This paper is devoted to the study of nonautonomous stochastic delay evolution equations in a Hilbert space framework. We investigate the existence, uniqueness and stability of mild solutions under <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(\mu ,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>–pseudo almost periodic assumptions in the p–th mean, formulated in the Stepanov sense. The main novelty of this work lies in the use of a measure–theoretic Stepanov framework, which allows the treatment of stochastic coefficients that are not uniformly bounded in time but remain controlled in an averaged <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^{p}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> sense. This setting strictly extends the classical pseudo almost periodic theory and includes, as particular cases, weighted and Lebesgue measure frameworks previously considered in the literature. We establish a complete and self–contained functional framework by proving the completeness of the associated Stepanov spaces, the invariance under translations, the uniqueness of the ergodic decomposition, and suitable composition properties. These results are then combined with deterministic and stochastic convolution estimates and a fixed point argument to obtain the existence and uniqueness of a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(\mu ,\nu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mo>,</mo> <mi>ν</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>–Stepanov pseudo almost periodic mild solution. In addition, we derive explicit exponential stability estimates in the p–th mean, showing that the recurrent solution attracts all other mild solutions at an exponential rate. Finally, two applications are presented, including a delayed stochastic differential equation and a stochastic heat equation, illustrating the relevance of the proposed framework.

Pseudo almost periodic solutions in the Stepanov sense for some nonautonomous stochastic evolution equations using measures theory

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发表日期：

2026年5月7日

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Lotfi Souden

摘要

This paper is devoted to the study of nonautonomous stochastic delay evolution equations in a Hilbert space framework. We investigate the existence, uniqueness and stability of mild solutions under $(\mu ,\nu )$ $(μ, ν)$ –pseudo almost periodic assumptions in the p–th mean, formulated in the Stepanov sense. The main novelty of this work lies in the use of a measure–theoretic Stepanov framework, which allows the treatment of stochastic coefficients that are not uniformly bounded in time but remain controlled in an averaged $L^{p}$ $L^{p}$ sense. This setting strictly extends the classical pseudo almost periodic theory and includes, as particular cases, weighted and Lebesgue measure frameworks previously considered in the literature. We establish a complete and self–contained functional framework by proving the completeness of the associated Stepanov spaces, the invariance under translations, the uniqueness of the ergodic decomposition, and suitable composition properties. These results are then combined with deterministic and stochastic convolution estimates and a fixed point argument to obtain the existence and uniqueness of a $(\mu ,\nu )$ $(μ, ν)$ –Stepanov pseudo almost periodic mild solution. In addition, we derive explicit exponential stability estimates in the p–th mean, showing that the recurrent solution attracts all other mild solutions at an exponential rate. Finally, two applications are presented, including a delayed stochastic differential equation and a stochastic heat equation, illustrating the relevance of the proposed framework.