second order hyperbolic s p d e s driven by homogeneous gaussian noise on a hyperplane we study a class of hyperbolic stochastic partial di erential equations in euclidean space that includes the wave equation and the telegraph equation driven by gaussian noise concentrated on a hyperplane the noise is assumed to be white in time but spatially homogeneous within the hyperplane two natural notions of solutions are function valued solutions and random field solutions for the linear form of the equations we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution and it turns out that the conditions di er in spatial dimensions 2 and 3 under the condition for existence of a random field solution to the linear form of the equation we prove existence and uniqueness of a random field solution to non linear forms of the equation

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

the stochastic heat equation feynman kac formula and intermittence we study in one space dimension the heat equation with a random potential that is a white noise in space and time this equation is a linearized model for the evolution of a scalar field in a space time dependent random medium it has also been related to the distribution of two dimensional directed polymers in a random environment to the kpz model of growing interfaces and to the burgers equation with conservative noise we show how the solution can be expressed via a generalized feynman kac formula we then investigate the statistical properties the two point correlation function is explicitly computed and the intermittence of the solution is proven this analysis is carried out showing how the statistical moments can be expressed through local times of independent brownian motions

tomographic imaging of dynamic objects with the ensemble kalman filter we address the image formation of a dynamic object from projections by formulating it as a state estimation problem the problem is solved with the ensemble kalman filter enkf a monte carlo algorithm that is computationally tractable when the state dimension is large in this paper we first rigorously address the convergence of the enkf then the effectiveness of the enkf is demonstrated in a numerical experiment where a highly variable object is reconstructed from its projections an imaging modality not yet explored with the enkf the results show that the enkf can yield estimates of almost equal quality as the optimal kalman filter but at a fraction of the computational effort further experiments explore the rate of convergence of the enkf its performance relative to an idealized particle filter and implications of modeling the system dynamics as a random walk

nonlinear noise excitation of intermittent stochastic pdes and the topology of lca groups consider the stochastic heat equation tu lu u where l denotes the generator of a levy process on a locally compact hausdor abelian group g r r is lipschitz continuous 1 is a large parameter and denotes space time white noise on r g the main result of this paper contains a near dichotomy for the expected squared energy e kutk 22 g of the solution roughly speaking that dichotomy says that in all known cases where u is intermittent the energy of the solution behaves generically as expfconst 2 g when g is discrete and expfconst 4 g when g is connected

intermittency for stochastic partial differential equations driven by strongly inhomogeneous space time white noises abstract in this paper the main topic is to investigate the intermittent property of the one dimensional stochastic heat equation driven by an inhomogeneous brownian sheet which is a noise deduced from the study of the catalytic super brownian motion under some proper conditions on the catalytic measure of the inhomogeneous brownian sheet we show that the solution is weakly full intermittent based on the estimates of moments of the solution in particular it is proved that the second moment of the solution grows at the exponential rate the novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the lebesgue measure on r

the existence and uniqueness of the local solution for a camassa holm type equation a shallow water equation of camassa holm type containing nonlinear dissipative effect is investigated using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself we establish the existence and uniqueness of its local solution in sobolev space h s r with s 32 meanwhile a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order sobolev space h s with 1

gaussian particle filtering sequential bayesian estimation for nonlinear dynamic state space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations this paper introduces a new filter called the gaussian particle filter it is based on the particle filtering concept and it approximates the posterior distributions by single gaussians similar to gaussian filters like the extended kalman filter and its variants it is shown that under the gaussianity assumption the gaussian particle filter is asymptotically optimal in the number of particles and hence has much improved performance and versatility over other gaussian filters especially when nontrivial nonlinearities are present simulation results are presented to demonstrate the versatility and improved performance of the gaussian particle filter over conventional gaussian filters and the lower complexity than known particle filters

the stochastic wave equation in two spatial dimensions keywords stochastic wave equation gaussian noise process solution reference prob article 1998 001view record in web of science record created on 2008 12 01 modified on 2017 05 12

approximation of the zakai equation by the splitting up method the objective of this article is to apply an operator splitting method to the time integration of zakai equation using this approach one can decompose the numerical integration into a stochastic step and a deterministic one both of them much simpler to handle than the original problem a strong convergence theorem is given in the spirit of existing results for deterministic problems

a numerical approach for computing flows by local transformations and domain decomposition using an optimization algorithm this paper presents theoretical information for numerical simulation of two and three dimensional flows based on the concepts of streamlines and local transformation functions associated with domain decomposition in this approach in addition to the pressure the primary unknowns are the local mapping functions between the physical sub domains and transformed domains where the mapped streamlines are parallel straight lines this makes it easy to handle time dependent constitutive equations for complex fluids to solve the governing equations that also involve compatibility equations between sub domains an optimization algorithm is set up in order to compute the unknowns related to the streamlines iteratively applications are given for two dimensional flows between eccentric cylinders using different constitutive equations

