Brownian motion on the Fubini extension space and applications

Abstract: The aim of this paper is to establish a certain equivalence between a Brownian motion on a Fubini extension space and a collection of essentially pairwise independent Brownian motions on the marginal sample spaces. This equivalence allows us to state a Girsanov theorem that preserves pairwise independence and to formulate graphon stochastic differential equations as a single McKean-Vlasov type equation driven by a standard Brownian motion.