Infinite-dimensional Noise Research Articles

Mean-square invariant manifolds for stochastic weak-damping wave equations with nonlinear noise

We study mean-square invariant manifolds for stochastic wave equations with nonlinear infinite-dimensional noise and weak damping, where all nonlinear terms satisfy the nonhomogeneous Lipschitz conditions. First, we consider the existence of a mean-square random dynamical system generated from the mild solution. Second, we give a careful analysis for the spectrum of the wave operator and show that the wave operator satisfies the pseudo exponential dichotomy. Finally, by using the Lyapunov-Perron method and analyzing the conditional expectation solution, we show the existence of a mean-square random invariant unstable manifold as well as a mean-square random invariant stable set.

Well-posedness of the 3D stochastic primitive equations with multiplicative and transport noise

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

Quasifree Stochastic Cocycles and Quantum Random Walks

The theory of quasifree quantum stochastic calculus for infinite-dimensional noise is developed within the framework of Hudson–Parthasarathy quantum stochastic calculus. The question of uniqueness for the covariance amplitude with respect to which a given unitary quantum stochastic cocycle is quasifree is addressed, and related to the minimality of the corresponding stochastic dilation. The theory is applied to the identification of a wide class of quantum random walks whose limit processes are driven by quasifree noises.

Difference Approximation of Stochastic Elastic Equation Driven by Infinite Dimensional Noise

AbstractAn explicit difference scheme is described, analyzed and tested for numerically approximating stochastic elastic equation driven by infinite dimensional noise. The noise processes are approximated by piecewise constant random processes and the integral formula of the stochastic elastic equation is approximated by a truncated series. Error analysis of the numerical method yields estimate of convergence rate. The rate of convergence is demonstrated with numerical experiments.

Dynamical Behavior of Delayed Reaction-Diffusion Hopfield Neural Networks Driven by Infinite Dimensional Wiener Processes.

In this paper, we focus on the long time behavior of the mild solution to delayed reaction-diffusion Hopfield neural networks (DRDHNNs) driven by infinite dimensional Wiener processes. We analyze the existence, uniqueness, and stability of this system under the local Lipschitz function by constructing an appropriate Lyapunov-Krasovskii function and utilizing the semigroup theory. Some easy-to-test criteria affecting the well-posedness and stability of the networks, such as infinite dimensional noise and diffusion effect, are obtained. The criteria can be used as theoretic guidance to stabilize DRDHNNs in practical applications when infinite dimensional noise is taken into consideration. Meanwhile, considering the fact that the standard Brownian motion is a special case of infinite dimensional Wiener process, we undertake an analysis of the local Lipschitz condition, which has a wider range than the global Lipschitz condition. Two samples are given to examine the availability of the results in this paper. Simulations are also given using the MATLAB.

On Mean-Variance Hedging of Bond Options with Stochastic Risk Premium Factor

We consider the mean-variance hedging problem for pricing bond options using the yield curve as the observation. The model considered contains infinite-dimensional noise sources with the stochastically- varying risk premium. Hence our model is incomplete. We consider mean-variance hedging under the real world measure and obtain an explicit form of the optimal hedging strategy.

Regularity Results for the Ordinary Product Stochastic Pressure Equation

We consider a stochastic pressure equation with log-normal coefficient and infinite dimensional noise. Using a white noise framework, we study spatial and stochastic regularity of solutions of the ...

The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove general existence and compactness theorems for Ck-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the stochastic Burgers equation with additive infinite-dimensional noise. In Part 2, stationary solutions are viewed as cocycle-invariant random points in the infinite-dimensional state space. The pathwise local structure of solutions of semilinear see’s and spde’s near stationary solutions is described in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see’s and spde’s (Theorems 2.4.1-2.4.4). These results give smooth stable and unstable manifolds in the neighborhood of a hyperbolic stationary solution of the underlying stochastic equation. The stable and unstable manifolds are stationary, live in a stationary tubular neighborhood of the stationary solution and are asymptotically invariant under the stochastic semiflow of the see/spde. Furthermore, the local stable and unstable manifolds intersect transversally at the stationary point, and the unstable manifolds has fixed finite dimension. The proof uses infinite-dimensional multiplicative ergodic theory techniques, interpolation and perfection arguments (Theorem 2.2.1).