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds we study a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectivelyp 1 discontinuous p 2 continuous p 1 continuous we use the method of lesaint raviart for the convection of the extra stress tensor we suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution we show by a fixed point method that the approximate problem has a solution and we give an error bound

second order linear hyperbolic spdes driven by isotropic gaussian noise on a sphere we study a class of linear hyperbolic stochastic partial differential equations in bounded domains which includes the wave equation and the telegraph equation driven by gaussian noise that is white in time but not in space we give necessary and sufficient conditions on the spatial correlation of the noise for the existence and uniqueness of square integrable solutions in the particular case where the domain is a ball and the noise is concentrated on a sphere we characterize the isotropic gaussian noises with this property we also give explicit necessary and sufficient conditions when the domain is a hypercube and the gaussian noise is concentrated on a hyperplane

boundary static output feedback control for nonlinear stochastic parabolic partial differential systems via fuzzy model based approach this article investigates a fuzzy boundary control problem of a class of stochastic nonlinear systems modeled by the ito type parabolic stochastic partial differential equation spde initially a takagi sugeno fuzzy spde model is proposed to accurately represent the nonlinear spde system then on the basis of the infinite dimensional infinitesimal operator a fuzzy boundary static output feedback controller is developed in terms of a set of linear matrix inequalities to locally exponentially stabilize the resulting system in the mean square sense by using an approximation argument and constructing a lyapunov function for the mild solution the local mean square exponential stability of the closed loop system is proved meanwhile the closed loop well posedness analysis is also given by virtue of the semigroup theory finally a simulation study on a belousov zhabotinsky reaction diffusion system with random parameter variation is presented to illustrate the effectiveness of the proposed method

spectral collocation domain decomposition method for viscoelastic flow simulations in model porous geometries abstract a domain decomposition spectral collocation ddsc method is developed for the solution of steady state non linear viscoelastic flow problems it is applied in simulations of viscoelastic flows through model porous media represented by arrays of cylinders the technique uses a splitting of the computational domain in sub elements in a fashion similar to finite elements at each element the solution is represented by using high order orthogonal polynomials static condensation is used for the solution of the linearized system of equations resulting from the application of newton s method this approach is suggested by the limited connectivity between neighboring elements mesh converged solutions have been obtained for the flow of an oldroyd b fluid through a square array of cylinders up to we 5 0 and for the flow past a single row of cylinders up to to we 4 0

strong invariance and noise comparison principles for some parabolic stochastic pdes we consider a system of interacting diffusions on the integer lattice by letting the mesh size go to zero and by using a suitable scaling we show that the system converges in a strong sense to a solution of the stochastic heat equation on the real line as a consequence we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities

a local time correspondence for stochastic partial differential equations it is frequently the case that a white noise driven parabolic and or hyperbolic stochastic partial differential equation spde can have random field solutions only in spatial dimension one here we show that in many cases where the spatial operator is the l generator of a levy process x a linear spde has a random field solution if and only if the symmetrization of x possesses local times this result gives a probabilistic reason for the lack of existence of random field solutions in dimensions strictly bigger than one in addition we prove that the solution to the spde is holder continuous in its spatial variable if and only if the said local time is holder continuous in its spatial variable we also produce examples where the random field solution exists but is almost surely unbounded in every open subset of space time our results are based on first establishing a quasi isometry between the linear l space of the weak solutions of a family of linear spdes on one hand and the dirichlet space generated by the symmetrization of x on the other hand we study mainly linear equations in order to present the local time correspondence at a modest technical level however some of our work has consequences for nonlinear spdes as well we demonstrate this assertion by studying a family of parabolic spdes that have additive nonlinearities for those equations we prove that if the linearized problem has a random field solution then so does the nonlinear spde moreover the solution to the linearized equation is holder continuous if and only if the solution to the nonlinear equation is and the solutions are bounded and unbounded together as well finally we prove that in the cases that the solutions are unbounded they almost surely blow up at exactly the same points date october 7 2007 2000 mathematics subject classification primary 60h15 60j55 secondary 35r60 35d05

