Stochastic Convolution Research Articles

Weak Error Estimates for Trajectories of SPDEs Under Spectral Galerkin Discretization

We consider stochastic semi-linear evolution equations which are driven by additive, spatially correlated, Wiener noise, and in particular consider problems of heat equation (analytic semigroup) and damped-driven wave equations (bounded semigroup) type. We discretize these equations by means of a spectral Galerkin projection , and we study the approximation of the probability distribution of the tra-jectories: test functions are regular, but depend on the values of the process on the interval [0, T ]. We introduce a new approach in the context of quantative weak error analysis for discretization of SPDEs. The weak error is formulated using a deterministic function (Ito map) of the stochastic convolution found when the nonlinear term is dropped. The regularity properties of the Ito map are exploited, and in particular second-order Taylor expansions employed, to transfer the error from spectral approximation of the stochastic convolution into the weak error of interest. We prove that the weak rate of convergence is twice the strong rate of convergence in two situations. First, we assume that the covariance operator commutes with the generator of the semigroup: the first order term in the weak error expansion cancels out thanks to an independence property. Second, we remove the commuting assumption, and extend the previous result, thanks to the analysis of a new error term depending on a commutator.

Time fractional and space nonlocal stochastic boussinesq equations driven by gaussian white noise

We present the time-spatial regularity of the nonlocal stochastic convolution for Caputo-type time fractional nonlocal Ornstein-Ulenbeck equations, and establish the existence and uniqueness of mild solutions for time fractional and space nonlocal stochastic Boussinesq equations driven by Gaussian white noise.

Well-posedness of the time-space fractional stochastic Navier-Stokes equations driven by fractional Brownian motion

The current paper is devoted to the time-space fractional Navier-Stokes equations driven by fractional Brownian motion. The spatial-temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of mild solution are obtained by Banach Fixed Point theorem and Mittag-Leffler families operators.

An $$L_{p}$$ L p -estimate for the stochastic heat equation on an angular domain in $$\mathbb {R}^2$$ R 2

We prove a weighted $L_p$-estimate for the stochastic convolution associated to the stochastic heat equation with zero Dirichlet boundary condition on a planar angular domain $\mathcal{D}_{\kappa_0}\subset\mathbb{R}^2$ with angle $\kappa_0\in(0,2\pi)$. Furthermore, we use this estimate to establish existence and uniqueness of a solution to the corresponding equation in suitable weighted $L_p$-Sobolev spaces. In order to capture the singular behaviour of the solution and its derivatives at the vertex, we use powers of the distance to the vertex as weight functions. The admissible range of weight parameters depends explicitly on the angle $\kappa_0$.

Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

Let $(E, \| \cdot\|)$ be a Banach space such that, for some $q\geq 2$, the function $x\mapsto \|x\|^q$ is of $C^2$ class and its first and second Fr\'{e}chet derivatives are bounded by some constant multiples of $(q-1)$-th power of the norm and $(q-2)$-th power of the norm and let $S$ be a $C_0$-semigroup of contraction type on $(E, \| \cdot\|)$. We consider the following stochastic convolution process \begin{align*} u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\; t\geq 0, \end{align*} where $\tilde{N}$ is a compensated Poisson random measure on a measurable space $(Z,\mathcal{Z})$ and $\xi:[0,\infty)\times\Omega\times Z\rightarrow E$ is an $\mathbb{F}\otimes \mathcal{Z}$-predictable function. We prove that there exists a c\`{a}dl\`{a}g modification a $\tilde{u}$ of the process $u$ which satisfies the following maximal inequality \begin{align*} \mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C \mathbb{E} \left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d} z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all $ q^\prime \geq q$ and $1<p\leq 2$ with $C=C(q,p)$.

On exponential stability of mild solutions for some stochastic partial integrodifferential equations

In this work, we study the existence and exponentially stability in p-mean square for some stochastic integrodifferential equation with delays. Also pathwise exponential stability is proved for p>2. We assume that the linear part has a resolvent operator in the sense given by Grimmer (1982). The delayed part is assumed to be continuous. Our results are proved by using the stochastic convolution.

