rough oscillatory singular integral operators of nonconvolution type abstract in this paper we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials we prove that such operators are bounded on l p provided their kernels satisfy a very weak condition in addition we also study the related truncated oscillatory singular integral operators moreover we present a class of unbounded oscillatory singular integral operators

long time existence for signed solutions of the heat equation with a noise term let be the circle 0 j with the ends identified we prove long time existence for the following equation

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

long time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise stratonovitch s case in this article we prove new results concerning the long time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non random global attractor which consists of two spatially and temporally homogeneous asymptotic states more precisely we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states or that they set out to oscillate between them in the first case we can also determine exactly the corresponding lyapunov exponents in the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one in both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states our methods of proof rest upon the use of a suitable regularization of the brownian motion along with a related wong zakai approximation procedure

weighted weak 1 1 and weighted estimates for oscillating kernels weak type 1 1 and strong type p p inequalities are proved for operators defined by oscillating kernels the techniques are sufficiently general to derive versions of these inequalities using weighted norms 0 introduction given a positive real number a o a 7 1 define the oscillating kernel ka by ka x 1 x leilxla and consider the convolution operator ka f in an earlier paper 2 we studied the boundedness properties of such operators on weighted lp spaces 1 1 we define l p f 1l p w a e f x lpw x dx r and say f e lp r if llfllp w 1 the multiplier operator t associated to ka is defixled by t d 0 1 l e 1tl f where 0 is a c function defined by i 0 4 0 if 1 when p 0

least squares shadowing sensitivity analysis of chaotic limit cycle oscillations abstract the adjoint method among other sensitivity analysis methods can fail in chaotic dynamical systems the result from these methods can be too large often by orders of magnitude when the result is the derivative of a long time averaged quantity this failure is known to be caused by ill conditioned initial value problems this paper overcomes this failure by replacing the initial value problem with the well conditioned least squares shadowing lss problem the lss problem is then linearized in our sensitivity analysis algorithm which computes a derivative that converges to the derivative of the infinitely long time average we demonstrate our algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations

sensitivity analysis of limit cycle oscillations many unsteady problems equilibrate to periodic behavior for these problems the sensitivity of periodic outputs to system parameters are often desired and must be estimated from a finite time span or frequency domain calculation sensitivities computed in the time domain over a finite time span can take excessive time to converge or fail altogether to converge to the periodic value additionally finite span outputs can exhibit local extrema in parameter space which the periodic outputs they approximate do not hindering their use in optimization we derive a theoretical basis for this error and demonstrate it using two examples a van der pol oscillator and vortex shedding from a low reynolds number airfoil we show that output windowing enables the accurate computation of periodic output sensitivities and may allow for decreased simulation time to compute both time averaged outputs and sensitivities we classify two distinct window types long time over a large not necessarily integer number of periods and short time over a small integer number of periods finally from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation respectively

a class of quasilinear stochastic partial differential equations of mckean vlasov type with mass conservation a system ofn particles inr d with mean field interaction and diffusion is considered assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by brownian sheets microscopic equation where all coefficients depend on the position of the particles and on the empirical mass distribution process this empirical mass distribution process satisfies a quasilinear stochastic partial differential equation spde this spde mezoscopic equation is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation starting with measures with densities inl 2 r d dr wheredr is the lebesgue measure the solution will have densities inl 2 r d dr and strong uniqueness in the ito sense is obtained finally it is indicated how to obtain macroscopic partial differential equations as limits of the so constructed spde s

superprocesses in a brownian environment superprocesses are idealized models for the evolution of a population with geographical structure motion and unisex reproductive behaviour in this paper the reproductive behaviour in particular the offspring distribution is related to an exterior process the environment we characterize these superprocesses as the unique solution of certain evolution equations we also prove that they can be obtained as limits of systems of branching particles with minimal variance branchings and faster non classical branching rates

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

comparison methods for a class of function valued stochastic partial differential equations a comparison theorem is derived for a class of function valued stochastic partial differential equations spde s with lipschitz coefficients driven by cylindrical and regular hilbert space valued brownian motions moreover we obtain necessary and sufficient conditions for the positivity of the mild solutions of the spde s where the sufficiency follows from the comparison theorem thereby it is e g possible to identify a class of spde s which can serve as stochastic space time models for the density of particles as a consequence we can construct unique mild solutions of spde s on the cone of positive functions with non lipschitz drift parts including the case of arbitrary polynomialsr x withr o o and leading negative coefficient

