Stepanov‐like doubly weighted pseudo almost automorphic processes and its application to Sobolev‐type stochastic differential equations driven by GG ‐Brownian motion

摘要

xml:id="mma4477-para-0001"> This paper investigates the properties of the p ‐mean Stepanov‐like doubly weighted pseudo almost automorphic ( S p DWPAA) processes and its application to Sobolev‐type stochastic differential equations driven by G ‐Brownian motion. We firstly prove the equivalent relation between the S p DWPAA and Stepanov‐like asymptotically almost automorphic stochastic processes based on ergodic zero set. We further establish the completeness of the space and the composition theorem for S p DWPAA processes. These results obtained improve and extend previous related conclusions. As an application, we show the existence and uniqueness of the S p DWPAA solution for a class of nonlinear Sobolev‐type stochastic differential equations driven by G ‐Brownian motion and present a decomposition of this unique solution. Moreover, an example is given to illustrate the effectiveness of our results.