a macroscopic multifractal analysis of parabolic stochastic pdes it is generally argued that the solution to a stochastic pde with multiplicative noise such as dot u frac12 u u xi where xi denotes space time white noise routinely produces exceptionally large peaks that are macroscopically multifractal see for example gibbon and doering arch ration mech anal 177 115 150 2005 gibbon and titi proc r soc a 461 3089 3097 2005 and zimmermann et al phys rev lett 85 17 3612 3615 2000 a few years ago we proved that the spatial peaks of the solution to the mentioned stochastic pde indeed form a random multifractal in the macroscopic sense of barlow and taylor j phys a 22 13 2621 2626 1989 proc lond math soc 3 64 125 152 1992 the main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio temporal peaks of the solution form infinitely many different multifractals on infinitely many different scales which we sometimes refer to as stretch factors a simpler though still complex such structure is shown to also exist for the constant coefficient version of the said stochastic pde

on existence and uniqueness of the global weak solution for a shallow water equation abstract a shallow water equation including the camassa holm and degasperis procesi equations as special cases is investigated provided that initial value u 0 h 1 r u 0 l 1 r and 1 x 2 u 0 does not change sign the existence and uniqueness of global weak solution for the equation are established

independent particle filters sequential monte carlo methods especially the particle filter pf and its various modifications have been used effectively in dealing with stochastic dynamic systems the standard pf samples the current state through the underlying state dynamics then uses the current observation to evaluate the sample s importance weight however there is a set of problems in which the current observation provides significant information about the current state but the state dynamics are weak and thus sampling using the current observation often produces more efficient samples than sampling using the state dynamics in this article we propose a new variant of the pf the independent particle filter ipf to deal with these problems the ipf generates exchangeable samples of the current state from a sampling distribution that is conditionally independent of the previous states a special case of which uses only the current observation each sample can then be matched with multiple samples of the previous states in

on the existence and position of the farthest peaks of a family of stochastic heat and wave equations we study the stochastic heat equation partial t u mathcal l u sigma u dot w in 1 1 dimensions where dot w is space time white noise r r is lipschitz continuous and mathcal l is the generator of a symmetric levy process that has finite exponential moments and u0 has exponential decay at we prove that under natural conditions on i the th absolute moment of the solution to our stochastic heat equation grows exponentially with time and ii the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting see however gartner et al in probab theory relat fields 111 17 55 1998 gartner et al in ann probab 35 439 499 2007 for the analysis of the location of the peaks in a different model finally we show that these results extend to the stochastic wave equation driven by laplacian

finite element approximation of viscoelastic fluid flow existence of approximate solutions and error bounds continuous approximation of the stress the author studies a finite element approximation of viscoelastic fluid flow obeying an oldroyd b type constitutive law the approximate stress velocity and pressure are respectively p 1 continuous p 2 continuous and p 1 continuous the streamline upwinding petrov galerkin method for the convection of the extra stress tensor is used suppose that the continuous problem admits a sufficiently smooth and sufficiently small solution it is shown by a fixed point method that the approximate problem has a solution and an error bound is given

comparison of interacting diffusions and an application to their ergodic theory a general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived as an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent and the components take values in an interval bounded on one side the technique also gives an alternative proof of clustering in the case of compact intervals

uniqueness for stochastic non linear wave equations abstract we prove uniqueness of solutions of the equation u t t u f u g u w u 0 u 0 u t 0 v 0 on r d in the class of processes with paths in h 1 here u 0 v 0 is a random variable in h 1 l 2 w is a spatially homogeneous wiener process with finite spectral measure and f g are locally lipschitz functions of subcritical growth

absolute continuity of the law of the solution to the 3 dimensional stochastic wave equation abstract we present new results regarding the existence of density of the real valued solution to a 3 dimensional stochastic wave equation the noise is white in time and with a spatially homogeneous correlation whose spectral measure satisfies that r 3 d 1 2 for some 0 1 2 our approach is based on the mild formulation of the equation given by means of dalang s extended version of walsh s stochastic integration we use the tools of malliavin calculus let s 3 be the fundamental solution to the 3 dimensional wave equation the assumption on the noise yields upper and lower bounds for the integral 0 t ds r 3 d f s 3 s 2 and upper bounds for 0 t ds r 3 d f s 3 s 2 in terms of powers of t these estimates are crucial in the analysis of the malliavin variance which can be done by a comparison procedure with respect to smooth approximations of the distribution valued function s 3 t obtained by convolution with an approximation of the identity

