a spectral mapping theorem for perturbed ornstein uhlenbeck operators on l2 rd abstract we consider ornstein uhlenbeck operators on l 2 r d perturbed by a radial potential v under weak assumptions on v we prove a spectral mapping theorem for the generated semigroup the proof relies on a perturbative construction of the resolvent based on angular separation and the gearhart pruss theorem

generalized two step maruyama methods for stochastic differential equations abstract in this paper we propose generalized two step maruyama methods for solving ito stochastic differential equations numerical analysis concerning consistency convergence and numerical stability in the mean square sense is presented we derive sufficient and necessary conditions for linear mean square stability of the generalized two step maruyama methods we compare the stability region of the generalized two step maruyama methods of adams type with that of the corresponding two step maruyama methods of adams type and show that our proposed methods have better linear mean square stability a numerical example is given to confirm our theoretical results

a transference principle for general groups and functional calculus on umd spaces let ia be the generator of a c 0 group u s s is an element of r on a banach space x and omega theta u the group type of u we prove a transference principle that allows to estimate parallel to f a parallel to in terms of the l p r x fourier multiplier norm of f i omega if x is a hilbert space this yields new proofs of important results of mcintosh and boyadzhiev de laubenfels if x is a umd space one obtains a bounded h 1 infinity calculus of a on horizontal strips related results for sectorial and parabola type operators follow finally it is proved that each generator of a cosine function on a umd space has bounded h infinity calculus on sectors

kernel estimates for nonautonomous kolmogorov equations abstract using time dependent lyapunov functions we prove pointwise upper bounds for the heat kernels of some nonautonomous kolmogorov operators with possibly unbounded drift and diffusion coefficients

some classes of non analytic markov semigroups we deal with markov semigroups tt corresponding to second order elliptic operators au u du f where f is an unbounded locally lipschitz vector field on rn we obtain new conditions on f under which tt is not analytic in cb rn in particular we prove that the one dimensional operator au u x3u with domain u c2 r u u x3u cb r is not sectorial in cb r under suitable hypotheses on the growth of f we introduce a class of non analytic markov semigroups in lp rn where is an invariant measure for tt

estimates of the derivatives for parabolic operators with unbounded coefficients we consider a class of second order uniformly elliptic operators a with unbounded coefficients in r n using a bernstein approach we provide several uniform estimates for the semigroup t t generated by the realization of the operator a in the space of all bounded and continuous or holder continuous functions in r n as a consequence we obtain optimal schauder estimates for the solution to both the elliptic equation u au f 0 and the nonhomogeneous dirichlet cauchy problem d t u au g then we prove two different kinds of pointwise estimates of t t that can be used to prove a liouville type theorem finally we provide sharp estimates of the semigroup t t in weighted l p spaces related to the invariant measure associated with the semigroup

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

toward an optimal solver for time spectral fluid dynamic and aeroelastic solutions on unstructured meshes abstract the time spectral method applied to the euler and coupled aeroelastic equations theoretically offers significant computational savings for purely periodic problems when compared to standard time implicit methods however attaining superior efficiency with time spectral methods over traditional time implicit methods hinges on the ability rapidly to solve the large non linear system resulting from time spectral discretizations which become larger and stiffer as more time instances are employed or the period of the flow becomes especially short i e the maximum resolvable wave number increases in order to increase the efficiency of these solvers and to improve robustness particularly for large numbers of time instances the generalized minimal residual method gmres is used to solve the implicit linear system over all coupled time instances the use of gmres as the linear solver makes time spectral methods more robust allows them to be applied to a far greater subset of time accurate problems including those with a broad range of harmonic content and vastly improves the efficiency of time spectral methods in previous work a wave number independent preconditioner that mitigates the increased stiffness of the time spectral method when applied to problems with large resolvable wave numbers has been developed this preconditioner however directly inverts a large matrix whose size increases in proportion to the number of time instances as a result the computational time of this method scales as the cube of the number of time instances in the present work this preconditioner has been reworked to take advantage of an approximate factorization approach that effectively decouples the spatial and temporal systems once decoupled the time spectral matrix can be inverted in frequency space where it has entries only on the main diagonal and therefore can be inverted quite efficiently this new gmres preconditioner combination is shown to be over an order of magnitude more efficient than the previous wave number independent preconditioner for problems with large numbers of time instances and or large reduced frequencies

