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 -