Convergence of \(p\)-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion.

*(English)*Zbl 1428.60056Summary: In this paper, we focus on fast-slow stochastic partial differential equations in which the slow variable is driven by a fractional Brownian motion and the fast variable is driven by an additive Brownian motion. We establish an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slow-varying process \(L^p\) (\(p \geq 2\)) converges to the solution of the corresponding averaging equation. To reduce the complexity, one can concentrate on the limit process instead of studying the original full fast-slow system.

##### MSC:

60G22 | Fractional processes, including fractional Brownian motion |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

##### Keywords:

averaging principle; mild solution; fractional Brownian motion; fast-slow; stochastic partial differential equations
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\textit{B. Pei} et al., Discrete Contin. Dyn. Syst., Ser. B 25, No. 3, 1141--1158 (2020; Zbl 1428.60056)

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