Contact Large Deviations of Stochastic Vector Bundles

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 $e=\frac{1}{2}{g}^{T}Ag-\sigma$, is the physical observable; the time-reversal involution $J:(t,y,\phi )\mapsto (t,y,-\phi -\mathrm{\nabla }\ln \rho )$ with reversed drift $v^{\mathrm{r}\mathrm{e}\mathrm{v}}=-v+Ag$ yields a Gallavotti–Cohen-type duality ${\lambda }_{\mathrm{f}\mathrm{w}\mathrm{d}}(q)={\lambda }_{\mathrm{r}\mathrm{e}\mathrm{v}}(q+1)+{\lambda }_{\mathrm{f}\mathrm{w}\mathrm{d}}(-1)$. The classical one-dimensional GC symmetry is recovered in the reversible case $Ag=0$.