Ergodicity of stochastic Magneto-Hydrodynamic equations driven by α-stable noise

The current paper is devoted to the ergodicity of stochastic Magneto-Hydrodynamic equations driven by α-stable noise with α∈(32,2). By the maximal inequality for the stochastic α-stable convolution and vorticity transformation, the well-posedness of the mild solution for stochastic Magneto-Hydrodynamic equation is established. Due to the discontinuous trajectories, the existence and uniqueness of the invariant measure for stochastic Magneto-Hydrodynamic equation are obtained by the strong Feller property and the accessibility to zero instead of the irreducibility.

A maximal inequality for pth power of stochastic convolution integrals

We prove an inequality for the pth power of the norm of a stochastic convolution integral in a Hilbert space. The inequality is stronger than analogous inequalities in the literature in the sense that it is pathwise and not in expectation.

Operator differential-algebraic equations with noise arising in fluid dynamics

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

$$G/{ GI}/N(+{ GI})$$ queues with service interruptions in the Halfin–Whitt regime

We study \(G/GI/N(+GI)\) queues with alternating renewal service interruptions in the Halfin–Whitt regime. The systems experience up and down alternating periods. In the up periods, the systems operate normally as the usual \(G/GI/N(+GI)\) queues with non-idling first-come–first-served service discipline. In the down periods, arrivals continue entering the systems, but all servers stop functioning while the amount of service that each customer has received will be conserved and services will resume when the next up period starts. For models with abandonment, interruptions do not affect customers’ patience times. We assume that the up periods are of the same order as the service times but the down periods are asymptotically negligible compared with the service times. We establish the functional central limit theorems for the queue-length processes and the virtual-waiting time processes in these models, where the limit processes are represented as stochastic integral convolution equations driven by jump processes. The convergence in these limit theorems is proved in the space \({\mathbb D}\) endowed with the Skorohod \(M_1\) topology.

On time regularity of stochastic evolution equations with monotone coefficients

We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity, we obtain a fractional Sobolev-type time regularity of order up to 12 for a certain functional G(u) of the solution. Namely, G(u)=∇u in the case of the heat equation and G(u)=|∇u|p−22∇u for the p-Laplacian. The motivation is twofold. On the one hand, it turns out that this is the natural time regularity result that allows us to establish the optimal rates of convergence for numerical schemes based on a time discretization. On the other hand, in the linear case, i.e. when the solution is given by a stochastic convolution, our result complements the known stochastic maximal space–time regularity results for the borderline case not covered by other methods.

A note on space–time Hölder regularity of mild solutions to stochastic Cauchy problems in $L^{p}$-spaces

This paper revisits the Holder regularity of mild solutions of parabolic stochastic Cauchy problems in Lebesgue spaces $L^{p}(\mathcal{O})$, with $p\geq2$ and $\mathcal{O}\subset\mathbb{R}^{d}$ a bounded domain. We find conditions on $p,\beta$ and $\gamma$ under which the mild solution has almost surely trajectories in $\mathcal{C}^{\beta}([0,T];\mathcal{C}^{\gamma}(\bar{\mathcal{O}}))$. These conditions do not depend on the Cameron–Martin Hilbert space associated with the driving cylindrical noise. The main tool of this study is a regularity result for stochastic convolutions in M-type 2 Banach spaces by Brzeźniak (Stochastics Stochastics Rep. 61 (1997) 245–295).

A Fair Policy for theG/GI/NQueue with Multiple Server Pools

We consider the G/GI/N queue with multiple server pools, each possessing a pool-specific service time distribution. The class of nonidling routing policies that we consider are referred to as u-greedy policies. These policies route incoming customers to the server pool with the longest weighted cumulative idle time to equitably spread incoming work amongst the server pools in the system. Our first set of results demonstrates that asymptotically in the Halfin-Whitt regime and under any u-greedy policy, the diffusion scaled cumulative idle time processes of each of the server pools are held in fixed proportion to one another. We next provide a heavy traffic limit theorem for the process keeping track of the total number of customers in the system. Our limit may be characterized as the solution to a stochastic convolution equation that is driven by a Gaussian process. To prove our main results, we introduce a new methodology for studying the G/GI/N queue in the Halfin-Whitt regime that has as its starting point a simple conservation of flow identity.

