Classical and generalized solutions of fractional stochastic differential equations

Lototsky, Sergey V.; Rozovsky, Boris L.

doi:10.1007/s40072-019-00158-2

Cited by 7 publications

(5 citation statements)

References 19 publications

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“…where we set η := α ∧ (H − 1 p ), which is less than η 1 . Combining ( 11), ( 27), (28), and noting that…”

Section: Existence Uniqueness and Regularity Results For Problem (P1)...mentioning

confidence: 96%

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New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

Tuấn

Caraballo

Thach

2023

Stochastic Processes and their Applications

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“…where we set η := α ∧ (H − 1 p ), which is less than η 1 . Combining ( 11), ( 27), (28), and noting that…”

Section: Existence Uniqueness and Regularity Results For Problem (P1)...mentioning

confidence: 96%

“…Some interesting works on SFDEs with Caputo derivative can be found in the work of S.V. Lotosky et al [28], Nane et al [61][62][63] and references therein.…”

mentioning

confidence: 99%

New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

Tuấn

Caraballo

Thach

2023

Stochastic Processes and their Applications

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“…Liu et al [19] studied the existence and uniqueness of the system (1) with more general elliptic operator. See also [12,20] for some analytic results of (1). For the deterministic counterpart of the model (1) in engineering, physics, and biology, we refer interested readers to References [18] and [22] for related works.…”

Section: Introductionmentioning

confidence: 99%

Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise

Hu,

Yan,

Sarwar

2023

Numerical Methods Partial

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Recently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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“…For analytical results, such as existence, uniqueness and regularity of various stochastic Volterra-type integro-differential equations driven by Wiener noise we refer to [ 3 , 5 , 8 – 10 , 13 , 14 , 32 – 34 ] and for results concerning asymptotic behaviour of solutions to [ 7 , 24 ]. The particular case of fractional order equations driven by Wiener noise are considered in [ 11 , 21 – 23 , 35 , 43 , 48 ]. Various integro-differential equations driven by Lévy noise are analysed in [ 20 , 30 ] with the particular case of fractional order equations in [ 6 ].…”

Section: Introductionmentioning

confidence: 99%

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

Fahim

Hausenblas

Kovács

2022

Stoch PDE: Anal Comp

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We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.

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Classical and generalized solutions of fractional stochastic differential equations

Cited by 7 publications

References 19 publications

New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise

Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise

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