On the Convergence of Slow-Fast Fractional McKean–Vlasov Stochastic Systems with Multivalued Operators

This article establishes a strong averaging principle for a class of slow-fast stochastic evolution systems that incorporate both McKean–Vlasov interactions and multivalued operators within a Hilbert-space variational framework. The slow variable $\upsilon$ evolves according to a Caputo fractional differential law of order $\varrho \in (0,1)$ and is driven by fractional Brownian motion (fBm) with Hurst index $H_1\in (1/2,1)$, interpreted as scalar or cylindrical according to the underlying state space, while the fast variable $\nu$ follows a standard Brownian-driven stochastic differential equation. This hybrid formulation preserves the memory effects and long-range dependencies of the slow dynamics while maintaining the Markov property of the fast subsystem, which is essential for the averaging argument. The analysis relies on a weighted Volterra–Young estimate for singular stochastic convolutions. Since $H_1>1/2$, the stochastic integral in the slow equation is interpreted pathwise in the Young sense rather than in the Itô sense. In contrast, the fast equation, driven by standard Brownian motion, is interpreted in the usual Itô sense, preserving its Markovian structure essential for the ergodic averaging argument. Under explicit Hölder, monotonicity, coercivity, compactness, and integrability assumptions, well-posedness is proved via a Hölder-space fixed-point argument combined with variational approximation and maximal-monotone limit identification. A key technical contribution of this work is the rigorous treatment of the multivalued fractional equation: an $L^{q^{\prime }}$-integrable measurable selector $\mathfrak{a}$ for the subdifferential term is constructed by identifying it as the weak limit of the regularized Yosida operators, and its membership in the multivalued graph is established through a Minty argument that requires a limsup inequality rather than merely weak convergence. Subject to the standard ergodicity of the Brownian-driven fast variable, strong convergence of the slow component to the solution of the averaged fractional equation is proved. The averaging principle is established for $\varrho \in (1/2,1)$. This restriction arises from the singular block-integral estimate used in the proof, which requires $\varrho >1/2$; the case $\varrho \le 1/2$ is therefore beyond the scope of the present argument and is left for future work. To demonstrate the broad applicability of the abstract framework, the general results are applied to two important classes of fractional stochastic systems. The first consists of finite-dimensional fractional slow-fast McKean–Vlasov (SF-MV) stochastic variational inequalities (SVIs), including reflected stochastic systems in convex domains as an important special case. The second consists of fractional SF-MV multivalued stochastic partial differential equations (SPDEs). For both classes, explicit averaging rates are obtained under suitable Lipschitz assumptions.