on the smoluchowski kramers approximation for a system with an infinite number of degrees of freedom according to the smolukowski kramers approximation we show that the solution of the semi linear stochastic damped wave equations utt t x u t x ut t x b x u t x qopen image in new window t u 0 u0 ut 0 v0 endowed with dirichlet boundary conditions converges as goes to zero to the solution of the semi linear stochastic heat equation ut t x u t x b x u t x qopen image in new window t u 0 u0 endowed with dirichlet boundary conditions moreover we consider relations between asymptotics for the heat and for the wave equation more precisely we show that in the gradient case the invariant measure of the heat equation coincides with the stationary distributions of the wave equation for any 0

harmonic analysis of random fractional diffusion wave equations this paper presents the green functions and spectral representations of the mean square solutions of the fractional diffusion wave equations with random initial conditions

on pathwise uniqueness for stochastic heat equations with non lipschitz coefficients 1 research supported in part by the israel science foundation grant no 116 01 10 0 2 research supported in part by an nserc research grant 3 research supported in part by an nserc research grant and by dfg priority program 1033

strong solutions to stochastic wave equations with values in riemannian manifolds let m be a d dimensional compact riemannian manifold we prove existence of a unique global strong solution of the stochastic wave equation dt tu dx xu yu tu xu w where y is a c1 smooth transformation and w is a spatially homogeneous wiener process on r whose spectral measure has finite moments up to order 2

augmented particle filters abstractparticle filters have been widely used for online filtering problems in state space models ssms the current available proposal distributions depend either only on the state dynamics or only on the observation or on both sources of information but are not available for general ssms in this article we develop a new particle filtering algorithm called the augmented particle filter apf for online filtering problems in ssms the apf combines two sets of particles from the observation equation and the state equation and the state space is augmented to facilitate the weight computation theoretical justification of the apf is provided and the connection between the apf and the optimal particle filter opf in some special ssms is investigated the apf shares similar properties as the opf but the apf can be applied to a much wider range of models than the opf simulation studies show that the apf performs similarly to or better than the opf when the opf is available and the apf can perform

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

invariant measures for stochastic nonlinear beam and wave equations existence of an invariant measure for a stochastic extensible beam equation and for a stochastic damped wave equation with polynomial nonlinearities is proved it is shown first that the corresponding transition semigroups map the space of all bounded sequentially weakly continuous functions on the state space into itself and then by a lyapunov functions approach solutions bounded in probability are found

regularity of the sample paths of a class of second order spde s we study the sample path regularity of the solutions of a class of spde s which are second order in time and that includes the stochastic wave equation non integer powers of the spatial laplacian are allowed the driving noise is white in time and spatially homogeneous continuing with the work initiated in dalang and mueller 2003 we prove that the solutions belong to a fractional l 2 sobolev space we also

the finite speed of propagation for solutions to nonlinear stochastic wave equations driven by multiplicative noise abstract we prove that the solutions to the stochastic wave equation in o r d d x x d t g x d t x d w for 1 d where g is a continuous function with polynomial growth of order less or equal to d d 2 and is lipschitz with 0 0 propagate with finite speed this result resembles the classical finite speed of propagation result for the solution to the klein gordon equation and extends to equations with dissipative damping a similar result follows for the equation with additive noise of the form f t d w where f t f t has compact support in for each t 0

nonlinear stochastic wave and heat equations we study nonlinear wave and heat equations on d driven by a spatially homogeneous wiener process for the wave equation we consider the cases of d 1 2 3 the heat equation is considered on an arbitrary d space we give necessary and sufficient conditions for the existence of a function valued solution in terms of the covariance kernel of the noise

a domain decomposition method for the oseen viscoelastic flow equations we study a non overlapping domain decomposition method for the oseen viscoelastic flow problem the data on the interface are transported through newmann and dirichlet boundary conditions for the momentum and constitutive equations respectively the discrete variational formulations of subproblems are presented and investigated for the existence of solutions we show convergence of the domain decomposition solution to a solution of the one domain problem convergence of an iterative algorithm and some numerical results are also presented