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

a theorem of the dore venni type for noncommuting operators a theorem of the dore venni type for the sum of two closed linear operators is proved where the operators are noncommuting but instead satisfy a certain commutator condition this result is then applied to obtain optimal regularity results for parabolic evolution equations u t l t u t f t and evolutionary integral equations u t t 0 a t s l s u s ds g t which are nonautonomous the domains of the involved operators l t may depend on t but l t 1 is required to satisfy a certain smoothness property the results are then applied to parabolic partial differential and integro differential equations

elliptic problems on ℝ n with unbounded coefficients in classical sobolev spaces we discuss the existence and uniqueness in h1 n and the h2 n regularity of the solutions of au f when f l2 n and a is a second order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity we also investigate whether a generates a c0 or analytic semigroup on l2 the approach in this nonweighted setting is based on a new and general method the idea consists in embedding a into a suitable one parameter family of operators as s with a0 a the properties of as when s 0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity this provides the needed estimates for uniqueness and regularity

numerical analysis of stochastic schemes in geophysics we present and study the stability convergence and order of convergence of a numerical scheme used in geophysics namely the stochastic version of a deterministic implicit leapfrog scheme which has been developed for the approximation of the so called barotropic vorticity model two other schemes which might be useful in the context of geophysical applications are also introduced and discussed

the navier stokes equations in ℝn with linearly growing initial data consider the equations of navier stokes on n with initial data u0 of the form u0 x u0 x mx where m is an n n matrix with constant real entries and u0 lp n it is shown that under these assumptions the equations of navier stokes admit a unique local solution in lp n moreover if etm 1 for all t 0 then this mild solution is even analytic in x this is surprising since the underlying semigroup of ornstein uhlenbeck type is not analytic in contrast to the stokes semigroup

gradient estimates for dirichlet parabolic problems in unbounded domains we consider autonomous parabolic dirichlet problems in a regular unbounded open set rn involving second order operator a with possibly unbounded coefficients we determine new conditions on the coefficients of a yielding global gradient estimates for the bounded classical solution

semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part abstract we study the cauchy problem for the semilinear nonautonomous parabolic equation u t a t u t u in s r d s in the spaces c b s r d and in l p s r d here is a borel measure defined via a tight evolution system of measures for the evolution operator g t s associated to the family of time depending second order uniformly elliptic operators a t sufficient conditions for existence in the large and stability of the null solution are also given in both c b and l p contexts the novelty with respect to the literature is that the coefficients of the operators a t are allowed to be unbounded

compactness and invariance properties of evolution operators associated with kolmogorov operators with unbounded coefficients abstract in this paper we consider nonautonomous elliptic operators a with nontrivial potential term defined in i r d where i is a right halfline possibly i r we prove that we can associate an evolution operator g t s with a in the space of all bounded and continuous functions on r d we also study the compactness properties of the operator g t s finally we provide sufficient conditions guaranteeing that each operator g t s preserves the usual l p spaces and c 0 r d

nonautonomous kolmogorov parabolic equations with unbounded coefficients we study a class of elliptic operators a with unbounded coeffi cients defined in i r d for some unbounded interval ir we prove that for any s 2 i the cauchy problem u s f 2 cb r d for the parabolic equation dtu au admits a unique bounded classical solution u this allows to associate an evolution family g t s with a in a natural way we study the main properties of this evolution family and prove gradient estimates for the function g t s f under suitable assumptions we show that there exists an evolution system of measures for g t s and we study the first properties of the extension of g t s to the l p spaces with respect to such measures

