Riemann\u2013Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion

Min Yang,Haibo Gu

doi:10.1186/s13660-020-02541-3

Abstract

This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.

Highlights

Fractional calculus has gained considerable popularity during the past decades since it has been recognized as one of the best tools to model physical systems possessing longterm memory and long-range spatial interactions, which plays an important role in diverse areas of science and engineering, as well as other applied sciences
We would like to emphasize that it is natural and important to study the existence of mild solutions for Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion since it has not yet been sufficiently studied in contrast with the integer-order case
In this paper we are concerned with the following Riemann–Liouville fractional stochastic evolution equations with nonlocal conditions driven by both Wiener process and fractional Brownian motion:

Summary

Introduction

Fractional calculus has gained considerable popularity during the past decades since it has been recognized as one of the best tools to model physical systems possessing longterm memory and long-range spatial interactions, which plays an important role in diverse areas of science and engineering, as well as other applied sciences. For more details about the existence of mild solutions of Riemann–Liouville fractional differential equations, one can refer to [12, 20]. The research of stochastic differential equations driven by fractional Brownian motion has been investigated by many authors, see [4, 9, 21, 28] and the references therein. We would like to emphasize that it is natural and important to study the existence of mild solutions for Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion since it has not yet been sufficiently studied in contrast with the integer-order case. In this paper we are concerned with the following Riemann–Liouville fractional stochastic evolution equations with nonlocal conditions driven by both Wiener process and fractional Brownian motion:.

L02 ds

L 2α1–1

L20 ds

L 2α1–1 x – y Cα

Conclusion