Cylindrical Wiener Process Research Articles

Gibbsian dynamics and ergodicity of magnetohydrodynamics and related systems forced by random noise

The magnetohydrodynamics system consists of the Navier-Stokes equations from fluid mechanics, coupled with the Maxwell’s equations from electromagnetism through multiples of non-linear terms involving derivatives. Following the approach of [1], we prove the existence of a unique invariant measure in case the forcing terms consist of the cylindrical Wiener processes with only low modes. Its proof requires taking advantage of the structure of the non-linear terms carefully and is extended to various other related models such as the magnetohydrodynamics-Boussinesq system from fluid mechanics in atmosphere and oceans, as well as the magneto-micropolar fluid system from the theory of microfluids.

Existence results for second-order stochastic differential inclusions driven by Lévy noise

In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with a standard cylindrical Wiener process and Poisson jumps. We consider the case in which the right hand side can be either convex-valued.

Coupled System of Second-Order Stochastic Neutral Differential Inclusions Driven by Wiener Process and Poisson Jumps

In this paper we prove the existence of mild solutions for a second-order impulsive semilinear stochastic differential inclusion with an infinite-dimensional standard cylindrical Wiener process and Poisson jumps. We consider non convex-valued cases.

Stochastic evolution equations in Banach spaces and applications to the Heath\u2013Jarrow\u2013Morton\u2013Musiela equations

The aim of this paper is threefold. Firstly, we study stochastic evolution equations (with the linear part of the drift being a generator of a C_{0}-semigroup) driven by an infinite-dimensional cylindrical Wiener process. In particular, we prove, under some sufficient conditions on the coefficients, the existence and uniqueness of solutions for these stochastic evolution equations in a class of Banach spaces satisfying the so-called H-condition. Moreover, we analyse the Markov property of the solutions.Secondly, we apply the abstract results obtained in the first part to prove the existence and uniqueness of solutions to the Heath–Jarrow–Morton–Musiela (HJMM) equations in weighted Lebesgue and Sobolev spaces.Finally, we study the ergodic properties of the solutions to the HJMM equations. In particular, we find a sufficient condition for the existence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations (when the coefficients are time-independent) in the weighted Lebesgue spaces.Our paper is a modest contribution to the theory of financial models in which the short rate can be undefined.

Well-posedness for a pseudomonotone evolution problem with multiplicative noise

Our aim is the study of well-posedness for the stochastic evolution equation $$\begin{aligned} \mathrm{d}u -\mathrm {div}\,(|\nabla u|^{p-2}\nabla u +F(u)) \,\mathrm{d} t = H(u) \,\mathrm{d}W, \end{aligned}$$ for $$T>0$$ , on a bounded Lipschitz domain $$D \subset \mathbb {R}^d$$ with homogeneous Dirichlet boundary conditions, initial values $$u_0\in L^2(D)$$ , $$p>2$$ and $$F:\mathbb {R}\rightarrow \mathbb {R}^d$$ Lipschitz continuous. W(t) is a cylindrical Wiener process in $$L^2(D)$$ with respect to a filtration $$(\mathcal {F}_t)$$ satisfying the usual assumptions on a complete, countably generated probability space $$(\varOmega ,\mathcal {F},\mathbb {P})$$ . We consider the case of multiplicative noise satisfying appropriate regularity conditions. By a semi-implicit time discretization, we obtain approximate solutions. Using the theorems of Skorokhod and Prokhorov, we are able to pass to the limit and show existence of martingale solutions. We establish pathwise $$L^1$$ -contraction and uniqueness and obtain existence and uniqueness of strong solutions.

Generalized KdV equation subject to a stochastic perturbation

We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving noise is a cylindrical Wiener process on $L^2(\mathbb{R})$ and the covariance operator is Hilbert-Schmidt in an appropriate Sobolev space, then the solutions with $H^1(\mathbb{R})$ data are globally well-posed in $H^1(\mathbb{R})$. This extends results obtained by A. de Bouard and A. Debussche for the stochastic KdV equation.

Conforming finite element methods for the stochastic Cahn–Hilliard–Cook equation

This paper is concerned with the finite element approximation of the stochastic Cahn–Hilliard–Cook equation driven by an infinite dimensional Wiener type noise. The Argyris finite elements are used to discretize the spatial variables while the infinite dimensional (cylindrical) Wiener process is approximated by truncated stochastic series spanned by the spectral basis of the covariance operator. The optimal strong convergence order in L2 and H˙−2 norms is obtained. Unlike the mixed finite element method studied in the existing literature, our method allows the covariance operator of the Wiener process to have an infinite trace, including the space–time white noise is allowed in our model. Numerical experiments are presented to illustrate the theoretical analysis.

Numerical solutions of stochastic Fisher equation to study migration and population behavior in biological invasion

In this paper, numerical solutions of the stochastic Fisher equation have been obtained by using a semi-implicit finite difference scheme. The samples for the Wiener process have been obtained from cylindrical Wiener process and Q-Wiener process. Stability and convergence of the proposed finite difference scheme have been discussed scrupulously. The sample paths obtained from cylindrical Wiener process and Q-Wiener process have also been shown graphically.

