Solutions Of Stochastic Evolution Equations Research Articles

The solution of stochastic evolution equation with the fractional derivative

In this paper, we consider the stochastic evolution equation driven by the Gaussian noise with white time and colored space, where the noise coefficient is the Marchaud fractional derivative. The key idea is that we transform our model into a stochastic space-fractional equation by taking the Marchaud fractional derivative, and then use Chaos expansion to prove the mild solution. There are three main results in this paper. First, we apply Chaos expansion to obtain the existence, uniqueness and Lyapunove exponent of the solution of the transformed equation. Second, we prove that there exists an unique mild solution of the original equation, the approach is taking the fractional integral operator into the transformed equation. Finally, we explore Hölder continuity of the mild solution.

The asymptotic behavior of solutions for stochastic evolution equations with pantograph delay

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the [Formula: see text]th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for [Formula: see text], for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the [Formula: see text]th moment and almost sure senses, of the nontrivial equilibrium solution are established.

Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

We are involved in this study with hybrid functions consisting of Taylor polynomials and block-pulse functions and use them as basis functions to achieve the numerical solution of stochastic evolution equation with fractional Brownian motion (FBM). First, we compute stochastic operational matrix based on hybrid Taylor block-pulse functions (HTBPFs) that this operator able us obtain an applicable procedure for solving stochastic integral equations (SIEs) and stochastic differential equations (SDEs) such as stochastic evolution equation. Then, we use this operator and HTBPFs ordinary operational matrix to convert solving stochastic evolution equations with FBM into solving a system of easily solvable algebraic equations. Under some mild conditions we prove that our proposed method is convergent when M, N → ∞, where N and M are the order of block-pulse functions and Taylor polynomials, respectively. Finally, we utilize the mentioned method for solving two test problems that their exact solution is available. Comparison obtained approximate solution with the exact solution demonstrate that the values of absolute error can be ignored and thus obtained solution has a good degree of accuracy. Furthermore, the theoretical discussions and numerical examples confirm that by increasing the values of N and M, the approximate solution tends to the exact solution.

Local mild solutions for rough stochastic partial differential equations

We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H∈(1/3,1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space.

Optimal regularity of stochastic evolution equations in M-type 2 Banach spaces

In this paper, we prove the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [9] dealing with Hilbert setting and a version of Burkholder inequality in M-type 2 Banach space. Applying our main results to the stochastic heat equation gives a positive answer to an open problem proposed in [10].

Existence and stability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.

Well-posedness and optimal regularity of stochastic evolution equations with multiplicative noises

In this paper, we establish the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized Lipschitz-type coefficients driven by general multiplicative noises. To ensure the well-posedness of the problem, the linear operator of the equations is only need to be a generator of a C0-semigroup and the proposed noises are quite general, which include space–time white noise and rougher noises. When the linear operator generates an analytic C0-semigroup, we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method.

Square mean almost automorphic solution of stochastic evolution equations with impulses on time scales

Square mean almost automorphic solution of stochastic evolution equations with impulses on time scales

On the differentiability of solutions of stochastic evolution equations with respect to their initial values

In this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations (SEEs) with respect to their initial values. We prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered SEEs are n-times continuously Fréchet differentiable, then the solutions of the considered SEEs are also n-times continuously Fréchet differentiable with respect to their initial values. Moreover, a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered SEE in the sense that the dominating linear operator appearing in the SEE smoothes the higher order derivative processes.

Global existence and asymptotic stability of mild solutions for stochastic evolution equations with nonlocal initial conditions

The aim of this paper is to discuss the global existence, uniqueness and asymptotic stability of mild solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. A sufficient condition is given for judging the relative compactness of a class of abstract continuous family of functions on infinite intervals. With the aid of this criteria the compactness of the solution operator for the problem studied on the half line is obtained. The theorems proved in this paper improve and extend some related results in this direction. Discussions are based on stochastic analysis theory, analytic semigroup theory, relevant fixed point theory and the well known Gronwall-Bellman type inequality. An example to illustrate the feasibility of our main results is also given.