an order n log n parallel solver for time spectral problems abstract the time spectral method is a fast and efficient scheme for computing the solution to temporal periodic problems compared to traditional backward difference implicit time stepping methods time spectral methods incur significant computational savings by using a temporal fourier representation of the time discretization and solving the periodic problem directly in the time spectral discretization all time instances are fully coupled to each other resulting in a dense temporal discretization matrix the evaluation of which scales as o n 2 where n denotes the number of time instances however by implementing the time spectral method based on the fast fourier transform fft the computational cost can be reduced to o n log n furthermore in parallel implementations where each time instance is assigned to an individual processor the wall clock time necessary to solve time spectral problems is reduced to o log n using the fft based approach as opposed to the o n weak scaling incurred by previous dense matrix or discrete fourier transform dft parallel time spectral solver implementations in this work first an fft based approximate factorization af scheme is used to solve time spectral problems with large numbers of time instances subsequently this solution strategy is reformulated as a preconditioner to be used in the context of a newton krylov method applied directly to the complete non linear space time time spectral residual the use of the generalized minimal residual method gmres krylov method enables additional coupling between the various time instances running on different processors resulting in faster overall convergence the gmres af scheme is shown to produce significantly faster convergence than the af scheme used as a solver alone and achieves orders of magnitude gains in efficiency compared to previous dft based implementations of similar solution strategies for large numbers of time instances

on a polynomial scalar perturbation of a schrodinger system in lp spaces abstract in the paper 10 the l p realization l p of the matrix schrodinger operator l u d i v q u v u was studied the generation of a semigroup in l p r d c m and characterization of the domain d l p has been established in this paper we perturb the operator l p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on l p r d c m with domain embedded in w 2 p r d c m we also study the analyticity compactness positivity and ultracontractivity of the semigroup and prove gaussian kernel estimates further kernel estimates and asymptotic behaviour of eigenvalues of the matrix schrodinger operator are investigated

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

kernel and eigenfunction estimates for some second order elliptic operators abstract for a potential v such that v x x with 2 we prove that the heat kernel k t x y associated to the uniformly elliptic operator a j k 1 n k a j k j v satisfies the estimate k t x y c e 0 t e c t b e 2 2 x 1 2 x 4 n 1 2 e 2 2 y 1 2 y 4 n 1 2 for large x y r n and all t 0 here 0 1 is an appropriate constant b 2 2 and 0 is the first eigenvalue of a we also obtain an estimate for large x of the eigenfunctions of a

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 numerical approximations of forward backward stochastic differential equations a numerical method for a class of forward backward stochastic differential equations fbsdes is proposed and analyzed the method is designed around the four step scheme j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 but with a hermite spectral method to approximate the solution to the decoupling quasi linear pde on the whole space a rigorous synthetic error analysis is carried out for a fully discretized scheme namely a first order scheme in time and a hermite spectral scheme in space of the fbsdes equally important a systematical numerical comparison is made between several schemes for the resulting decoupled forward sde including a stochastic version of the adams bashforth scheme it is shown that the stochastic version of the adams bashforth scheme coupled with the hermite spectral method leads to a convergence rate of frac 32 in time which is better than those in previously published work for the fbsdes

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

compact implementation strategy for a harmonic balance method within implicit flow solvers a two step approximate factorization technique for implementing a computationally stable nonlinear unsteady frequency domain harmonic balance solution method within existing implicit computational fluid dynamic flow solver codes is presented the approach uses an explicit discretization of the harmonic balance source term and no new implicit code development is required both of these features enable the harmonic balance method to be implemented within existing implicit flow solver codes with minimal modification necessary to the underlying flow solver code the resulting harmonic balance solver can then be used for modeling nonlinear periodic unsteady flows the methodology is applied to the nasa overflow flow solver code and results are presented for transonic viscous flow past an unsteady pitching airfoil section