forward and adjoint sensitivity computation of chaotic dynamical systems this paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems such as the lorenz attractor the algorithms compute the derivative of long time averaged statistical quantities to infinitesimal perturbations of the system parameters the algorithms are demonstrated on the lorenz attractor we show that sensitivity derivatives of statistical quantities can be accurately estimated using a single short trajectory over a time interval of 20 on the lorenz attractor

a framework for simultaneous aerodynamic design optimization in the presence of chaos abstract integrating existing solvers for unsteady partial differential equations into a simultaneous optimization method is challenging due to the forward in time information propagation of classical time stepping methods this paper applies the simultaneous single step one shot optimization method to a reformulated unsteady constraint that allows for both forward and backward in time information propagation especially in the presence of chaotic and turbulent flow solving the initial value problem simultaneously with the optimization problem often scales poorly with the time domain length the new formulation relaxes the initial condition and instead solves a least squares problem for the discrete partial differential equations this enables efficient one shot optimization that is independent of the time domain length even in the presence of chaos

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

stability in mean of partial variables for stochastic reaction diffusion systems abstract it is with the aid of the lyapunov method that stochastic ordinary differential equations and stochastic functional differential equations have been investigated intensively in the past decades however for stochastic reaction diffusion equations this useful technique seems to feel helpless on account of the emptiness of its own ito s formula to get over this difficulty we will manage to regard the integral of the considered trajectory with respect to spatial variables as the solution of the corresponding stochastic ordinary differential equations and then employ the ito formula to study stability in mean of partial variables for ito stochastic reaction diffusion equations some sufficient conditions for uniform stability in mean asymptotic stability in mean uniformly asymptotic stability in mean exponential stability in mean of partial variables are presented

convergence of the least squares shadowing method for computing derivative of ergodic averages for a parameterized hyperbolic system u i 1 f u i s the derivative of an ergodic average langle j rangle lim n rightarrow infty frac1n sum 1 n j u i s to the parameter s can be computed via the least squares shadowing method this method solves a constrained least squares problem and computes an approximation to the desired derivative frac d langle j rangle ds from the solution this paper proves that as the size of the least squares problem approaches infinity the computed approximation converges to the true derivative

the stampacchia maximum principle for stochastic partial differential equations and applications abstract stochastic partial differential equations spdes are considered linear and nonlinear for which we establish comparison theorems for the solutions or positivity results a e and a s for suitable data comparison theorems for spdes are available in the literature the originality of our approach is that it is based on the use of truncations following the stampacchia approach to maximum principle we believe that our method which does not rely too much on probability considerations is simpler than the existing approaches and to a certain extent more directly applicable to concrete situations among the applications boundedness results and positivity results are respectively proved for the solutions of a stochastic boussinesq temperature equation and of reaction diffusion equations perturbed by a non lipschitz nonlinear noise stabilization results to a chafee infante equation perturbed by a nonlinear noise are also derived

characterizing dynamics with covariant lyapunov vectors a general method to determine covariant lyapunov vectors in both discrete and continuous time dynamical systems is introduced this allows us to address fundamental questions such as the degree of hyperbolicity which can be quantified in terms of the transversality of these intrinsic vectors for spatially extended systems the covariant lyapunov vectors have localization properties and spatial fourier spectra qualitatively different from those composing the orthonormalized basis obtained in the standard procedure used to calculate the lyapunov exponents

discrete second order adjoints in atmospheric chemical transport modeling atmospheric chemical transport models ctms are essential tools for the study of air pollution for environmental policy decisions for the interpretation of observational data and for producing air quality forecasts many air quality studies require sensitivity analyses i e the computation of derivatives of the model output with respect to model parameters the derivatives of a cost functional defined on the model output with respect to a large number of model parameters can be calculated efficiently through adjoint sensitivity analysis while the traditional first order adjoint models give the gradient of the cost functional with respect to parameters second order adjoint models give second derivative information in the form of products between the hessian of the cost functional and a vector representing a perturbation in sensitivity analysis a search direction in optimization an eigenvector etc in this paper we discuss the mathematical foundations of the discrete second order adjoint sensitivity method and present a complete set of computational tools for performing second order sensitivity studies in three dimensional atmospheric ctms the tools include discrete second order adjoints of runge kutta and of rosenbrock time stepping methods for stiff equations together with efficient implementation strategies numerical examples illustrate the use of these computational tools in important applications like sensitivity analysis optimization uncertainty quantification and the calculation of directions of maximal error growth in three dimensional atmospheric ctms

white noise driven spdes with reflection we study reflected solutions of a nonlinear heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by space time white noise the nonlinearity appears both in the drift and in the diffusion coefficient roughly speaking at any point t x where the solutionu t x is strictly positive it obeys the equation and at a point t x whereu t x is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities existence of a minimal solution is proved the construction uses a penalization argument a new existence theorem for spdes whose coefficients depend on the past of the solution and a comparison theorem for solutions of white noise driven spdes