Stochastic evolution of regulation errors in a boundary-actuated desalination plant

A desalination plant – considered in two configurations (once-through and brine recirculation) – is modelled and controlled using a system of coupled PDEs that describe the desalination processes. The analysis is conducted in two separate parts. First, the operating point of the plant is obtained based on the deterministic process models of the plant. The steady-state distillate production is optimized with respect to a reference pressure head (operating point) that is achieved by applying relatively simple boundary controls. Both deterministic plant configurations are compared in term of characteristic numbers that evaluates the energy-efficient operation of the plant. In particular, those are the thermal ratio and the specific flow rate, where gains of roughly 5.5% and 21.5% are obtained in favour of the brine recirculation plant. The pressure head is subject to turbulence phenomena that disturb its surface so that a deterministic model is an insufficient representation of the real-case scenario. Concerning the second part of the paper, the effects of turbulence are incorporated through stochastic elements given as generalized and cylindrical Wiener processes located on the boundaries and throughout the plant (subdomain), respectively. The pressure head residual is defined as the difference between the deterministic and stochastic system. As both systems are actuated by the same type of boundary controls, the residual field is interpreted as a measure of a regulation error. It is statistical characterization is done spatially by means of the first four statistical moments (sampled) and temporarily with the autocorrelation function. It is found that the applied boundary controls are robust enough to keep the regulation error within tight bounds throughout the whole subdomain of the plant. Throughout the plant, the spatial standard deviation (std) is less than 0.3.

Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form dXt−divγ(∇Xt)dt+β(Xt)dt∋B(t,Xt)dWt, where γ and β are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on Rd and R respectively, and W is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray–Lions conditions on γ and with no restrictive smoothness or growth assumptions on β. The operator B is assumed to be Hilbert–Schmidt and to satisfy some classical Lipschitz conditions in the second variable.

Stochastic differential equations in a Banach space driven by the cylindrical Wiener process

Generalized stochastic integral from predictable operator-valued random process with respect to a cylindrical Wiener process in an arbitrary Banach space is defined. The question of existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The sufficient condition of existence of the stochastic integral in terms of p-absolutely summing operators is given. The stochastic differential equation for generalized random processes is considered and existence and uniqueness of the solution is developed. As a consequence, the corresponding results of the stochastic differential equations in an arbitrary Banach space are given.

Linear approximation of nonlinear Schrödinger equations driven by cylindrical Wiener processes

In this paper the existence and uniqueness of the solution for a stochastic nonlinear Schrödinger equation, which is perturbed by a cylindrical Wiener process is investigated. The existence of the variational solution and of the generalized weak solution are proved by using sequences of successive approximations, which are the solutions of certain linear problems.

Strong averaging principle for two-time-scale non-autonomous stochastic FitzHugh-Nagumo system with jumps

In this paper, we study an averaging principle for stochastic FitzHugh-Nagumo system with different time scales driven by cylindrical Wiener processes and Poisson jumps, where the slow equation is non-autonomous and the fast equation is autonomous case. Under suitable assumptions, we show that the slow component mean-square strongly converges to the solution of the corresponding averaging equation, and the rate of the convergence as a by-product is also affirmed. Finally, we give some open problems which are derived from this paper.

The stochastic p(ω,t,x)-Laplace equation with cylindrical Wiener process

We propose the analysis of a non-linear parabolic problem of p(ω,t,x)-Laplace type in the framework of Orlicz Lebesgue and Sobolev spaces with variable random exponents and a stochastic forcing by a cylindrical Wiener process. We give a result of well-posedness: existence, uniqueness and stability of the solution, for additive and multiplicative problems.

Degenerate parabolic stochastic partial differential equations: Quasilinear case

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized It\^{o} formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem.

Regularity of solutions of abstract linear evolution equations*

In this paper, we study regularity of solutions to linear evolution equations of the form $dX+AXdt=F(t)dt$ in a Banach space $H$, where $A$ is a sectorial operator in $H$ and $A^{-\alpha} F \, (\alpha>0)$ belongs to a weighted H\"{o}lder continuous function space. Similar results are obtained for linear evolution equations with additive noise of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in a separable Hilbert space $H$, where $W(t)$ is a cylindrical Wiener process. Our results are applied to a model arising in neurophysiology, which has been proposed by Walsh \cite{Walsh}.

Ornstein-Uhlenbeck Processes Driven by Cylindrical Lévy Processes

In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical Levy processes in separable Banach spaces. Here, a cylindrical Levy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical Levy processes and several examples are considered.

Strong averaging principle for slow-fast SPDEs with Poisson random measures

This work concerns the problem associated with an averaging principle for two-time-scales stochastic partial differential equations (SPDEs) driven by cylindrical Wiener processes and Poisson random measures. Under suitable dissipativity conditions, the existence of an averaging equation eliminating the fast variable for the coupled system is proved, and as a consequence, the system can be reduced to a single SPDE with a modified coefficient. Moreover, it is shown that the slow component mean-square strongly converges to the solution of the corresponding averaging equation.

Impulsive neutral stochastic functional integro-differential equations with infinite delay driven by fBm

In this paper, we study a class of impulsive neutral stochastic functional integro-differential equations with infinite delay driven by a standard cylindrical Wiener process and 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 mild solution for this kind of equations with the coefficients satisfying some non-Lipschitz conditions, which include the classical Lipschitz conditions as special case. An example is provided to illustrate the theory. Some well-known results are generalized and extended.

A Large Deviation Principle of Retarded Ornstein-Uhlenbeck Processes Driven by Lévy Noise

In this article, we develop a large deviation principle (LDP) for a class of retarded Ornstein-Uhlenbeck processes driven by Lévy processes. We first present a LDP result for time delay systems driven by cylindrical Wiener processes based on the large deviations of Gaussian processes. By using a contraction technique and passing on a finite-dimensional approximation, an LDP is obtained for stochastic time delay evolution equations driven by additive Lévy noise, whose solutions are generally not Lévy processes any more.