asymptotic behavior in time periodic parabolic problems with unbounded coefficients abstract we study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in r r d our results generalize and improve asymptotic behavior results for markov semigroups having an invariant measure we also study spectral properties of the realization of the parabolic operator u a t u u t in suitable l p spaces

fully implicit chebyshev time spectral method for general unsteady flows an efficient fully implicit chebyshev time spectral method tsm is developed to solve general unsteady flows with or without periodicity high accuracy of the chebyshev time spectral operator is d

on exponential mean square stability of two step maruyama methods for stochastic delay differential equations we are concerned with the exponential mean square stability of two step maruyama methods for stochastic differential equations with time delay we propose a family of schemes and prove that it can maintain the exponential mean square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation numerical results for linear and nonlinear equations show that this family of two step maruyama methods exhibits better stability than previous two step maruyama methods

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

on mean square stability of two step maruyama methods for nonlinear neutral stochastic delay differential equations the asymptotic mean square stability of two step maruyama methods is considered for nonlinear neutral stochastic differential equations with constant time delay nsddes under the one sided lipschitz condition and the linear growth condition it is proved that a family of implicit two step maruyama methods can preserve the stability of the analytic solution in mean square sense numerical results for both a nonlinear nsdde and a system show that the family of two step maruyama methods have better stability than previous two step maruyama methods

block jacobi implicit algorithms for the time spectral method many industrial applications involve flows periodic in time such flows are not simulated with enough efficiency when using classical unsteady techniques as a transient regime must be bypassed new techniques dedicated to time periodic flows and based on fourier analysis have been developed recently among these the time spectral method casts a time periodic flow computation in several coupled steady computations corresponding to a uniform sampling of the period up until now the steady states were reached using an explicit pseudotime algorithm thus very small time steps were needed and the convergence rate slowed down when increasing the number of harmonics in this paper a block jacobi approach is presented to solve the stationary problems with an implicit algorithm numerical simulations show on one hand the good quality of the results and on the other hand the interest of the proposed method to reduce the sensitivity of computations to a large number of harmonics

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

long time behavior of heat kernels of operators with unbounded drift terms abstract given a second order elliptic operator on r d with bounded diffusion coefficients and unbounded drift which is the generator of a strongly continuous semigroup on l 2 r d represented by an integral we study the time behavior of the integral kernel and prove estimates on its decay at infinity if the diffusion coefficients are symmetric a local lower estimate is also proved

on diagonal estimates on schrodinger semigroup kernels and reduced heat kernels we prove various estimates for the kernels of semigroups generated by schrodinger operators with magnetic field and potential of polynomial growth we also investigate the reduced heat kernels

global properties of generalized ornstein uhlenbeck operators on lp rn rn with more than linearly growing coefficients we show that the realization ap of the elliptic operator au div q u f u vu in lp rn rn p 1 generates a strongly continuous semigroup and we determine its domain d ap u w2 p rn rn f u vu lp rn rn if 1 p the diffusion coefficients q qij are uniformly elliptic and bounded together with their first order derivatives the drift coefficients f can grow as x log x and v can grow logarithmically our approach relies on the monniaux pruss theorem on the sum of noncommuting operators we also prove lp lq estimates and under somewhat stronger assumptions we establish pointwise gradient estimates and smoothing of the semigroup in the spaces w p rn rn 0 1 where 1 p

intrinsic taylor formula for kolmogorov type homogeneous groups abstract we consider a class of ultra parabolic kolmogorov type operators satisfying the hormander s condition we prove an intrinsic taylor formula with global and local bounds for the remainder given in terms of the norm in the homogeneous lie group naturally associated to the differential operator

lp estimates for parabolic systems with unbounded coefficients coupled at zero and first order abstract we consider a class of nonautonomous parabolic first order coupled systems in the lebesgue space l p r d r m d m 1 with p 1 sufficient conditions for the associated evolution operator g t s in c b r d r m to extend to a strongly continuous operator in l p r d r m are given some l p l q estimates are also established together with l p gradient estimates