Contact Large Deviations of Stochastic Vector Bundles

Abstract: Large deviation theory lacks a geometric foundation for its rate functions and fluctuation symmetries. This paper develops a contact large deviation theory on stochastic vector bundles, in which the rate function, the scaled cumulant generating function, and a Gallavotti--Cohen-type fluctuation duality all follow from the contact 1-form. The constraint function acts as a generalized Lagrangian; the least constraint principle yields the dynamics, and the contact potential is built order by order from the master equation, producing a coupled Hamilton--Jacobi--transport system governed by the invariant density, drift, and fluctuation tensor. The contact path measure satisfies a large deviation principle with rate function given by the constraint action; the scaled cumulant generating function obeys a stationary eigenvalue equation with a Donsker--Varadhan variational characterization. The entropy production rate, the fluctuation--dissipation combination $e=\tfrac12 g^T Ag-σ$, is the physical observable; the time-reversal involution $J:(t,y,φ)\mapsto(t,y,-φ-\nabla\lnρ)$ with reversed drift $v<sup>{\mathrm{rev}}=-v+Ag$ yields a Gallavotti--Cohen-type duality $$λ<em>{\mathrm{fwd}}(q)=λ</em>{\mathrm{rev}}(q+1)+λ_{\mathrm{fwd}}(-1)$$. The classical one-dimensional GC symmetry is recovered in the reversible case $Ag=0$.