Large deviation principle for a class of SPDE with locally monotone coefficients

This work aims to prove the large deviation principle for a class of stochastic partial differential equations with locally monotone coefficients under the extended variational framework, which generalizes many previous works. Using stochastic control and the weak convergence approach, we prove the Laplace principle,which is equivalent to the large deviation principle in our framework. Instead of assuming compactness of the embedding in the corresponding Gelfand triple or finite dimensional approximation of the diffusion coefficient in some existing works, we only assume some temporal regularity in the diffusion coefficient.     

Stochastic partial functional differential equations with locally monotone coefficients,locally Lipschitz non-linearity and delay

In this paper, we study the existence and uniqueness of strong solutions for stochastic partial functional differential equations with locally monotone coefficients, locally Lipschitz non-linearity, and time delay. Our results extend previous results obtained by Liu–Röckner, Caraballo et al. and Taniguchi et al. Examples are given to illustrate the wide applicability of our results.     

Asymptotic error distribution for the Euler scheme with locally Lipschitz coefficients

In traditional works on numerical schemes for solving stochastic differential equations (SDEs), the globally Lipschitz assumption is often assumed to ensure different types of convergence. In practice, this is often too strong a condition. Brownian motion driven SDEs used in applications sometimes have coefficients which are only Lipschitz on compact sets, but the paths of the SDE solutions can be arbitrarily large. In this paper, we prove convergence in probability and a weak convergence result under a less restrictive assumption, that is, locally Lipschitz and with no finite time explosion. We prove if a numerical scheme converges in probability uniformly on any compact time set (UCP) with a certain rate under a global Lipschitz condition, then the UCP with the same rate holds when a globally Lipschitz condition is replaced with a locally Lipschitz plus no finite explosion condition. For the Euler scheme, weak convergence of the error process is also established. The main contribution of this paper is the proof of $\sqrt{n}$ weak convergence of the normalized error process and the limit process is also provided. We further study the boundedness of the second moments of the weak limit process and its running supremum under both global Lipschitz and locally Lipschitz conditions.     

Convergence of stationary sequences for variational inequalities with maximal monotone operators

LetT be a maximal monotone operator defined on ^N . In this paper we consider the associated variational inequality 0 T(x ^*) and stationary sequences {x _k ^* for this operator, i.e., satisfyingT(x _k ^* 0. The aim of this paper is to give sufficient conditions ensuring that these sequences converge to the solution setT ^–1(0) especially when they are unbounded. For this we generalize and improve the directionally local boundedness theorem of Rockafellar to maximal monotone operatorsT defined on ^N .     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Conditions for existence of a global strong solution to one class of nonlinear evolution equations in Hilbert space. II

We obtain a criterion of global strong solvability for one class of nonlinear evolution equations in Hilbert space.     

Stochastic evolution equations in Hilbert spaces

In this paper we consider Ito's stochastic differential equation in Hilbert spaces. A strong solution is generated by the difference approximation. A regularity result is obtained for solutions to a class of parabolic stochastic partial differential equations. Hyperbolic stochastic evolution equations are also discussed.     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

Solution of monotone complementarity problems with locally Lipschitzian functions

The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥ ₂ ² , we extend results which hold for sufficiently smooth functionsF to the nonsmooth case. In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed. To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the order of 1+p.     

The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ( $ \tfrac{1} {2} Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ (, 1). Namely, the following assertion is true. Let α ∈ (, 1), < p < 2, a sequence a ∈ M _α, and . Then the series cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π). Original Russian Text ? M.I. D’yachenko, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2008, No. 5, pp. 38–47.     

Gradient estimates for stochastic evolution equations with non-Lipschitz coefficients

By using finite-dimensional approximations and a recent result on gradient estimates for singular diffusions on Rd, gradient estimates are derived for the semigroups of solutions to a class of stochastic evolution equations on Hilbert spaces with non-Lipschitz coefficients.     

Stochastic Volterra equations in Banach spaces and stochastic partial differential equation

In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier-Stokes equations are also investigated.     

Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space

Let be a real Hilbert space. Let , be bounded monotone mappings with , where and are closed convex subsets of satisfying certain conditions. Suppose the equation has a solution in . Then explicit iterative methods are constructed that converge strongly to such a solution. No invertibility assumption is imposed on , and the operators and need not be defined on compact subsets of .

     

Pseudo-monotone complementarity problems in Hilbert space

In this paper, some existence results for a nonlinear complementarity problem involving a pseudo-monotone mapping over an arbitrary closed convex cone in a real Hilbert space are established. In particular, some known existence results for a nonlinear complementarity problem in a finite-dimensional Hilbert space are generalized to an infinite-dimensional real Hilbert space. Applications to a class of nonlinear complementarity problems and the study of the post-critical equilibrium state of a thin elastic plate subjected to unilateral conditions are given.This research was partially supported by the National Science Foundation Grant DMS-89-13089, Department of Energy Grant DE-FG03-87-ER-25028, and Office of Naval Research Grant N00014-89-J-1659. The authors would like to express their sincere thanks to Professor S. Schaible, School of Administration, University of California, Riverside, for his helpful suggestions and comments. They also thank the referees for their comments and suggestions that improved this paper substantially.     

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.     

Strong convergence of new iterative projection methods with regularization for solving monotone variational inequalities in Hilbert spaces

In this paper, we introduce two new numerical methods for solving a variational inequality problem involving a monotone and Lipschitz continuous operator in a Hilbert space. We describe how to incorporate a regularization term depending on a parameter in the projection method and then establish the strong convergence of the resulting iterative regularization projection methods. Unlike known hybrid methods, the strong convergence of the new methods comes from the regularization technique. The first method is designed to work in the case where the Lipschitz constant of cost operator is known, whereas the second one is more easily implemented without this requirement. The reason is because the second method has used a simple computable stepsize rule. The variable stepsizes are generated by the second method at each iteration and based on the previous iterates. These stepsizes are found with only one cheap computation without line-search procedure. Several numerical experiments are implemented to show the computational effectiveness of the new methods over existing methods.     

Hilbert空间中双曲型随机线性发展方程的显示解(英文)

本文研究了Hilbert空间中的双曲型随机线性发展方程 ,证明了 ,在某些条件下得到了用“样本广义解方法”得到这类方程的显示解 ,并利用显示解证明了解的存在性与唯一性 .     

Non-densely defined impulsive neutral stochastic functional differential equations driven by fBm in Hilbert space with infinite delay

We study a class of non-densely defined impulsive neutral stochastic functional differential equations driven by an independent cylindrical fractional Brownian motion (fBm) with Hurst parameter H∈ (1/2, 1) in the Hilbert space. We prove the existence and uniqueness of the integral solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions. The results are obtained by using the method of successive approximation.