Infinite dimensional weak Dirichlet processes and convolution type processes

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process.The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs).In particular the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t,X(t)) where f:[0,T]×H→R is a C0,1 function and X a convolution type process.

Square-mean pseudo almost automorphic mild solutions for stochastic evolution equations driven by G-Brownian motion

ABSTRACTIn this article, we prove the existence and uniqueness of the square-mean pseudo almost automorphic mild solutions for a class of stochastic evolution equations driven by G-Brownian motion (G-SSEs). Our results are established by means of the fixed point theorem. An example is given to illustrate the theory.

Measurable solutions for stochastic evolution equations without uniqueness

A method for obtaining measurable solutions to stochastic evolution equations in which there is no uniqueness for the corresponding non-stochastic equation is presented. It involves a technique based on a measurable selection theorem for set-valued functions. No assumptions are needed on the underlying probability space. An application is given to the stochastic Navier–Stokes problem in arbitrary dimensions. We also show the existence of measurable solutions to stochastic ordinary differential equations in which there is no uniqueness. A finite-dimensional generalization is given to adapted solutions in the case of a normal filtration and path uniqueness.

On signed measure valued solutions of stochastic evolution equations

We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized two-dimensional Navier–Stokes equations in vorticity form introduced by Kotelenez.

A variational approach to the construction and Malliavin differentiability of strong solutions of SDE’s

In this article we develop a new approach to construct solutions of stochastic equations with merely measurable drift coefficients. We aim at demonstrating the principles of our technique by analyzing strong solutions of stochastic differential equations driven by Brownian motion. An important and rather surprising consequence of our method which is based on Malliavin calculus is that the solutions derived by Veretennikov (Theory Probab Appl 24:354–366, 1979) for Brownian motion with bounded and measurable drift in \(\mathbb{R }^{d}\) are Malliavin differentiable. Further, a strength of our approach, which does not rely on a pathwise uniqueness argument, is that it can be transferred and applied to the analysis of various other types of (stochastic) equations: We obtain a Bismut–Elworthy–Li formula (Elworthy and Li, J Funct Anal 125:252–286, 1994) for spatial derivatives of solutions to the Kolmogorov equation with bounded and measurable drift coefficients. To derive the formula, we use that our approach can be applied to obtain Sobolev differentiability in the initial condition in addition to Malliavin differentiability of the associated stochastic differential equations. Another application of our technique is the construction of unique solutions of the stochastic transport equation with irregular vector fields. Moreover, our approach is also applicable to the construction of solutions of stochastic evolution equations on Hilbert spaces.

Compensated fractional derivatives and stochastic evolution equations

We are interested in developing a pathwise theory for mild solutions of stochastic evolution equations when the noise path is β-Hölder continuous for β∈(1/3,1/2). From the point of view of the Rough Path Theory, stochastic integrals related to the solution of ordinary differential equations contain area-elements from a tensor space. Based on (compensated) fractional derivatives we are able to derive a second mild equation for these area components. We formulate sufficient conditions for the existence and uniqueness of a pathwise mild solution by using the Banach fixed point theorem provided that the coefficients of the system are sufficiently regular.

Robust stabilization with a general decay of mild solutions of stochastic evolution equations

In this paper we study the question of robust stabilization of infinite dimensional stochastic systems against uncertainty induced by relatively bounded perturbations of the principal operator determining the system. We present results on state feedback robust stabilization with a general decay. Two examples are included to illustrate the theory.

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton–Jacobi–Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.

Large deviation principle for semilinear stochastic evolution equations with monotone nonlinearity and multiplicative noise

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplicative noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An Itô-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.

Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids

In this Note we show that under suitable conditions on the data we can construct a sequence of solutions of the stochastic second grade fluid that converges to the probabilistic strong solution of the stochastic Navier–Stokes equations when the stress modulus α tends to zero.