic heat equation of the following form t ut x lut x b ut x ut x t x for t 0 x r where l is the generator of a levy process and is a spatially colored temporally white gaussian noise we will be concerned mainly with the long term behavior of the mild solution to this stochastic pde for the most part we work under the assumptions that the initial data u0 is a bounded and measurable function and is nonconstant and lipschitz continuous in this case we find conditions under which the preceding stochastic pde admits a unique solution which is also weakly intermittent in addition we study the same equation in the case that lu is replaced by its massive dispersive analogue lu u where r and we describe accurately the effect of the parameter on the intermittence of the solution in the case that u is proportional to u the parabolic anderson model we also look at the linearized version of our stochastic pde that is the case when is identically equal to one any other constant works also in this case we study not only the existence and uniqueness of a solution but also the regularity of the solution when it exists and is unique

l 1 and l uniform convergence of a difference scheme for a semilinear singular perturbation problem a nonlinear difference scheme is given for solving a semilinear singularly perturbed two point boundary value problem without any restriction on turning points the solution of the scheme is shown to be first order accurate in the discretel1 norm uniformly in the perturbation parameter when turning points are excluded the scheme is first order accurate in the discretel norm uniformly in the perturbation parameter

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

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

fractional stochastic volterra equation perturbed by fractional brownian motion in this paper we consider a class of fractional stochastic volterra equation of convolution type driven by infinite dimensional fractional brownian motion with hurst index h 0 1 base on the explicit formula for the scalar resolvent function and the properties of the mittag leffler s function the existence and regularity results of the stochastic convolution process are established separate proofs are required for the cases of hurst parameter above and below 1 2 and it will turn out that the regularity of the solution increases with hurst parameter h

scattering of em waves from a slightly rough surface of a generally anisotropic plane layered halfspace the problem of plane wave scattering by a slightly rough boundary of an arbitrarily layered medium with general anisotropy of electric and magnetic properties is considered using boundary conditions transference method and a small perturbation approximation combined with the green s functions technique it is shown that in the born approximation the second statistical moments of the fluctuation field scattered from an inhomogeneous anisotropic halfspace can be expressed exclusively in terms of the external parameters referring to a flat boundary and the spectral density of roughness in the present circumstance the set of the external parameters includes the four plane wave reflection coefficients from a flat boundary and the limiting values of constitutive parameters of the adjacent media at said boundary within the framework of the outlined approach the covariance dyads poynting s vector scattering coefficients and stokes parameters of the fluctuation field and the mueller matrix for a rough boundary are calculated

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

intermittency for the wave and heat equations with fractional noise in time in this article we consider the stochastic wave and heat equations driven by a gaussian noise which is spatially homogeneous and behaves in time like a fractional brownian motion with hurst index h 1 2 the solutions of these equations are interpreted in the skorohod sense using malliavin calculus techniques we obtain an upper bound for the moments of order p 2 of the solution in the case of the wave equation we derive a feynman kac type formula for the second moment of the solution based on the points of a planar poisson process this is an extension of the formula given by dalang mueller and tribe trans amer math soc 360 2008 4681 4703 in the case h 1 2 and allows us to obtain a lower bound for the second moment of the solution these results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time

on a modelled rough heat equation we use the formalism of hairer s regularity structures theory to study a heat equation model with non linear perturbation driven by a space time fractional noise different regimes are observed depending on the global pathwise roughness of the noise to this end and following the procedure exhibited in hairer invent math 198 2 269 504 2014 the equation is first lifted into some abstract regularity structure and therein solved through a fixed point argument then we construct a consistent model above the fractional noise by relying on a smooth fourier type approximation of the process

adaptivity and error estimation for the finite element method applied to convection diffusion problems a detailed analysis is performed for a finite element method applied to the general one dimensional convection diffusion problem piecewise polynomials are used for the trial space the test space is formed by locally projecting l spline basis functions onto upwinded polynomials the error is measured in the l p mesh dependent norm the method is shown to be quasi optimal provided that the input data is piecewise smooth a reasonable assumption in practice a posteriors error estimates are derived having the property that the effectivity index theta error estimate true error converges to one as the maximum mesh size goes to zero these error estimates are composed of locally computable error indicators providing for an adaptive mesh refinement strategy numerical results show that theta is nearly one even on coarse meshes and optimal rates of convergence are attained by the adaptive procedure the robustness of the algorithm is tested on a nonlinear turning point problem modeling flow th