Linear stochastic degenerate Sobolev equations and applications†

In this paper, a general class of linear stochastic degenerate Sobolev equations with additive noise is considered. This class of systems is the infinite-dimensional analogue of linear descriptor systems in finite dimensions. Under appropriate assumptions, the mild and strong well-posedness for the initial value problem are studied using elements of the semigroup theory and properties of the stochastic convolution. The final value problem is also examined and it is proved that this is uniquely strongly solvable and the solution is continuously dependent on the final data. Based on the results of the forward and backward problem, the conditions for the exact controllability are investigated for a special but important class of these equations. The abstract results are illustrated by applications in complex media electromagnetics, in the one-dimensional stochastic Dirac equation in the non-relativistic limit and in a potential application in input–output analysis in economics.

Non Autonomous Semilinear Stochastic Evolution Equations

In this article, we first give a version with continuous paths for stochastic convolution ∫t0U(t, s)φ(s)dW(s) driven by a Wiener process W in a Hilbert space under weaker conditions. Based on the Picard approximation and the factorization method, we prove the existence, uniqueness and regularity of mild solutions for non-autonomous semilinear stochastic evolution equations with more general assumptions on the coefficients. As an application, we obtain the Feller property of the associated semigroup.

Non-Standard Skorokhod Convergence of Lévy-Driven Convolution Integrals in Hilbert Spaces

We study the convergence in probability in the non-standard M1 Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Lévy process L. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes with diagonalizable generators.

Fractional stochastic Volterra equation perturbed by fractional Brownian motion

In this paper, we consider a class of fractional stochastic Volterra equation of convolution type driven by infinite dimensional fractional Brownian motion with Hurst index h∈(0,1). Base on the explicit formula for the scalar resolvent function and the properties of the Mittag–Leffler’s function, the existence and regularity results of the stochastic convolution process are established. Separate proofs are required for the cases of Hurst parameter above and below 12 and it will turn out that the regularity of the solution increases with Hurst parameter h.

A Regularity Result for Quasilinear Stochastic Partial Differential Equations of Parabolic Type

We consider a nondegenerate quasilinear parabolic stochastic partial differential equation (SPDE) with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Holder continuous in time and possesses spatial regularity. Our proof is based on an efficient method of increasing regularity: the solution is rewritten as the sum of two processes, one solves a linear parabolic SPDE with the same noise term as the original model problem whereas the other solves a linear parabolic PDE with random coefficients. This way, the required regularity can be achieved by repeatedly making use of known techniques for stochastic convolutions and deterministic PDEs.

A Series Approach to Perturbed Stochastic Volterra Equations of Convolution Type

In the paper, perturbed stochastic Volterra Equations with noise terms driven by series of independent scalar Wiener processes are considered. In the study, the resolvent approach to the equations under consideration is used. Sufficient conditions for the existence of strong solution to the class of perturbed stochastic Volterra Equations of convolution type are given. Regularity of stochastic convolution is supplied, as well.

Stochastic modified Boussinesq approximate equation driven by fractional Brownian motion

Abstract The current paper is devoted to the dynamics of a stochastic modified Boussinesq approximate equation driven by fractional Brownian motion with H ∈ ( 1 2 , 1 ) . Based on the different diffusion operators △ 2 and −△ in the stochastic system, we combine two types of operators Φ 1 = I and a Hilbert-Schmidt operator Φ 2 to guarantee the convergence of the corresponding Wiener-type stochastic integrals. Then the existence and regularity of the stochastic convolution for the corresponding additive linear stochastic equation can be shown. By the Banach modified fixed point theorem in the selected intersection space, the existence and uniqueness of the global mild solution are obtained. Finally, the existence of a random attractor for the random dynamical system generated by the mild solution for the modified Boussinesq approximation equation is also established. MSC:35B40, 35Q35, 76D05.