Square mean pseudo almost automorphic solutions of class r involving the measure theory for some stochastic partial differential equation driven by Lévy noise
Khalil Ezzinbi 1, Djendode Mbainadji 2 and Issa Zabsonré 3
1
Faculté des Sciences Semlalia, Département de Mathématiques, Université Cadi Ayyad, Marrakesh B.P. 2390, Morocco
2
Département de Mathématiques-Informatiques, Faculté des Sciences Fondamentales, Université Polytechnique de Mongo, Mongo, B.P 4377, Tchad
3
Departement de Mathématiques, Université Joseph KI-ZERBO,
Unite de Recherche et de Formation en Sciences Exactes et Appliquées, B.P.7021 Ouagadougou 03, Burkina Faso
Received on October 13, 2025, revised on May 26, 2026
Accepted on May 31, 2026
Communicated by Mamadou Moustapha Mbaye
| Abstract. In this paper, we investigate the existence and uniqueness of square-mean (m,n)-pseudo almost automorphic solutions of class r for a class of stochastic partial functional differential equations driven by Lévy noise. The main novelty of this work lies in the combination of the measure-theoretic framework associated with (m,n)-ergodicity, the class r setting, and the Lévy noise structure. This framework allows us to extend several existing results on square-mean pseudo almost automorphic solutions to stochastic systems involving jumps. First, we introduce and study new classes of square-mean (m,n)-ergodic processes and Poisson square-mean (m,n)-pseudo almost automorphic processes of class r, and establish their main properties. Then, by using the spectral decomposition of the phase space developed by Adimy et al., together with semigroup techniques and the Banach contraction principle, we establish existence and uniqueness results for mild solutions. Finally, an example is provided to illustrate the applicability of the obtained theoretical results. | |
| Keywords: | Measure theory; Poisson square-mean (m, n)-pseudo almost automorphy; composition theorem, stochastic processes, stochastic evolution equations, Lévy noise. |
| 2010 Mathematics Subject Classification: 34K14; 34K30; 34K50; 43A60; 60G20. | |
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