a uniform finite element method for a conservative singularly perturbed problem abstract we examine the problem p x u q x u r x u f x for 0 x p 0 q 0 r 0 p q r and f in c 2 0 1 in 0 1 u 0 and u 1 given existence of a unique solution u and bounds on u and its derivatives are obtained using finite elements on an equidistant mesh of width h we generate a tridiagonal difference scheme which is shown to be uniformly second order accurate for this problem i e the nodal errors are bounded by ch 2 where c is independent of h and with a natural choice of trial functions uniform first order accuracy is obtained in the l 0 1 norm using trial functions which interpolate linearly between the nodal values generated by the difference scheme gives uniform first order accuracy in the l 1 0 1 norm

a uniformly accurate finite element method for a singular perturbation problem in conservative form a petrov galerkin finite element method with exponential basis elements is applied to a nonselfadjoint singularly perturbed two point boundary value problem in conservative form it is shown to be uniformly first order accurate in l infty and is uniformly second order accurate at the nodes i e the error constants are independent of the mesh size and the perturbation parameter

nonlinear singular perturbation problems and one sided difference schemes we consider the boundary value problem varepsilon y a y y b x y f x 1 leqq x leqq 1 for y 1 and y 1 given and b x 0 equiv 0 b y x y geqq 0 we construct a simple difference scheme approximating varepsilon u xx a u u x b x u f x u t for t geqq 0 1 leqq x leqq 1 with the same boundary conditions and an arbitrary initial guess as the numerical artificial time approaches infinity we prove convergence to a unique steady state solution as the mesh size delta x and the parameter varepsilon approach zero in any order we prove convergence to the solution of the corresponding o d e we use one sided differences to approximate the transport term the method applies equally well to multidimensional analogues we present numerical results verifying the theory

electromagnetic scattering from 3d layered structures with randomly rough interfaces analysis with the small perturbation method and the small slope approximation we propose a theoretical study on the electromagnetic wave scattering from three dimensional layered structures with an arbitrary number of rough interfaces by using the small perturbation method and the small slope approximation the interfaces are characterized by gaussian height distributions with zero mean values and gaussian correlation functions they can be correlated or not isotropic or not the electromagnetic field in each medium is represented by a continuum of plane waves and a perturbation theory is used for solving the boundary value problem and determining the first order scattering amplitudes by recurrence relations the scattering amplitudes under the first order small slope approximation are deduced from results derived from the first order small perturbation method we analyze with the small slope approximation model the combined influence of the anisotropy and cross correlation upon the electromagnetic signature of a natural stratified structure

second order perturbative solution of cross polarized scattering from multilayered rough surfaces an analytical solution for the cross polarized scattering from multilayered media with an arbitrary number of rough interfaces is presented the second order perturbative solutions are summations of the fourier transform pair products of the roughness profiles and hence the problem is initially reduced to two generic problems one with just a single rough surface with two stratified media above and below it and another with two rough interfaces with three stratified media above between and below them the former helps find the second order contribution of an individual rough interface while the latter is intended to determine the interaction contribution of two arbitrary rough interfaces in this paper the second generic problem is solved using the small perturbation method spm based on the extended boundary conditions ebc in this spm formulation the stratifications are easily accounted for using the generalized coefficients inserted in the dyadic green s functions dgfs then the expressions are simplified to find the solution of the first generic problem which is a special case of the second one next the explicit expressions of the generalized coefficients are introduced to the solutions subsequently the interaction terms of the second and the self term of the first generic problem are combined to find the desired second order solution finally using the solution derived a few numerical examples are presented

stochastic evolution equations with fractional brownian motion in this paper linear stochastic evolution equations driven by infinite dimensional fractional brownian motion are studied a necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed separate proofs are required for the cases of hurst parameter above and below 1 2 the particular case of the laplacian on the circle is discussed in detail