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Ito integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.

Variational solutions for a class of fractional stochastic partial differential equations

In this Note we present new results regarding the existence, the uniqueness and the equivalence of two notions of variational solution related to a class of non autonomous, semilinear, stochastic partial differential equations defined on an open bounded domain D ⊂ R d . The equations we consider are driven by an infinite-dimensional noise derived from an L 2 ( D ) -valued fractional Wiener process W H with Hurst parameter H ∈ ( 1 γ + 1 , 1 ) , where γ ∈ ( 0 , 1 ] denotes the Hölder exponent of the derivative of the nonlinearity that appears in the stochastic term. To cite this article: D. Nualart, P.-A. Vuillermot, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Non-random Invariant Sets for Some Systems of Parabolic Stochastic Partial Differential Equations

In this article we investigate a class of non-autonomous, semilinear, parabolic systems of stochastic partial differential equations defined on a smooth, bounded domain 𝒪 ⊂ ℝ n and driven by an infinite-dimensional noise defined from an L 2(𝒪)-valued Wiener process; in the general case the noise can be colored relative to the space variable and white relative to the time variable. We first prove the existence and the uniqueness of a solution under very general hypotheses, and then establish the existence of invariant sets along with the validity of comparison principles under more restrictive conditions; the main ingredients in the proofs of these results consist of a new proposition concerning Wong–Zakaï approximations and of the adaptation of the theory of invariant sets developed for deterministic systems. We also illustrate our results by means of several examples such as certain stochastic systems of Lotka–Volterra and Landau–Ginzburg equations that fall naturally within the scope of our theory.

Equivalence and Hölder–Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations

In this article we investigate a class of non-autonomous, semilinear, stochastic partial differential equations defined on a smooth, bounded, convex domain of R d and driven by an infinite-dimensional noise; this noise is colored relative to the space variable and white relative to the time variable. Under an appropriate integrability condition regarding the covariance operator of the associated Wiener process, we introduce three notions of solution for them and prove their indistinguishability. We then prove the existence, the uniqueness and the pointwise boundedness of the moments, along with the spatial Sobolev regularity and the joint space-time Hölder regularity of such solutions. Moreover, we show how to weaken some requirements regarding the covariance operator in order to generalize the notions we alluded to above by introducing a fourth type of solution, whose existence and regularity properties we also analyze in detail. Our results represent a preliminary step toward the analysis of the support and the smoothness properties of the corresponding laws.

Lévy area of Wiener processes in Banach spaces

The goal of this paper is to construct canonical Lévy area processes for Banach space valued Brownian motions via dyadic approximations. The significance of the existence of canonical Lévy area processes is that a (stochastic) integration theory can be established for such Brownian motions (in Banach spaces). Existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of Lyons and Lyons and Qian. This investigation involves a careful analysis on the choice of tensor norms, motivated by the applications to infinite dimensional stochastic differential equations.

Invariant measures for dichotomous stochastic differential equations in Hilbert spaces

We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. We consider infinite dimensional noise that is white in time and colored in space and we assume that the nonlinearities are Lipschitz continuous. We show that if the equation is dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure. To cite this article: O. Van Gaans, S. Verduyn Lunel, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1083–1088.

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form govern stochastic flows on a C *-algebra ?. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on ?. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when ? is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and *-homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations: in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions.

A Support Theorem for the Filter under Infinite Dimensional Noise and Unbounded Observation Coefficients

In this paper we consider a nonlinear filtering problem with an unbounded observation coefficient, correlated noises, and a signal process driven by an infinite dimensional Brownian motion. We prove that the unnormalized filter admits a smooth density which is in the Schwartz space and we give a description of the support of the law of this density.

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n≤4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.

Existence of a smooth density for the filter in nonlinear filtering with infinite dimensional noise

In this paper, we prove by means of Malliavin calculus that under a local Hörmander condition, the unnormalized filter associated with a nonlinear filtering problem in which the system process depends on an infinite dimensional noise admits a smooth density with respect to the Lebesgue measure.