$ p $th moment exponential stability and convergence analysis of semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion

Xueqi Wen; Zhi Li; Xueqi Wen; Zhi Li

doi:10.3934/math.2022806

AIMS Mathematics

2022, Volume 7, Issue 8: 14652-14671. doi: 10.3934/math.2022806

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Research article

$ p $th moment exponential stability and convergence analysis of semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion

Xueqi Wen ,
Zhi Li ^,

School of Information and Mathematics, Yangtze University, Jingzhou, Hubei 434023, China

Received: 24 February 2022 Revised: 23 April 2022 Accepted: 05 May 2022 Published: 08 June 2022
MSC : 60H15, 60G15

Many works have been done on Brownian motion or fractional Brownian motion, but few of them have considered the simpler type, Riemann-Liouville fractional Brownian motion. In this paper, we investigate the semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion with Hurst parameter $ H < 1/2 $. First, we prove the $ p $th moment exponential stability of mild solution. Then, based on the maximal inequality from Lemma 10 in ^[1], the uniform boundedness of $ p $th moment of both exact and numerical solutions are studied, and the strong convergence of the exponential Euler method is established as well as the convergence rate. Finally, two multi-dimensional examples are carried out to demonstrate the consistency with theoretical results.
Citation: Xueqi Wen, Zhi Li. $ p $th moment exponential stability and convergence analysis of semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion[J]. AIMS Mathematics, 2022, 7(8): 14652-14671. doi: 10.3934/math.2022806

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Abstract

Many works have been done on Brownian motion or fractional Brownian motion, but few of them have considered the simpler type, Riemann-Liouville fractional Brownian motion. In this paper, we investigate the semilinear stochastic evolution equations driven by Riemann-Liouville fractional Brownian motion with Hurst parameter $ H < 1/2 $. First, we prove the $ p $th moment exponential stability of mild solution. Then, based on the maximal inequality from Lemma 10 in ^[1], the uniform boundedness of $ p $th moment of both exact and numerical solutions are studied, and the strong convergence of the exponential Euler method is established as well as the convergence rate. Finally, two multi-dimensional examples are carried out to demonstrate the consistency with theoretical results.

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