Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise
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In: Stochastics and Partial Differential Equations: Analysis and Computations, Vol. 2022, No. 11, 11, 2023, p. 1044-1088.
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TY - JOUR
T1 - Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise
AU - Fahim, Kistosil
AU - Hausenblas, Erika
AU - Kovács, Mihaly
N1 - Publisher Copyright: © 2022, The Author(s).
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Stochastic partial differential equation
KW - Stochastic integro-differential equation
KW - Wiener process
KW - Fractal Wiener process
KW - Stochastic Volterra equation
KW - Finite element method
KW - Spectral Galerkin method
KW - Fractional partial differential equation
U2 - 10.1007/s40072-022-00250-0
DO - 10.1007/s40072-022-00250-0
M3 - Article
VL - 2022
SP - 1044
EP - 1088
JO - Stochastics and Partial Differential Equations: Analysis and Computations
JF - Stochastics and Partial Differential Equations: Analysis and Computations
SN - 2194-041x
SN - 2194-041X
IS - 11
M1 - 11
ER -