This talk develops mean-field limits for heterogeneous stochastic interacting particle systems with states in separable Hilbert spaces and interaction structures described by digraph measures. The finite system is formulated in mild form, and its well-posedness is established using infinite-dimensional stochastic-analysis and semigroup methods.

The main result extends the digraph-measure mean-field framework to Hilbert-space-valued dynamics. The limit is a label-dependent family of path-space laws rather than a single exchangeable McKean--Vlasov law. We show existence and uniqueness of this limiting process, weak convergence in probability of empirical path measures to its averaged law, and Lipschitz regularity of in the label variable with respect to the Wasserstein distance. Under analytic-semigroup assumptions, the convergence further improves to stronger fractional-domain path topologies.

Finally, the theory is applied to two large heterogeneous learning systems, namely, an RKHS-valued diffusion-KLMS model and a delayed recurrent-neural-network model formulated on an infinite-dimensional Sobolev phase space. These examples provide continuum descriptions of distributed kernel learning and recurrent neural dynamics on large networks.