Abstract
We establish a Sanov type large deviation principle for an ensemble of interacting Brownian rough paths. As application a large deviations for the (k-layer, enhanced) empirical measure of weakly interacting diffusions is obtained. This in turn implies a propagation of chaos result in a space of rough paths and allows for a robust analysis of the particle system and its McKean–Vlasov type limit, as shown in two corollaries.
Highlights
We introduce the space of rough paths and the space where our empirical measures live
We show an large deviation principle (LDP) for the enhanced empirical measures, whose proof uses the idea for the double-layer empirical measures but exploits the extended contraction principle, together with approximation lemmata coming from rough paths theory
We close the section with the convergence of the enhanced empirical measures, which follows from the LDP
Summary
We introduce the space of rough paths and the space where our empirical measures live. That is why we introduce the space Cg0,α([0, T ]; Re) of geometric rough paths This is the subspace of Cα([0, T ]; Re) obtained as the closure, with respect to the ρα distance, of the space of smooth Re-valued paths and their iterated integrals (see [12], Section 2.2). The space of geometric rough paths has the following geometrical interpretation (taken for example from [12, Section 2.3]): it can be identified with the space C0,α([0, T ]; G2(Re)) of the closure of smooth paths, with respect to the α-Holder topology, over the (free step-2 nilpotent) Lie group G2(Re). Let f be a function in Cb2(Re) and let X be a geometric α-Holder rough path on Re. Given a partition ∆ of the interval [0, T ], define the approximated integral on ∆ as. The rough integral ∫T f (B) dBStrat coincide P-a.s
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