Matrix-valued path transformations over conditional transmissions

Abstract This paper introduces what we call matrix-valued contiguous transmissive processes as non-linear multivariate structures in which any given system (defined as a random matrix) dynamically transforms its behaviour in relation to another system, if/when a certain set of conditions is satisfied within transmission zones. In light of the underlying spatiotemporal drivers, represented by path-dependent functional stochastic differential equations, every matrix can morph itself, not only across time but also in an interactive manner with other matrices, insomuch as any behavioural data can be shared between ‘networks of agents’ rather than solely between individuals, not to mention that the latter example can be extracted as a specific use-case from the proposed framework. The class of differential equations under study, by virtue of being defined on an infinite-dimensional space of matrix-valued càdlàg paths, further enables us to associate informational transmission with collective memory that records and retains historical states, and in turn governs future transformations through reciprocal adoptions and adaptations. By allowing an assembly of interconnected random systems that can dynamically influence one another towards either partial or complete metamorphoses, the proposed construct may find various applications in the study of time-evolving network graphs that are in ‘conversation’ with one another.