Introduction to Infinite Dimensional Stochastic Analysis

Huang, Zhiyuan; Yan, Jiajie

doi:10.1007/978-94-011-4108-6

Cited by 79 publications

(90 citation statements)

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“…There have been rapid progresses in this area. 12,22,9,26,2,14 More recently, SPDEs have been investigated in the context of random dynamical systems ͑RDSs͒; 1 see, for example Refs. 3,5,4,8,7,23,10, and 11, among others.…”

Section: Introductionmentioning

confidence: 99%

A dynamical approximation for stochastic partial differential equations

Wang

Duan

2007

Journal of Mathematical Physics

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Random invariant manifolds provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential system on the random invariant manifold, under suitable conditions. As an application, stationary states ͑invariant measures͒ are considered for a class of stochastic hyperbolic partial differential equations.

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Section: Introductionmentioning

confidence: 99%

A dynamical approximation for stochastic partial differential equations

Wang

Duan

2007

Journal of Mathematical Physics

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“…It is worth noting that a martingale usually does not share the good properties of Brownian motion, so there are many difficulties when we replace the Brownian motion in the stochastic integral with a martingale, and maybe this is one of the reasons why biological models driven by martingales have not been widely studied. For detailed information on martingales, readers can refer to [25,10]. Throughout this paper, we assume that M is independent of N.…”

Section: The Classical Nonautonomous Logistic Equation Is Dx(t) = X(tmentioning

confidence: 99%

Population dynamical behaviors of stochastic logistic system with jumps

Wu¹,

Ke²

2014

Turk J Math

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This paper is concerned with a stochastic logistic model driven by martingales with jumps. In the model, generalized noise and jump noise are taken into account. This model is new and more feasible. The explicit global positive solution of the system is presented, and then sufficient conditions for extinction and persistence are established.The critical value of extinction, nonpersistence in the mean, and weak persistence in the mean are obtained. The pathwise and moment properties are also investigated. Finally, some simulation figures are introduced to illustrate the main results.

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“…Example 3 (Gaussian Probability Spaces [13,16]) Let (Ω, F, P ; H) be a Gaussian probability space, so H is a real separable Hilbert space and there exists an isometric embedding…”

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confidence: 99%

“…Note that classical and abstract Wiener spaces and also white noise spaces all fit into this context (see [13], pp.60-61).…”

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confidence: 99%

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A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

Applebaum¹

2007

Journal of Applied Analysis

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We introduce "probabilistic" and "stochastic Hilbertian structures". These seem to be a suitable context for developing a theory of "quantum Gaussian processes". The Schauder system is utilised to give a Lévy-Cielsielski representation of quantum Brownian motion as operators in Fock space over a space of square summable sequences. Quantum Brownian bridges are defined and a number of representations of these are given.

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Introduction to Infinite Dimensional Stochastic Analysis

Cited by 79 publications

References 0 publications

A dynamical approximation for stochastic partial differential equations

A dynamical approximation for stochastic partial differential equations

Population dynamical behaviors of stochastic logistic system with jumps

A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

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