Besicovitch almost automorphic stochastic processes in distribution and an application to Clifford-valued stochastic neural networks

In this paper, firstly, the concept of Besicovitch almost automorphic stochastic processes in distribution is proposed, and some basic properties and relations with concepts of almost automorphy in other senses are given. As an application, we study the existence and stability of Besicovitch almost automorphic solutions in distribution for a class of Clifford-valued stochastic neural networks with time-varying delays. Because the space composed of Besicovitch almost automorphic stochastic processes in distribution has no linear structure, so, we first prove that the system under consideration has a unique L-p -bounded and uniformly L-p-continuous solution by using the Banach fixed point theorem, and then prove that this solution is also a Besicovitch almost automorphic solution in distribution by using a variant of Gronwall inequality. Secondly, we use inequality techniques to prove that the Besicovitch almost automorphic solution in distribution is globally exponentially stable. Even when the system we consider in this paper is a real-valued system, our results are new. Finally, we give an example to illustrate the effectiveness of our results. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces the concept of Besicovitch almost automorphic stochastic processes in distribution and explores the existence and stability of Besicovitch almost automorphic solutions in distribution for a class of Clifford-valued stochastic neural networks with time-varying delays. By utilizing the Banach fixed point theorem and a variant of Gronwall inequality, the unique and bounded solutions of the system are proven to be uniformly continuous and also Besicovitch almost automorphic solutions in distribution. Furthermore, it is demonstrated that the Besicovitch almost automorphic solution in distribution is globally exponentially stable. These results are novel even for real-valued systems, and their effectiveness is illustrated through a specific example.