Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Min Yang; Guanggan Chen

doi:10.3934/dcdsb.2019240

Article Contents

2020, Volume 25, Issue 4: 1565-1581. Doi: 10.3934/dcdsb.2019240

This issue Previous Article Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations Next Article G-neutral stochastic differential equations with variable delay and non-Lipschitz coefficients

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

Min Yang and
Guanggan Chen^,

School of Mathematical Science, and V.C. & V.R. Key Lab, Sichuan Normal University, Chengdu 610068, China

^* Corresponding author: Guanggan Chen
^* Corresponding author: Guanggan Chen

Received: January 2019

Revised: April 2019

Early access: November 2019

Published: April 2020

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Abstract

This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

Keywords:

Stochastic partial differential equations,
finite dimensional reducing,
random invariant manifolds,
invariant random cone,
Wong-Zakai approximations.

Mathematics Subject Classification: Primary: 37L55, 60H15, 58J65; Secondary: 37D10.

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References

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