stochastic burgers equation driven by fractional brownian motion abstract in this paper we consider the stochastic burgers equation driven by a genuine cylindrical fractional brownian motion with hurst parameter h 1 4 we first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic burgers equation then we obtain the local and global existence and uniqueness results for the stochastic burgers equation

self intersection local time critical exponent large deviations and laws of the iterated logarithm if t is renormalized self intersection local time for planar brownian motion we characterize when ee 1 is finite or infinite in terms of the best constant of a gagliardo nirenberg inequality we prove large deviation estimates for 1 and 1 we establish lim sup and lim inf laws of the iterated logarithm for t as t

scattering from multilayer rough surfaces based on the extended boundary condition method and truncated singular value decomposition electromagnetic scattering from rough surfaces has been investigated extensively for the past decades there exist analytical solutions to rough surface scattering such as small perturbation method spm or kirchhoff approximation ka these solutions however are limited to either small or large roughness regimes recent efforts have been put forth to study electromagnetic scattering of multilayer rough surface in the application of deep soil moisture and ice property estimation in this paper multilayer rough surface scattering is analyzed based on the extended boundary condition method ebcm and truncated singular value decomposition the method of extended boundary condition can accurately handle scattering from a slightly rough surface for mild and large roughness regime ill conditioned matrices can arise from ebcm in this paper we present a regularization scheme to mitigate this ill conditioning effect via truncated singular value decomposition tsvd the truncation parameters are set to enforce power conservation relations across all rough interfaces bistatic scattering coefficients are then obtained by incoherently averaging the power computed from the resulting floquet modes in the scattering directions therefore the electromagnetic wave interactions in multilayer rough surfaces are analyzed in a systematic way finally the results are validated against the analytical spm solution to two rough interface rough surface scattering in addition the formulation is extended to solve scattering from three rough interface rough surfaces several results are reported

intermittency and multifractality a case study via parabolic stochastic pdes let denote space time white noise and consider the following stochastic partial dierential equations i u 1 u 00 u started identically at one and ii z 1 z 00 started identically at zero it is well known that the solution to i is intermittent whereas the solution to ii is not and the two equations are known to be in dierent universality classes we prove that the tall peaks of both systems are multifractals in a natural large scale sense some of this work is extended to also establish the multifractal behavior of the peaks of stochastic pdes on r r d with d 2 g lawler has asked us if intermittency is the same as multifractality the present work gives a negative answer to this question as a byproduct of our methods we prove also that the peaks of the brownian motion form a large scale monofractal whereas the peaks of the ornstein uhlenbeck process on r are multifractal throughout we make extensive use of the macroscopic fractal theory of m t barlow and s j taylor 3 4 we expand on aspects of the barlow taylor theory as well

on the chaotic character of the stochastic heat equation ii consider the stochastic heat equation partial t u varkappa 2 delta u sigma u dot f where the solution u u t x is indexed by t x in 0 infty times bf r d and dot f is a centered gaussian noise that is white in time and has spatially correlated coordinates we analyze the large x fixed t behavior of the solution u in different regimes thereby study the effect of noise on the solution in various cases among other things we show that if the spatial correlation function f of the noise is of riesz type that is f x propto x alpha then the fluctuation exponents of the solution are psi for the spatial variable and 2 psi 1 for the time variable where psi 2 4 alpha moreover these exponent relations hold as long as alpha in 0 d wedge 2 that is precisely when dalang s theory dalang electron j probab 4 paper no 6 29 1999 implies the existence of a solution to our stochastic pde these findings bolster earlier physical predictions kardar et al phys rev lett 58 20 889 892 1985 kardar and zhang phys rev lett 58 20 2087 2090 1987

gradient type noises ii systems of stochastic partial differential equations abstract the present paper is the second and main part of a study of partial differential equations under the influence of noisy perturbations existence and uniqueness of function solutions in the mild sense are obtained for a class of deterministic linear and semilinear parabolic boundary initial value problems if the noise data are random the results may be seen as a pathwise approach to spde s for typical examples such as spatially one dimensional stochastic heat equations with additive or multiplicative perturbations of fractional brownian type we recover and extend known results in addition we propose to consider partial noises of low order