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

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

a stochastic log laplace equation we study a nonlinear stochastic partial differential equation whose solution is the conditional log laplace functional of a superprocess in a random environment we establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by kurtz and xiong stochastic process appl 83 1999 103 126 we also derive the wong zakai type approximation for this equation as an application we give a direct proof of the moment formulas of skoulakis and adler ann appl probab 11 2001 488 543

sensitivity analysis on chaotic dynamical systems by non intrusive least squares shadowing nilss abstract this paper develops the non intrusive least squares shadowing nilss method which computes the sensitivity for long time averaged objectives in chaotic dynamical systems in nilss we represent a tangent solution by a linear combination of one inhomogeneous tangent solution and several homogeneous tangent solutions next we solve a least squares problem using this representation thus the resulting solution can be used for computing sensitivities nilss is easy to implement with existing solvers in addition for chaotic systems with many degrees of freedom but few unstable modes nilss has a low computational cost nilss is applied to two chaotic pde systems the lorenz 63 system and a cfd simulation of flow over a backward facing step in both cases the sensitivities computed by nilss reflect the trends in the long time averaged objectives of dynamical systems

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

a stochastic pitchfork bifurcation in a reaction diffusion equation we study in some detail the structure of the random attractor for the chafee infante reaction diffusion equation perturbed by a multiplicative white noise d u u u u 3 d t u o d w t x d r m first we prove for m 5 a lower bound on the dimension of the random attractor which is of the same order in as the upper bound we derived in an earlier paper and is the same as that obtained in the deterministic case then we show for m 1 that as passes through 1 the first eigenvalue of the negative laplacian from below the system undergoes a stochastic bifurcation of pitchfork type we believe that this is the first example of such a stochastic bifurcation in an infinite dimensional setting central to our approach is the existence of a random unstable manifold

minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation adjoint equations of differential equations have seen widespread applications in optimization inverse problems and uncertainty quantification a major challenge in solving adjoint equations for time dependent systems has been the need to use the solution of the original system in the adjoint calculation and the associated memory requirement in applications where storing the entire solution history is impractical checkpointing methods have frequently been used however traditional checkpointing algorithms such as revolve require a priori knowledge of the number of time steps making these methods incompatible with adaptive time stepping we propose a dynamic checkpointing algorithm applicable when the number of time steps is a priori unknown our algorithm maintains a specified number of checkpoints on the fly as time integration proceeds for an arbitrary number of time steps the resulting checkpoints at any snapshot during the time integration have the optimal repetition number the efficiency of our algorithm is demonstrated both analytically and experimentally in solving adjoint equations this algorithm also has significant advantage in automatic differentiation when the length of execution is variable

on quasi linear stochastic partial differential equations we prove existence and uniqueness of the solution of a parabolic spde in one space dimension driven by space time white noise in the case of a measurable drift and a constant diffusion coefficient as well as a comparison theorem

harmonic analysis on nilpotent groups and singular integrals i oscillatory integrals abstract this paper is devoted to the study of the operator given by where k is a standard calderon zygmund kernel and p is a real polynomial on r n r n we show t is bounded on lp to itself when 1

singular integrals with rough kernels along real analytic submanifolds in r n l p mapping properties will be established in this paper for singular radon transforms with rough kernels defined by translated of a real analytic submanifold in r n

singular integrals related to the radon transform and boundary value problems abstract two classes of integral operators are introduced that combine features of calderon zygmund singular integrals the hilbert integral and the radon transform examples and applications arising from integral geometry and several complex variables are discussed

high density limit theorems for nonlinear chemical reactions with diffusion the solutionx of a nonlinear reaction diffusion equation on then dimensional unit cubes is approximated by a space time jump markov processx v n law of large numbers lln x v n is constructed on a grids n ons ofn cells wherev is proportional to the initial number of particles in each cell the deviation ofx v n fromx is computed by a central limit theorem clt the assumptions on the parametersv n are for the lln asn and for the clt frac n upsilon to 0 asn the limity y x in the clt which is a generalized ornstein uhlenbeck process is represented as the mild solution of a linear stochastic partial differential equation spde and its best possible state spaces are described the problem of stationary solutions ofy x in dependence ofx is also investigated