Cylindrical fractional Brownian motion in Banach spaces

Issoglio, Elena; Riedle, Markus

doi:10.1016/j.spa.2014.05.010

Cited by 6 publications

(6 citation statements)

References 26 publications

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“…[10]). For cylindrical Brownian motions in a separable Banach space Y, the interested readers are referred to see sections 4 and 5, [18]. For stochastic integrals in Y, a series expansion similar to (4.6) is available, where the Hilbert-Schmidt operators from U to Y are replaced by γ-radonifying operators from U to Y (see [18] for more details).…”

Section: Fractional Brownian Motionmentioning

confidence: 99%

“…In this subsection, we provide a brief description of fBm and its stochastic integral representation in separable Hilbert spaces (cf. sections 4 and 5, [18] for separable Banach spaces). Let us consider a time interval [0, T ], where T is an arbitrary fixed time horizon.…”

Section: 1mentioning

confidence: 99%

“…For cylindrical Brownian motions in a separable Banach space Y, the interested readers are referred to see sections 4 and 5, [18]. For stochastic integrals in Y, a series expansion similar to (4.6) is available, where the Hilbert-Schmidt operators from U to Y are replaced by γ-radonifying operators from U to Y (see [18] for more details). One can refer [5,20], etc for the local solvability in L p -spaces for some mathematical models like semilinear heat equation, Hardy-Hénon parabolic equations, etc perturbed by fBm.…”

Section: Definition 41 a Fractional Brownian Motion (Fbm) With Hurst ...mentioning

confidence: 99%

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Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Mohan

2020

Stochastics

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This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.

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Section: Fractional Brownian Motionmentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Definition 41 a Fractional Brownian Motion (Fbm) With Hurst ...mentioning

confidence: 99%

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Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Mohan

2020

Stochastics

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“…One of the main motivations is the use of these random objects to construct stochastic integrals and as the driving noise to a stochastic partial differential equation (see e.g. [1,12,16,29,31,32,43]). Due to the great importance of the semimartingales in the theory of stochastic calculus, it is only natural to consider cylindrical semimartingales as the driving force for these types of stochastic equations.…”

Section: Introductionmentioning

confidence: 99%

Semimartingales on duals of nuclear spaces

Fonseca-Mora¹

2020

Electron. J. Probab.

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This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual Φ ′ of a nuclear space Φ to have a Φ ′ -valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a Φ ′ -valued semimartingale to be a Φ ′ -valued Lévy process.

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“…See also, e.g., [15,58] for some more recent results. On the other hand, while the literature on integration in Banach spaces for infinite-dimensional fractional processes is much more scarce, we refer to the papers [13] and [25] for some results in this direction. In the present article, however, different additional assumptions are put on the driving noise and on the considered Banach space than those usually considered.…”

Section: Introductionmentioning

confidence: 99%

Stochastic integration with respect to fractional processes in Banach spaces

Čoupek¹,

Maslowski²,

Ondreját³

2020

Preprint

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In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of γ-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order 2m with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.

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Cylindrical fractional Brownian motion in Banach spaces

Cited by 6 publications

References 26 publications

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Semimartingales on duals of nuclear spaces

Stochastic integration with respect to fractional processes in Banach spaces

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