implicit sampling for particle filters we present a particle based nonlinear filtering scheme related to recent work on chainless monte carlo designed to focus particle paths sharply so that fewer particles are required the main features of the scheme are a representation of each new probability density function by means of a set of functions of gaussian variables a distinct function for each particle and step and a resampling based on normalization factors and jacobians the construction is demonstrated on a standard ill conditioned test problem

a multigrid method for viscoelastic fluid flow we study a multigrid finite element method for a viscoelastic fluid flow obeying an oldroyd b type constitutive law the multigrid method is a time saving method in which the full nonlinear system is solved on a coarse grid and subsequent approximations are generated on a succession of refined grids by solving a linearized problem we show that the linearized problem has an approximate solution and present an error bound for a two grid method we also numerically demonstrate that the multigrid method is significantly more efficient than the standard one grid finite element method

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

approximation of the kushner equation for nonlinear filtering in this paper we discuss the well posedness and approximation of solutions to the kushner equation in nonlinear filtering problems we develop and analyze a time integration method based on the operator splitting also we discuss its relation to the operator splitting method for the zakai equation

stability and absence of overflow oscillations for 2 d discrete time systems it is shown that a 2 d linear time invariant discrete time system could be bounded input bounded output bibo unstable even if the 2 d system which is implemented by finite word length is free of overflow oscillations with zero input according to a previous criterion for the absence of overflow oscillations in 2 d discrete time systems an improved criterion is presented to ensure both stability properties

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

a nonoverlapping domain decomposition method for stabilized finite element approximations of the oseen equations abstract a nonoverlapping domain decomposition algorithm of robin robin type is applied to the discretized oseen equations using stabilized finite element approximations of velocity and pressure thus allowing in particular equal order interpolation as a crucial result we have to inspect the proof of a modified inf sup condition in particular the dependence of the stability constant with respect to the reynolds number cf appendix after proving coercivity and strong convergence of the method we derive an a posteriori estimate which controls convergence of the discrete subdomain solutions to the global discrete solution provided that jumps of the discrete solution converge at the interface furthermore we obtain information on the design of some free parameters within the robin type interface condition which essentially influence the convergence speed some numerical results confirm the theoretical ones

long time existence for the wave equation with a noise term we consider the equationut uxx u w forx on a finite interval with dirichlet boundary conditions w is spacetime white noise the initial condition is continuous and nonnegative we show existence and uniqueness for all time provided 1 3 2

on the splitting up method and stochastic partial differential equations we consider two stochastic partial differential equations du varepsilon t l ru varepsilon t f r t dv varepsilon t r m k u varepsilon t g k t circ dy t k qquad hspace 5pt varepsilon 0 1 driven by the same multidimensional martingale y yk and by different increasing processes vr0 vr1 r 1 2 d1 where lr and mk are second and first order partial differential operators and stands for the stratonovich differential we estimate the moments of the supremum in t of the sobolev norms of u1 t u0 t in terms of the supremum of the differences break vr0t vr1t hence we obtain moment estimates for the error of a multistage splitting up method for stochastic pdes in particular for the equation of the unnormalized conditional density in nonlinear filtering

stochastic wave equation with critical nonlinearities temporal regularity and uniqueness we consider a nonlinear wave equation utt u f u g u w on r d driven by a spatially homogeneous wiener process w with a finite spectral measure and with nonlinear terms f g of critical growth we study pathwise uniqueness and norm continuity of paths of u ut in h 1 rd l2 rd under the hypothesis that there exists a local solution u such that u ut has weakly continuous paths in h 1

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

limit of fluctuations of solutions of wigner equation we consider fluctuations of the solution w e t x k of the wigner equation which describes energy evolution of a solution of the schrodinger equation with a random white noise in time potential the expectation of w e t x k converges as e 0 to bar w t x k which satisfies the radiative transport equation we prove that when the initial data is singular in the x variable that is w e 0 x k x f k and f in mathcal s mathbb r d then the laws of the rescaled fluctuation z varepsilon t varepsilon 1 2 w varepsilon t x k bar w t x k converge as e 0 to the solution of the same radiative transport equation but with a random initial data this complements the result of 6 where the limit of the covariance function has been considered

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments

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